Ss Statistical Terms & Charts

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Histograms

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Purpose of a Histogram

A histogram is used to graphically summarize and display the distribution of a process data set.

Sample Histogram Depiction
Sample Histogram Depiction
How to Construct a Histogram

A histogram can be constructed by segmenting the range of the data into equal sized bins (also called segments, groups or classes). For example, if your data ranges from 1.1 to 1.8, you could have equal bins of 0.1 consisting of 1 to 1.1, 1.2 to 1.3, 1.3 to 1.4, and so on.

The vertical axis of the histogram is labeled Frequency (the number of counts for each bin), and the horizontal axis of the histogram is labeled with the range of your response variable.

You then determine the number of data points that reside within each bin and construct the histogram. The bins size can be defined by the user, by some common rule, or by software methods (such as Minitab).

What Questions the Histogram Answers

What is the most common system response?
What distribution (center, variation and shape) does the data have?
Does the data look symmetric or is it skewed to the left or right?
Does the data contain outliers?

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Box Plots

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Boxplot
A box plot, also known as a box and whisker diagram, is a basic graphing tool that displays centering, spread, and distribution of a continuous data set.

A box and whisker plot provides a 5 point summary of the data.
1) The box represents the middle 50% of the data.

2) The median is the point where 50% of the data is above it and 50% below it. (Or left and right depending on orientation).

3) The 25th quartile is where, at most, 25% of the data fall below it.

4) The 75th quartile is where, at most, 25% of the data is above it.

5) The whiskers cannot extend any further than 1.5 times the length of the inner quartiles. If you have data points outside this they will show up as outliers.

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Stern and Leaf Plot

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Stem and Leaf Plot
Stem and Leaf Plot

Using the data set’s numbers themselves to form a diagram, the stem and leaf plot (or simply, stemplot) is a histogram-style tabulation of data developed by John Tukey.

Consider the following data set, sorted in ascending order: 8, 13, 16, 25, 26, 29, 30, 32, 37, 38, 40, 41, 44, 47, 49, 51, 54, 55, 58, 61, 63, 67, 75, 78, 82, 86, 95

A stem and leaf plot of this data can be constructed by writing the first digits in the first column, then writing the second digits of all the numbers in that range to the right.

Stem and Leaf Plot

0|8
1|3 6
2|5 6 9
3|0 2 7 8
4|0 1 4 7 9
5|1 4 5 8
6|1 3 7
7|5 8
8|2 6
9|5

The result is a histogram turned on its side, constructed from the digits of the data. The term “stem and leaf” is used to describe the diagram since it resembles the right half of a leaf, with the stem at the left and the outline of the edge of the leaf on the right. Alternatively, some people consider the rows to be stems and their digits to be leaves.

If a larger number of bins is desired, the stem may be 2 digits for larger numbers, or there may be two stems for each first digit – one for 2nd digits of 0 to 4 and the other for 2nd digits of 5 to 9.

Stem and Leaf Plot Advantages

The stem and leaf plot essentially provides the same information as a histogram, with the following added benefits:

The plot can be constructed quickly using pencil and paper.
The values of each individual data point can be recovered from the plot.
The data is arranged compactly since the stem is not repeated in multiple data points.
The stem and leaf plot offers information similar to that conveyed by a histogram, and easily can be constructed without a computer.

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Comparing Samples vs. Population

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Basic Statistics


There are primarily two branches in which statistics are studied:

Descriptive Statistics
Applied to describe the data using numbers, charts, and graphs. Terms such as mean, median, mode, variance, standard deviation are values that summarize data. Descriptive statistics describe the entire group for which the numbers were obtained. These are the actual values for the entire group.

Inferential Statistics
Uses sample statistics to infer relationships of the population parameters.This is most often done in the Analyze and Improve phases using hypothesis testing, correlation analysis, regression analysis, and design of experiments (DOE). Rarely is it possible to be able to analyze the population (such as the average weight of all sharks in the ocean) so a sampling strategy is employed. The analysis of the sample is used to apply inferences to the entire population. The values may or may not be the same values for the entire population, but they are often used with a confidence interval applied.

Inferential Statistics


A comparison (of means, variance, proportions) is initiated with a hypothesis statement about a population or populations. The sample statistics are studied to determine, with a certain level of confidence and power, that the hypothesis (null hypothesis) is to be proven false or not false (but not necessarily true), see Hypothesis Testing for more information.

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Short Term vs. Long Term Sampling

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Short Term Sample / Long Term Sample
Long term and Short Term Comparison
SHORT term sample:

1) Free from assignable or special cause

2) Represents random causes only

3) Group of similar things

4) Collected across a narrow inference space

5) Data from one lot of material, on one shift, one part, one machine, one operator

LONG term sample:

1) Consists of random and assignable causes

2) Collected across a broad inference space

3) Data from several lots, many shifts, many machines and operators

Date across many short term (within) samples is shown below in the illustration that when the data is combined it exhibits the long term distribution. The long term distribution includes all the short term distributions.

Short Term and Long Term Sample
Applying Measurements
As a general rule, six sigma performance is a long term (future) process that creates a level of 3.4 defects per million opportunities (DPMO).

If the area under the normal curve represents one million opportunities then approximately 3.4 of them would be outside of the customer specification limit(s) when shifted 1.5 sigma to account for all the short term shifts.

A six sigma process refers to the process short-term performance or how it is performing currently. When referring to DPMO of the process, we are referring to long-term or projected performance behavior. DPMO is a more exact and informative measurement than PPM.

A six sigma level of performance has 3.4 defects per million opportunities (3.4 DPMO). A current six sigma process now will have a estimated shift of 1.5 sigma (lower) in the future and will perform at a 4.5 sigma level, which produces 3.4 DPMO.

A typical process has been proven to have a shift in its average performance of up to +/- 1.5 sigma over the long term. A long term Six Sigma process that is rated at 4.5 sigma is considered to have a short term sigma score of 6 sigma. The combination of all the short term samples that make up the long term performance will create no more than 3.4 defects per million opportunities.

A process, product, or service would need to create conformance

999,996.6 times for every 1,000,000 opportunities

and sustaining a process mean shift of up to 1.5 standard deviations (sigma).

The Six Sigma methodology focuses on variation reduction within a process and designing new processes or products that will perform at a near perfect and consistent level over the long term. The idea is to have the best term performance be the actual long term performance, the long term performance does not have to be -1.5 sigma lower, but studies show this is usually the case

SHORT TERM process capability metrics

DPU or DPO
Short term Sigma
Cpk
Cp (the best a process can perform)
LONG TERM process capability metrics

DPMO or PPM (be consistent, use one to describe short and long term)
Long term sigma
Ppk
Pp
DPMO is NOT the same as PPM since it is possible that each unit (part) being appraised may be found to have multiple defects of the same type or may have multiple types of defects. A part is defective if it has one or more defects. Defectives can never exceed defects.

IF each part only has one characteristic that can be a defect, then DPMO and PPM will be the same.

DPMO will always exceed or equal PPM for a given yield or sigma level of performance.

Get the DPMO and Sigma Calculator where you can enter values and scenarios and several metrics are calculated with the formulas shown with the cells.

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Data Classification

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Description:
Data classification is necessary to ensure the correct statistical tools are used to analyze the baseline and final performance.

Objective:

Six Sigma projects can start out with the wrong baseline sigma scores or control charts as a result of improper data classification. The goal is not only selecting the correct data type but to collect data that provides the most information at the least expense.

DISCRETE DISTRIBUTIONS:

1) Binomial Distribution

2) Poisson Distribution

3) Hypergeometric Distribution

CONTINUOUS DISTRIBUTIONS:

1) Uniform Distribution

2) Normal Distribution

3) Exponential Distribution

4) t Distribution

5) Chi-square Distribution

6) F Distribution

Continuous Data:

Theoretically has an infinite number of measurements depending on the resolution of the measurement system. There are no limits to the gaps between the measurements. It is data that can be expressed on an infinitely divisible scale.
Even if the measurements range from 0-1 there may be an infinite number of measurements within (0.000000000000... to 0.999999999999...)
The continuous random variables can be any of the infinite number of values over a given interval. These variables generally represent things that are measured, NOT counted.

Examples are:

Temperature
Height
Money
Weight
Pressure
Force
Lumens
Hardness
Length
Decibels
Time
Ohms
Watts
Amperage
Voltage
Torque
Tension
Distance
Volume
Area
Tensile Strength
Discrete Data:

Data types that have a finite number of measurements and are based on counts. Data that can be sorted into distinct, countable, and in completely separate categories. The count value can not be divided further on an infinite scale with meaning.

Example: How many people can comfortably fit into an airplane? It doesn't make sense to say 129.7632213 people. It is either 129 or 130, in this case you would round down to 129. Attribute and discrete do not mean exactly the same when describing data, discrete has more than two outcomes.

rating 1-10 (whole numbers with 1 being LOWEST - 10 being HIGHEST)
ratings provided on a FMEA for Severity, Occurrence, and Detection
color designation
gender
rating 1-10
race
# of defects on an order form or in a batch of parts
political party affiliation
types of defects on an order form
number of late deliveries
Attribute Data:

Used to represent the presence or lack of a certain characteristic. A binomial measurement has two characteristics. This is the lowest level of data type due to low level of information provided.

An attribute data measurement systems analysis (MSA) compares how often each appraiser repeats his/her own answer each time analyzing the same unit/part and how often the answer matches an known or master answer (when one exists), and how often the appraiser response reproduces the other appraiser(s) responses.

go/no-go
pass/fail
on/off
correct/incorrect
full/empty
hot/cold
small/big
paper/plastic
The measures of central tendency are the mean, variance, and standard deviation.

The mean is the expected value over a long run of occurrences. The standard deviation is simple the square root of the variance. The mean is the most commonly used measure of central tendency. It is the measurement used when analyzing a normal distribution of data.

Discrete Distribution Formulas

Locational Data:

Data that uses concentration charts and answers the question “where”. Such as the concentrations of home foreclosures by regions in the United States.

COMPARING DATA CLASSIFICATION TYPES

Continuous data is more precise than discrete data.
Continuous data provides more informative than discrete data.
Continuous data can remove estimation and rounding of measurements.
Continuous data often more time consuming to obtain.
NOTE: Convert to Continuous Data when possible as shown in the table below are a few examples to obtain a higher level of information and detail:

Converting Data

Example One:

Instead of a shipment being late or on time, it is better to know how late of how early the shipment arrived from the due date. It may be acceptable if shipments are within +/- 2 days, but the best score is given when shipments arrive at due date or +1 day. Recording YES/NO for on-time delivery will not provide the level of detail to make the best decisions.

Example Two:

Instead of recording dollars or pieces scrapped, it is more valuable to know the scrap per unit or sales.

If Plant A had molding scrap cost of $63,000/month and Plant B scraps $48,000/month, which performed better? With a denominator such as sales dollars a better conclusion can be made. If Plant A had $1,000,000 in sales in the same month, and Plant B had $50,000 in sales in the month, it is obvious that Plant B scrapped a much higher percentage of its product.

4 Levels of Data Measurement
1. Nominal Data:

The lowest level of data classification. A numerical label that represents a qualitative description. These numbers are labels or assignments of numbers that represent a category or classification. This is also referred to a categorical data usually of more than two categories and is a form of discrete data and should apply nonparametric test to analyze. The number assignment does not reflect that one category is better or worse than another.

Political Party Affiliation
1 = Independent
2 = Democratic
3 = Republican
Gender
1 = Male
2 = Female
Geographical Location
1 = Midwest
2 = South
3 = Northeast
4 = East coast
Other types of variables that often result in nominal data are religion, zip code numbers, birth dates, telephone numbers, federal tax ID number, ethnicity, and social security numbers. There limited statistical techniques to analyze this type of data, but chi-square statistic is most common.

The average of the data or variance of the data is meaningless and values and quantitative descriptions are not appropriate. There is also no priority or rank based on these numbers.

2. Ordinal Data:

The next level higher of data classification than nominal data. Numerical data where number is assigned to represent a qualitative description similar to nominal data. However, these numbers can be arranged to represent worst to best or vice-versa. Ordinal data is a form of discrete data and should apply nonparametric test to analyze.

ratings provided on a FMEA for Severity, Occurrence, and Detection
DETECTION
1 = detectable every time
5 = detectable about 50% of the time
10 = not detectable at all
(All whole numbers from 1 - 10 represent levels of detection capability that are provided by team, customer, standards, or law)

classifying households as low income, middle-income, and high income
Nominal and ordinal data are from imprecise measurements and are referred to as non metric data, sometime referred to as qualitative data.

Ordinal data is also round when ranking sports teams, ranking the best cities to live, most popular beaches, and survey questionnaires.

3. Interval Data:

The next higher level of data classification. Numerical data where the data can be arranged in a order and the differences between the values are meaningful but not necessarily a zero point. Interval data can be both continuous and discrete. Zero degrees Fahrenheit does not mean it is the lowest point on the scale, it is just another point on the scale.

The lowest appropriate level for the mean is interval data.

Parametric AND nonparametric statistical techniques can be used to analyze interval data.

Examples in temperature readings, percentage change in performance of machine, and dollar change in price of oil/gallon.

4. Ratio Data:

Similar to interval data EXCEPT has a defined absolute zero point and is the highest level of data measurement. Ratio data can be both continuous and discrete.

Ratio level data has the highest level of usage and can be analyzed in more ways than the other three types of data.

Interval data and ratio data are considered metric data, also called quantitative data.

Examples include time, volume, weight, voltage, height, pieces/hour, force, defects per million opportunities, resistance, watts, per capita income, and lumens.

Classifying data measurements

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Kurtosis

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Kurtosis is the degree of 'peakedness' of a continuous distribution. It is calculated from the formula:
kurtosis equation

The '3' is included in the formula to give the normal distribution a kurtosis of zero (some published versions do not include it).
A distribution with a kurtosis greater than zero is more peaky than a normal distribution and is 'Leptokurtic'. A distribution that is flatter than a normal distribution is 'Platykurtic'.

Kurtosis
Kurtosis characterizes the relative peakedness or
flatness of a distribution compared with the normal distribution. Positive
kurtosis indicates a relatively peaked distribution. Negative kurtosis
indicates a relatively flat distribution. [Microsoft Excel Help File, 2003.]

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Multi-variate Analysis

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Confidence Intervals

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Measures of Central Tendency: Mean, Median, Mode

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Measures of Dispersion: Range, Standard Deviation, Variance

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Find

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Probability Density Function

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Find

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Alpha and Beta Risks ( Type l Error and Type ll Error)

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Find

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Hypothesis Testing

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Hypothesis Testing:
Selecting the appropriate comparison test can be challenging especially in the learning stages. There are more than listed here. A Six Sigma project manager should understand the formulas and computations within the commonly applied tests.

Statistical software has simplified the work to the point where comprehension of these tests is convenient to overlook. REMEMBER: A statistical difference doesn't always imply a practical difference, numbers don't always reflect reality.

Parametric Tests are used when:

Normally distributed data
Non-normal distribution but transformable
Sample size is large enough to satisfy the Central Limit Theorem
Require that the data be interval or ratio data
Nonparametric Tests are used when:

The above criteria are not met or if distribution is unknown:
These test are used when analyzing nominal or ordinal data.
Nonparametric test can also analyze interval or ratio data.

Comparison of Means using parametric tests

Comparison of Sample Means


Comparison of Variances:
Use the F-Test or ANOVA for >2 variances. The F-test assumes the data is normal.

Levene's test is an option to compare variances of non-normal data.

Hypothesis Testing Steps
1) Define the Problem
2) State the Objectives
3) Establish the Hypothesis
4) State the Null Hypothesis (Ho)
5) State the Alternative Hypothesis (Ha)
6) Select the appropriate statistical test
7) State the Alpha Risk level
8) State the Beta Risk level
9) Establish the Effect Size
10) Create Sampling Plan, determine sample size

Now, the physical part of the test:

11) Gather samples
12) Collect and record data
13) Calculate the test statistic
14) Determine the p-value

If p-Value is < than alpha-risk, reject Ho and accept Ha
If p-Value is > than alpha-risk, fail to reject the Null, Ho

Try to re-run the test (if practical) to further confirm results. The next step is to take the statistical results and translate it to a practical solution.

It is also possible to determine the critical value of the test and use to calculated test statistic to determine the results. Either way, using the p-value approach or critical value should provide the same result.

Create a Visual Aid of the Test
To make the learning process easier, it is recommended that the problem be broken into four smaller steps.

Create a table similar to the one below and begin by completing the top two boxes. The bottom-left is the results from the test and then then coverting those numbers into meaning is the practical result that belongs in the bottom-right box.

Hypothesis Test Chart



The null hypothesis is refered to as "Ho".

The alternate hypothesis is referred to as "Ha".

This is the hypothesis being tested or the claim being tested. The null hypothesis is either "rejected" or "failed to reject". Rejecting the null hypothesis means accepting the alternative hypothesis.

The null hypothesis is valid until it is proven wrong. This is done by collecting data and using statistics with a specified amount of certainty. The more samples of data usually equates as more evidence and reduces the risk of an improper decision.

The null hypothesis is never accepted, it can be "failed to reject" due to lack of evidence, just as a defendant is not proven guilty due to lack of evidence. The defendant is not necessarily innocent but is determined "not guilty".

There is simply not enough evidence and the decision is made that no change exists so the defendant started the trial as not guilty and leaves the trial not guilty.

Selecting the Hypothesis Test
If you have One X and One Y variable and......
Hypothesis Test Matrix 1 Y and 1 X

If you have >1 X and One Y variable and......

Hypothesis Testing on TI-83 or TI-84 Calculator

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Analysis of Variance ( ANOVA)

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ANOVA

Six Sigma ANOVA
ANOVA in six sigma stands for analysis of variation in six sigma courses. Six sigma is an aid for the organizations that need improvements in various departments and this is why, all the time-tested strategies of management are included in the methodologies of six sigma. The black belts and master black belts are coached thoroughly on this particular topic as this involves the statistical analysis of the data fluctuations. The consultants should understand the concept of ANOVA by heart in order to become successful in six sigma project implementations.

Six sigma is a compilation of all the time-tested methods that has proved beneficial in bringing out certain improvements in the organizations. There are certain applications and tools that help one assess and execute various six sigma projects. ANOVA is one of the most useful tools available with six sigma. This is described as the statistical estimation of the collected or sample data which aids in deciding whether there exists any difference among the various data sets. A variation of ratio, obtained by assumptions, is used to calculate the fluctuation in averages.
Professionals who have mastered the techniques of six sigma are all aware of the ANOVA and its usages. The full form of ANOVA is Analysis of Variance. The training institutes use sample materials to make the professionals understand various facts about ANOVA. In solving the practical problems, ANOVA proves really helpful. As ANOVA is based on the variation of data, the six sigma consultants use it at the analysis and statistical control phase. Those who do not belong to a background of statistics are often not aware of the importance of variants and small fluctuation in data. With a complete study of ANOVA they can easily make out when and how these data should be used.
ANOVA is also known as hypothesis testing. It helps in determining the differences, if any, in two samples of collected data. When the averages are equal to each other, the situation is denoted as “Null Hypothesis”. When the samples have differing values, it is described in the results which are measured at a certain level of confidence. This analysis can be done with Excel. The two available forms are single and double factor analysis. The concept of ANOVA is used in the six sigma applications in various industries, especially in the health care field.

Assumptions of ANOVA:

The data collected in various phases is generally analyzed by the double analysis method of ANOVA. This calculation is based on the results shown on ratio scales or interval of the same experiments.
Collected data is sampled without fail on random basis.
Data collection is based on variations.
Variation of data is noted and measured carefully for thorough analysis.
Benefits of ANOVA: There are quite a number of benefits of ANOVA. It helps a black belt or master black to estimate the credibility of data collected. The variation in data is important to execute various levels of DMAIC methods. In the final step of statistical controlling, ANOVA helps a professional significantly.

Front 16

DPU - Defects Per Unit

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DPO - Defects Per Opportunity

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DPMO - Defects Per Million Opportunities

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Process Yiels Metrics:
FY - Final Yield

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Process Yield Metrics:
TYP - Throughput Yield

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Process Yield Metrics:
RTY - Rolled Throughput Yield

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Process Yield Metrics:
NY - Normalized Yield

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Z-score, Z-Value

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Z-Score, Z-Value

Standard Normal Z Score

Purpose

Z-Score, or z-value, is used to determine the probability of defect for a random variable critical characteristic which is measured in continuous data. The Z transform is an important tool because it enables this calculation without the complex mathematical calculations which would otherwise be required.



Anatomy

Z Score
Standard Normal Curve - Normal Distribution

Reference: Juran's Quality Control Handbook

Terminology

A. The standard transform, Z, transforms a set of data such that the z-value mean is always zero (0) and the z-value standard deviation is always one (1.0). By virtue of this transformation, the raw units of measure (e.g. inches, etc.) are eliminated or lost so the Z measurement scale is without units.

B. X is the value that a random variable CT characteristic can take.

C. The mean of the population. When using a sample, an estimate of mu such as Xbar will be used to substitute mu.

D. The standard deviation of the population. When using a sample, an estimate of sigma such as "s", will be used to substitute mu.

E. The area under the z-value Normal Curve Table, for normal distributions where mu=0 and standard deviation =1.0, is the reference used to find the surface that lies beyond the value of X.

F. This area represents the probability of a defect. When X takes the value of a performance limit, for example a specification limit (SL), the area under the normal curve which lies beyond the Z value is the probability of producing a defect.

Major Considerations

The use of the z-value Standard Normal Deviate assumes that the underlying distribution is Normal. When establishing a rate of nonconformance with the Z value, if the actual distribution is markedly skewed (i.e. non-normal), the likelihood of grossly distorted estimates is quite high. To avoid such distortion, it is often possible to mathematically transform the raw data.

Application Cookbook

1. To calculate the Z value from sample data, apply the formula and replace population average and the population standard deviation with Xbar and "s" respectively.

2. Use the following Excel function (NORMSDIST) to obtain the probability related to a Z value. Note that Excel gives the probability to be lower than the Z value. In order to obtain the probability of being greater than a Z value, simply use 1-NORMSDIST.

3. In Minitab use the function "Calc>Probability Distribution>Normal" to obtain the probability to be lower than a Z value.

Cumulative Distribution Function

Normal: mean = 0 and standard deviation = 1.00000

X--------P(X < SL)
2.91----0.9982

1-0.9982= .0018 (.18%) which represents the probability of having a value greater than Z.

Z-Value: Short-Term

Purpose

To be able to evaluate short-term process performance. To rate performance based on benchmarking.

Anatomy

Z Value
Z-Score Short Term

Reference: Mikel J. Harry, The vision of six sigma

Terminology

A. General equation of Z

B. Z short-term for Upper specification limit

C. Z short-term for Lower specification limit

D. Specification limits, which are an expression of the CTs

E. Central tendency to be used in the calculation; Target = T for short-termartificially centered process

F. Short-term standard deviation

Major Considerations

Z is a metric
Z is always in term of "how many short-term sigma"
Z makes a bridge between process and Normal probability
The Six Sigma objective is to achieve a ZST level of 6 or higher
Application Cookbook

1. Take the reference for the central tendency. Target value is used because Z short-term reflects "process capability" under the assumption of random variation.

2. Estimate the standard deviation short-term

3. Compute both specification limits

4. Apply the proper formula to compute Upper and Lower Z short-term
Note: Z upper = Z lower if the target is the specification mid point.

Z-Value: Long-Term

Purpose

To be able to evaluate long-term process performance. To statistically estimate the PPM, DPMO.

Anatomy

Z Value
Z-Score Long Term

Reference: Mikel J. Harry, The vision of six sigma

Terminology

A. General equation of Z

B. Z long-term for Upper specification limit

C. Z long-term for Lower specification limit

D. Specification limits, which are an expression of the CTs

E. Central tendency to be used in the calculation

F. Long-term standard deviation

Major Considerations

Z is a metric
Z long-term is always in term of "how many long-term sigma"
Z makes a bridge between process and Normal probability
The Six Sigma objective is to achieve a ZLT level of 4.5 or higher
Application Cookbook

1. Choose the reference for the central tendency

for the actual distribution, use m (i.e. the overall mean)
for artificially centering on the mid point of the specification, use T(i.e. the target )
2. Estimate the standard deviation long-term

3. Compute both specification limits

4. Apply the proper formula to compute Upper and Lower Z long-term

Z-Test – One Sample

Purpose

To compare the population mean of a continuous CT characteristic with a value such as the target. Since we don't know the population mean, an analysis of a data sample is required. This test is usually used to determine if the mean of a CT characteristic is on target when the sample size is greater or equal to 30.

Anatomy

Z Test
One Sample Z-Test

Reference: Basic Statistics by Kiemele, Schmidt and Berdine

Terminology

A. Null (H0) and alternative (Ha) hypotheses, where the population mean (mu) is compared to a value such as a target.

For the alternative hypothesis, one of the three hypotheses has to be chosen before collecting the data to avoid being biased by the observations of the sample.

B. Minitab Session Window output.

C. Hypotheses tested: H0 is equal to 10 vs. Ha is not equal to10.

D. Descriptive Statistics - Name of the column that contains the data, sample size, sample mean, sample standard deviation (StDev) and standard error of the mean.

E. Computed Z statistic using the formula below to obtain the z-value.

Z-Score Formula

Calculating the Z Score
Calculating The Z-Score

F. P-Value – This value has to be compared with the alpha level and the following decision rule is used: if P < alpha, reject H0 and accept Ha with (1-P) 100% confidence; if P is equal to or greater than alpha, don't reject H0.

Major Considerations

The assumption for using this test is that the data comes from a random sample with a size greater or equal to 30. It can also be used when the standard deviation of the population is known but this case is quite rare in the practice.

Application Cookbook

1. Define problem and state the objective of the study.

2. Establish hypothesis - State the null hypothesis (H0) and the alternate hypothesis (Ha).

3. Establish alpha level. Usually alpha is 0.05.

4. Establish sample size (see tool Sample Size - Continuous Data – Z Test - One Sample).

5. Select a random sample.

6. Measure the CT characteristic.

7. Analyze data with Minitab:

Use the function under Stat>Basic Statistics> 1-Sample z.
Select the Test mean option, input the target value and the desired alternative hypothesis (>, <, not equal: the default setting).
Enter a sigma value. This value can be the sample standard deviation (s) or a known value of the population standard deviation.
8. Make statistical decision from the session window output of Minitab. Either accept or reject H0.

9. Translate statistical conclusion to practical decision about the CT characteristic.

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Yield to Sigma Relationships

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Front 25

Process Capability Indicies (PCI):
Pp

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PCI:
Ppk

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PCI:
Cp

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PCI:
Cpk

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PCI:
Cpm

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Discrete Dustribution:
Binomial Distribution

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Binomial Distribution

Figure 7: Binomial Distribution Shape
Figure 7: Binomial Distribution Shape
Basic assumptions:

Discrete distribution
Number of trials are fixed in advance
Just two outcomes for each trial
Trials are independent
All trials have the same probability of occurrence
Uses include:

Estimating the probabilities of an outcome in any set of success or failure trials
Sampling for attributes (acceptance sampling)
Number of defective items in a batch size of n
Number of items in a batch
Number of items demanded from an inventory

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Discrete Distribution:
Poisson Distribution

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Poisson Distribution

Figure 10: Poisson Distribution Pdf
Figure 10: Poisson Distribution Pdf
Basic assumptions:

Discrete distribution
Length of the observation period (or area) is fixed in advance
Events occurs at a constant average rate
Occurrences are independent
Rare event
Uses include:

Number of events in an interval of time (or area) when the events are occurring at a constant rate
Number of items in a batch of random size
Design reliability tests where the failure rate is considered to be constant as a

Front 32

Discrete Distribution:
Hypergeometric Distribution

Back 32

Hypergeometric

Shape is similar to Binomial/Poisson distribution.

Basic assumptions:

Discrete distribution
Number of trials are fixed in advance
Just two outcomes for each trial
Trials are independent
Sampling without replacement
This is an exact distribution – the Binomial and Poisson are approximations to this
Other Distributions

Front 33

Continuous Distribution:
Uniform Distribution

Back 33

Find

Front 34

Continuous Distribution:
Normal Distribution

Back 34

Normal Distribution (Gaussian Distribution)

Figure 3: Normal Distribution Shape
Figure 3: Normal Distribution Shape
Basic assumptions:

Symmetrical distribution about the mean (bell-shaped curve)
Commonly used in inferential statistics
Family of distributions characterized is by m and s
Uses include:

Probabilistic assessments of distribution of time between independent events occurring at a constant rate
Mean is the inverse of the Poisson distribution
Shape can be used to describe failure rates that are constant as a function of usage

Front 35

Continuous Distribution:
Exponential Distribution

Back 35

Exponential Distribution

Figure 4:Exponential Distribution Shape
Figure 4:Exponential Distribution Shape
Basic assumptions:

Family of distributions characterized by its m
Distribution of time between independent events occurring at a constant rate
Mean is the inverse of the Poisson distribution
Shape can be used to describe failure rates that are constant as a function of usage
Uses include probabilistic assessments of:

Mean time between failure (MTBF)
Arrival times
Time, distance or space between occurrences of the events of interest
Queuing or wait-line theories

Front 36

Continuous Distribution:
t Distribution

Back 36

distribution


The various t-tests are applied during the ANALYZE and CONTROL phase. You should be very familiar with these test and able to explain the results.

William Sealy Gosset is credited with first publishing the data of the test statistic and became known as the Student's t-distribution.

The t-test is generally used when:

Sample sizes less than 30 (n<30)
Standard Deviation is UNKNOWN
The t-distribution bell curve gets flatter as the Degrees of Freedom (DF) decrease. Looking at it from the other perspective, as the DF increases, the number of samples (n) must be increasing thus the sample is becoming more representative of the population and the sample statistics approach the population parameters. So as these values come closer to one another the "z" calculation and "t" calculation get closer and closer to same value.

The table below explains each test in more detail.

Various t-test to apply in hypothesis testing

Hypothesis Testing
Before running test, try to visualize your data to get a better understanding of the projected outcome of expected result. Using tools such as Box Plot can provide a wealth of information.

Also, if the confidence interval contains the value of zero then insufficient evidence exist to suggest their is a statistical difference between the null and alternative hypothesis and accept the null.

One Sample t-test
This test compares a sample to a known population mean, historical mean, or targeted mean. The population standard deviation is unknown and the data must satisfy normality assumptions.

Given:
n = sample size
The degrees of freedom (DF) = n-1

Most statistical software will allow a variety of options to be examined from how large a sample must be to detect a certain size difference given a desired level of Power (= 1 - Beta Risk). You can also select various levels of Significance or Alpha Risk.

For a given difference that you are interested in, the amount of samples required increases if you want to reduce Beta Risk (which seems logical). However, gathering more samples has a cost and that is the job of the GB/BB to balance getting the most info to get more Power and highest Confidence Level without too much cost or tying up too many resources.

Two Sample t-test
This test is used when comparing the means of:
1) Two random independent samples are drawn, n1 and n2
2) Each population exhibit normal distribution
3) Equal standard deviations assumed for each population
The degrees of freedom (DF) = n1 + n2 - 2
Example:
The overall length of a sample of a part running of two different machines is being evaluated. The hypothesis test is to determine if there is a difference between the overall lengths of the parts made of the two machines using 95% level of confidence.

Machine 1:
Sample Size: 22 parts
Mean: 28.4mm
Sample standard deviation: 3.4mm
Machine 2:
Sample Size: 20 parts
Mean: 27.6mm
Sample standard deviation: 2.2mm
DF = n1 + n2 - 2 = 22+20-2 = 40
Alpha-risk = 1-CI = 1-0.95 = 0.05

Establish the hypothesis test:

NULL: Mean A equals the Mean B
ALTERNATIVE: Mean A does not equal Mean B
This is two-tailed example since the direction (shorter or longer) is not relevant. All that is relevant is if there is a statistical difference or not.
Now, determine the range for which t-statistic and any values outside these ranges will result in rejecting the null and inferring the Alternative hypothesis.
Using the t-table below notice that:

-t(0.975) to t(0.975) with 40 DF equals a range of -2.021 to 2.021.


If the calculated t-value from our example falls within this range then accept the Null hypothesis.
NOTE: Keep in mind the table below is a one-tailed table so use the column 0.025 that corresponds to 40 DF and include both the positive and negative value.

Paired t-test
Use this test when analyzing the samples of a BEFORE and AFTER situation and the number of samples must be the same. Also referred to as pre-post test and consist of two measurements taken on the same subjects such as machines, people, process, etc. This option is selected to test the hypothesis of no difference between two variables. The data may consist of two measurements taken on the same machine (or subject) or one before and after measurement taken on a matched pair of subjects.

For example, if the Six Sigma team has implemented improvements from the IMPROVE phase they are expecting a favorable change to the outputs (Y). If the improvements had no effect the average difference between the measurements is equal to 0 and the null hypothesis is inferred. If the team did a good job making improvements to address the critical inputs (X's) to the problem (Y's) that were causing the variation (and/or to shift the mean in unfavorable direction) then their should be a statistical difference and the alternative hypothesis should be inferred.

DF = n - 1

The "Sd" is the standard deviation of the difference in all of the samples. The data is recorded in pairs and each pair of data has a difference, d.

Another application may be to measure the weight or cholesterol levels of a group of people that are given a certain diet over a period of time. The before data of each person (weight or cholesterol levels) are recorded to serve as the basis for the null hypothesis. With as many of the other variable controlled and maintained consistent for all people for the duration of the study, then the after measurements are taken. The null hypothesis infers that there in not a significant difference in the weights (or cholesterol levels) of the patients.

Again, this test assumes the data set are normally distributed and continuous.

t-Distribution Table
t-distribution table for given alpha values and degrees of freedom
Z test
The Z test uses a set of data and test a point of interest. An example is shown below using Excel. This function returns the one-tailed probability.

The sigma value is optional. If the population standard deviation is known enter it in, if not the test defaults to the sample standard deviation.

Running the Z test at the mean of the data set returns a value of 0.5, or 50%.

The example below also uses a point of interest for the hypothesized population mean of 105 corresponds to a Z test value of 0.272 indicating that there is a 27.2% chance that 105 is greater than the average of actual data set assuming data set meets normality assumptions.

The Z test (as shown in the example below) value represents the probability that the sample mean is greater than the actual value from the data set when the underlying population mean is μ0.

The Z-test value IS NOT the same as a z-score. The z-score involves the Voice of the Customer and the USL, LSL specification limits.

Six Sigma projects should have a baseline z-score after the Measure phase is completed and before moving into Analyze. The final Z-score is also calculated after the Improve phase and the Control phase is about instituting actions to maintain the gains.

There other metrics such as RTY, NY, DPMO, PPM, RPN, can be used in Six Sigma projects as the Big "Y" but usually they can be converted to a z-score.

Using Excel to run the Z test

Confidence Interval Formula
Calculating the confidence interval for t-distribution

T-test or Z-test?

Front 37

Continuous Distribution:
Chi-square Distribution

Back 37

Chi-square distribution
The Chi-square distribution is most often for hypothesis tests and in determining confidence intervals.

1) Chi-square test for independence in an "Row x Column" contingency table.

2) Chi-square test to determine if the standard deviation of a population is equal to a specified value.

Unlike the normal distribution, the chi-square distribution is not symmetric. Separate tables exist for the upper and lower tails of the distribution.

This statistical test can be used to examine the hypothesis of independence between two attribute variables and determine if the attribute variables are related.

Chi square

Assumptions
Chi-Square is the underlying distribution for these tests
Attribute data (X data and Y data are attribute)
Observations must be independent
Ideally used when comparing more than two samples otherwise use the 2-Proportions Test (with two samples) or 1-Proportion Test (with one sample).

Formula
The sum of the expected frequencies is always equal to the sum of the observed frequencies

The chi-squared statistic can be used to:

1) Test how well the distribution fits the population

2) Test association of two attribute variables

Chi-Squared

Possible Applications
Test to see if a particular region of the country is an important factor in the number wins for a baseball team
Determine if the number of injuries among a few facilities is statistically different
Determine if a coin or a set of dice is biased or fair.
Goodness of Fit (GOF) Hypothesis Test
The GOF test compares the frequency of occurrence from an observed sample to the expected frequency from the hypothesized distribution.

As in all Hypothesis Tests, craft a statement (without numbers) and use simple terms for the team's understanding and then create the numerical or statistical version of the problem statement.

State the practical problem
State the statistical problem
Develop null and alternative hypotheses
Create table of observed and expected frequencies
Calculate the test statistic
The Degrees of Freedom = (# of Rows - 1) * (# of Columns - 1)

Two methods can be applied to test the hypotheses. The decision to reject the null (and infer the alternative hypothesis) if:

1) Calculate the critical value the chi-squared test statistic and reject the null hypothesis if the calculated value is GREATER THAN the critical value

OR

2) Reject the null hypothesis if the p-value is LESS THAN the alpha-risk. Your team determines the alpha-risk. For a Confidence Level of 95% the alpha-risk = 5% or 0.05.

Using EXCEL to determine the p-value is done by:

P-value = CHIDIST (Chi-Square calculated, degrees of freedom)

OR

P-value = CHITEST(Observed Range, Expected Range)

Observed and Expected Values of Attribute Data
Create the table of Observed values and create the table of Expected values. Creating a table helps visualize the values and ensure each condition is calculated correctly and then the sum of those is equal to actual Chi-Square calculated value.

Calculate Expected values for each condition (fe).

fe = (row total * column total) / grand total.

The Chi-Square calculated value is compared to the Chi-Square critical value depending on the Confidence Level desired (usually 95% which is alpha-risk of 5% or 0.05)

Chi-Squared Table
Chi-Square Table

Chi-Square "Goodness of Fit" for TI-83/84 Calculators


Chi-Square "Independence" Tests for TI 84 Calculator

Front 38

Continuous Distribution:
F Distribution

Back 38

F Distribution
The F Distribution is a continuous probability distribution based on the ratio of two random variances. It is formed from the ratios of two chi-squared variables with the following formula.
F-distribution

Rarely is the probability density function calculated mathematically. Thankfully that work has been done and F-tables with the most common levels of significance and degrees of freedom exist for reference.

BE CAREFUL when using tables. Some tables are for one-tailed test and others cover two-tailed test.

For a one-tailed test, the null hypothesis is rejected when the test statistic is greater than the value given in the F-table. value.

Front 39

What is the Difference Between Kurtosis and Skewness?

Back 39

What is the Difference Between Kurtosis and Skewness?
Categorized in: Six Sigma Implementation, Six Sigma Tools & Metrics


A Six Sigma review of any operation or process will involve the analysis of large sets of data to come to sound decisions. It is a well-established business method that has been used for the past 20 years to save companies millions of dollars and make operations much more efficient.

The goal in Six Sigma is to be able to run a nearly flawless operation. There should be no variance whatsoever in the function that is being performed. Whether it is a manufacturing line or a call center, the goal is to be able to complete the task in an error-free way every time. When a data sample is charted and there are big variations in the numbers, that can signal a problem. A chart with big peaks is called kurtosis. The word comes from a Greek word which means bulging.

Analyzing the data that is collected is the job of Six Sigma Black Belts who lead the reviews and use the charts and graphs produced to identify flaws that need to be corrected. Kurtosis and skewness are two of the distributions that the black belt will look for to highlight where there is too much variance in the process.

In a perfect process, there would be negative kurtosis because the graph would be almost a flat line. When there is positive kurtosis however, you have a huge swing in data values that can be an indication of a problem. If the sample size is large enough to be a true reflection on the operation, it is imperative to figure out why there is such huge variance. If you are dealing with a small sample size, do not read too much into kurtosis.

Skewness is another statistical term that can indicate too much variance. Like kurtosis, the values are unevenly spread out on a graph. Skewness measures the asymmetry of the distribution. A true symmetrical distribution would put an equal number of values on either side of the mean. When too many values fall to the left, you have negative symmetry, and when more numbers go to the right of the mean, you have positive symmetry.

When numbers are skewed in either direction, the Six Sigma black belt knows that this could be a problem. The goal is to reduce variance, and any skewness that is shown means that the process is failing to produce the same results over and over again.

Front 40

Factor Analysis

Back 40

Multivariate Analysis is concerned with analyzing processes that have several inputs and/or outputs:

multivariate analysis

There are several types of multivariate analysis:

Discriminant Analysis many inputs but only one categorical output
Factor Analysis converts many input factors into a smaller number of input factors
MANOVA Multiple Analysis of Variance
Principal Components Analysis similar to factor analysis

Front 41

Discriminant Analysis

Back 41

Multivariate Analysis is concerned with analyzing processes that have several inputs and/or outputs:

multivariate analysis

There are several types of multivariate analysis:

Discriminant Analysis many inputs but only one categorical output
Factor Analysis converts many input factors into a smaller number of input factors
MANOVA Multiple Analysis of Variance
Principal Components Analysis similar to factor analysis

Front 42

Manova Analysis

Back 42

Multivariate Analysis is concerned with analyzing processes that have several inputs and/or outputs:

multivariate analysis

There are several types of multivariate analysis:

Discriminant Analysis many inputs but only one categorical output
Factor Analysis converts many input factors into a smaller number of input factors
MANOVA Multiple Analysis of Variance
Principal Components Analysis similar to factor analysis

Front 43

Multivariate Analysis

Back 43

Multivariate Analysis is concerned with analyzing processes that have several inputs and/or outputs:

multivariate analysis

There are several types of multivariate analysis:

Discriminant Analysis many inputs but only one categorical output
Factor Analysis converts many input factors into a smaller number of input factors
MANOVA Multiple Analysis of Variance
Principal Components Analysis similar to factor analysis

Front 44

Discrete Distribution
Geometric Distribution

Back 44

Geometric

Figure 8: Geometric Distribution Pdf
Figure 8: Geometric Distribution Pdf
Basic assumptions:

Discrete distribution
Just two outcomes for each trial
Trials are independent
All trials have the same probability of occurrence
Waiting time until the first occurrence
Uses include:

Number of failures before the first success in a sequence of trials with probability of success p for each trial
Number of items inspected before finding the first defective item – for example, the number of interviews performed before finding the first acceptable candidate

Front 45

Continuous Distribution
Lognormal Distribution

Back 45

Lognormal Distribution

Figure 5: Lognormal Distribution Shape
Figure 5: Lognormal Distribution Shape
Basic assumptions:

Asymmetrical and positively skewed distribution that is constrained by zero.

Distribution can exhibit many pdf shapes
Describes data that has a large range of values
Can be characterized by m and s
Uses include simulations of:

Distribution of wealth
Machine downtimes
Duration of time
Phenomenon that has a positive skew (tails to the right)

Front 46

Continuous Distribution
Weibull Distribution

Back 46

Weibull Distribution

Figure 6: Weibull Distribution Pdf
Figure 6: Weibull Distribution Pdf
Basic assumptions:

Family of distributions
Can be used to describe many types of data
Fits many common distributions (normal, exponential and lognormal)
The differing factors are the scale and shape parameters
Uses include:

Lifetime distributions
Reliability applications
Failure probabilities that vary over time
Can describe burn-in, random, and wear-out phases of a life cycle (bathtub curve)

Front 47

Discrete Distribution
Negative Binomial

Back 47

Negative Binomial

Figure 9: Negative Binomial Distribution Pdf
Figure 9: Negative Binomial Distribution Pdf
Basic assumptions:

Discrete distribution
Predetermined number of occurrences – s
Just two outcomes for each trial
Trials are independent
All trials have the same probability of occurrence
Uses include:

Number of failures before the sth success in a sequence of trials with probability of success p for each trial
Number of good items inspected before finding the sth defective item

Front 48

Understanding Statistical Distributions for Six Sigma

Back 48

Understanding Statistical Distributions for Six Sigma
Profile photo of J. DeLayne StroudJ. DeLayne Stroud 2
Many consultants remember the hypothesis testing roadmap, which was a great template for deciding what type of test to perform. However, think about the type of data one gets. What if there is only summarized data? How can that data be used to make conclusions? Having the raw data is the best case scenario, but if it is not available, there are still tests that can be performed.

In order to not only look at data, but also interpret it, consultants need to understand distributions. This article discusses how to:

Understand different types of statistical distributions.
Understand the uses of different distributions.
Make assumptions given a known distribution.
Six Sigma Green Belts receive training focused on shape, center and spread. The concept of shape, however, is limited to just the normal distribution for continuous data. This article will expand upon the notion of shape, described by the distribution (for both the population and sample).

Getting Back to the Basics

With probability, statements are made about the chances that certain outcomes will occur, based on an assumed model. With statistics, observed data is used to determine a model that describes this data. This model relates to the distribution of the data. Statistics moves from the sample to the population while probability moves from the population to the sample.

Inferential statistics is the science of describing population parameters based on sample data. Inferential statistics can be used to:

Establish a process capability (determine defects per million).
Utilize distributions to estimate the probability of a variable occurring given known parameters.
Inferential statistics are based on a normal distribution.

Figure 1: Normal Curve and Probability Areas
Figure 1: Normal Curve and Probability Areas
Normal curve distribution can be expanded on to learn about other distributions. The appropriate distribution can be assigned based on an understanding of the process being studied in conjunction with the type of data being collected and the dispersion or shape of the distribution. It can assist with determining the best analysis to perform.

Types of Distributions

Distributions are classified in the same ways as data is classified – continuous and discrete:

Continuous probability distributions are probabilities associated with random variables that are able to assume any of an infinite number of values along an interval.
Discrete probability distributions are listings of all possible outcomes of an experiment, along with their respective probabilities of occurrence.
Distribution Descriptions

Probability mass function (pmf) - For discrete variables, the pmf is the probability that a variate takes the value x.

Probability density function (pdf) - For continuous variables, the pdf is the probability that a variate assumes the value x, expressed in terms of an integral between two points.

In the continuous sense, one cannot give a probability of a specific x on a continuum – it will be some specific (and small) range. For additional insight, think of x + Dx where Dx is small.

The notation for the pdf is f(x). For discrete distributions:

f(x) = P(X = x)

Some refer to this as the probability mass function, since it is evaluating the probability upon that one discrete mass. For continuous distributions, one mass cannot be established.

Cumulative density function (cdf) - The probability that a variable takes a value less than or equal to x.

Figure 2: Normal Distribution Cdf
Figure 2: Normal Distribution Cdf
Cdf progresses to a value of 1 because there cannot be a probability greater than 1. Once again, cdf is F(x) = P(X < x).This holds for both continuous and discrete.

Parameters

Parameter is a population description. Consultants rely on parameters to characterize the distributions. There are three parameters:

Location parameter – the lower or midpoint (as prescribed by the distribution) of the range of the variate (think of the mean)
Scale parameter – determines the scale of measurement for x (magnitude of the x-axis scale) (think of the standard deviation)
Shape parameter – defines the pdf shape within a family of shapes
Not all distributions have all the parameters. For example, the normal distribution parameters have just the mean and standard deviation. Just those two need to be known to describe a normal population.

Summary of Distributions

The remaining portion of this article will summarize the various shapes, basic assumptions and uses of distributions. Keep in mind that there is a different pdf and different distribution parameters associated with each.

Normal Distribution (Gaussian Distribution)

Figure 3: Normal Distribution Shape
Figure 3: Normal Distribution Shape
Basic assumptions:

Symmetrical distribution about the mean (bell-shaped curve)
Commonly used in inferential statistics
Family of distributions characterized is by m and s
Uses include:

Probabilistic assessments of distribution of time between independent events occurring at a constant rate
Mean is the inverse of the Poisson distribution
Shape can be used to describe failure rates that are constant as a function of usage
Exponential Distribution

Figure 4:Exponential Distribution Shape
Figure 4:Exponential Distribution Shape
Basic assumptions:

Family of distributions characterized by its m
Distribution of time between independent events occurring at a constant rate
Mean is the inverse of the Poisson distribution
Shape can be used to describe failure rates that are constant as a function of usage
Uses include probabilistic assessments of:

Mean time between failure (MTBF)
Arrival times
Time, distance or space between occurrences of the events of interest
Queuing or wait-line theories
Lognormal Distribution

Figure 5: Lognormal Distribution Shape
Figure 5: Lognormal Distribution Shape
Basic assumptions:

Asymmetrical and positively skewed distribution that is constrained by zero.

Distribution can exhibit many pdf shapes
Describes data that has a large range of values
Can be characterized by m and s
Uses include simulations of:

Distribution of wealth
Machine downtimes
Duration of time
Phenomenon that has a positive skew (tails to the right)
Weibull Distribution

Figure 6: Weibull Distribution Pdf
Figure 6: Weibull Distribution Pdf
Basic assumptions:

Family of distributions
Can be used to describe many types of data
Fits many common distributions (normal, exponential and lognormal)
The differing factors are the scale and shape parameters
Uses include:

Lifetime distributions
Reliability applications
Failure probabilities that vary over time
Can describe burn-in, random, and wear-out phases of a life cycle (bathtub curve)
Binomial Distribution

Figure 7: Binomial Distribution Shape
Figure 7: Binomial Distribution Shape
Basic assumptions:

Discrete distribution
Number of trials are fixed in advance
Just two outcomes for each trial
Trials are independent
All trials have the same probability of occurrence
Uses include:

Estimating the probabilities of an outcome in any set of success or failure trials
Sampling for attributes (acceptance sampling)
Number of defective items in a batch size of n
Number of items in a batch
Number of items demanded from an inventory
Geometric

Figure 8: Geometric Distribution Pdf
Figure 8: Geometric Distribution Pdf
Basic assumptions:

Discrete distribution
Just two outcomes for each trial
Trials are independent
All trials have the same probability of occurrence
Waiting time until the first occurrence
Uses include:

Number of failures before the first success in a sequence of trials with probability of success p for each trial
Number of items inspected before finding the first defective item – for example, the number of interviews performed before finding the first acceptable candidate
Negative Binomial

Figure 9: Negative Binomial Distribution Pdf
Figure 9: Negative Binomial Distribution Pdf
Basic assumptions:

Discrete distribution
Predetermined number of occurrences – s
Just two outcomes for each trial
Trials are independent
All trials have the same probability of occurrence
Uses include:

Number of failures before the sth success in a sequence of trials with probability of success p for each trial
Number of good items inspected before finding the sth defective item
Poisson Distribution

Figure 10: Poisson Distribution Pdf
Figure 10: Poisson Distribution Pdf
Basic assumptions:

Discrete distribution
Length of the observation period (or area) is fixed in advance
Events occurs at a constant average rate
Occurrences are independent
Rare event
Uses include:

Number of events in an interval of time (or area) when the events are occurring at a constant rate
Number of items in a batch of random size
Design reliability tests where the failure rate is considered to be constant as a function of usage
Hypergeometric

Shape is similar to Binomial/Poisson distribution.

Basic assumptions:

Discrete distribution
Number of trials are fixed in advance
Just two outcomes for each trial
Trials are independent
Sampling without replacement
This is an exact distribution – the Binomial and Poisson are approximations to this
Other Distributions

There are other distributions – for example, sampling distributions and X2, t and F distributions.

Summary

Distribution refers to the behavior of a process described by plotting the number of times a variable displays a specific value or range of values rather than by plotting the value itself. It is often said that a picture is worth a thousand words. Viewing data graphically will make a much greater impact to an audience. Becoming familiar with the various distributions can help consultants to better interpret their data.

Front 49

Student's T Distribution

Back 49

STUDENT’S T DISTRIBUTION, STUDENT’S T TEST
Six Sigma Study Guide
A kind of Paired T Test, Student’s T distribution is used for finding confidence intervals for the population mean when the sample size is less than 30 and the population standard deviation is unknown. If you need to evaluate something with a population greater than 30, use the Z distribution => t distribution is flatter and wider than the z distribution. The t distribution becomes narrower (taller) as sample sizes increase, and gradually becomes very close to the Normal Distribution. Both z and t distributions are symmetric and bell-shaped, and both have a mean of zero.

*The more degrees of freedom, the better.

Student t-test would be used to compare two population means using samples from each. See this example.

Also used to test hypotheses about population means based on sample data.

http://stattrek.com/probability-distributions/t-distribution.aspx

Student’s T Distribution Example Questions
Confidence of a mean 1
At a soda bottling factory, the normal filling specification (goal) is 16 fluid ounces. A sample of 20 bottles is tested with the following results: x = 16.13 fl oz and s = 0.24 fl oz. What interval would allow you to say, with 95% confidence, that the interval contains the actual mean of the filling process?

Student 1

Student 1 chart

Confidence of a mean 2
An automated computer attendant is supposed to respond to all inquiries within 15 seconds. The company claims with 99 percent confidence that the average response time is less than 17 seconds. A random sample of 25 calls reveals that X = 13.5 seconds and s = 0.95 seconds. Based on this data is the company correct?

Student 2 chart Student 2

Confidence of a mean 3
A photography developing machine uses ink to produce photographs. A sample of 15 pictures showed an average of 1.5 mL of ink used per picture, with a standard deviation o 0.075 mL. Develop a 99% confidence band for the actual average amount of ink used by the machine to produce each photograph.

Student 3 chart Student 3

Front 50

Understanding the Central Limit Theorem

Back 50

Understanding the Central Limit Theorem
Tumbling dice and birthdays
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Story update 8/27/2009: An error was spotted and corrected by author in paragraph starting with "The population mean for a six-sided die..."

Mark Twain famously quipped that there were three ways to avoid telling the truth: lies, damned lies, and statistics. The joke works because statistics frequently seem like a black box—it can be difficult to understand how statistical theorems make it possible to draw conclusions from data that, on their face, defy easy analysis.

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But because data analysis plays a critical role in everything from jet engine reliability to determining the shows we see on television, it’s important to acquire at least a basic understanding of statistics. One of the most important concepts to understand is the central limit theorem.

In this article, we will explain the central limit theorem and show how to demonstrate it using common examples, including the roll of a die and the birthdays of Major League Baseball players.

Defining the central limit theorem
A typical textbook definition of the central limit theorem goes something like this:

As the sample size increases, the sampling distribution of the mean, X-bar, can be approximated by a normal distribution with mean µ and standard deviation σ/√n where:

µ is the population mean
σ is the population standard deviation
n is the sample size

In other words, if we repeatedly take independent random samples of size n from any population, then when n is large, the distribution of the sample means will approach a normal distribution.

How large is large enough? Generally speaking, a sample size of 30 or more is considered to be large enough for the central limit theorem to take effect. The closer the population distribution is to a normal distribution, the fewer samples needed to demonstrate the theorem. Populations that are heavily skewed or have several modes may require larger sample sizes.

Why does it matter?
The field of statistics is based upon the fact that it is rarely feasible or practical to collect all of the data from an entire population. Instead, we can gather a subset of data from a population, and then use statistics for that sample to draw conclusions about the population.

For example, we can collect random samples from an industrial process, then use the means of our samples to make conclusions about the stability of the overall process.

Two common characteristics used to define a population are the mean and standard deviation. When data follow a normal distribution, the mean indicates where the center of that distribution is, and the standard deviation reveals the spread.

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Michelle Paret’s picture
Michelle Paret
Michelle Paret is a product marketing manager at Minitab Inc., developer of statistical analysis/process improvement software. She loves the field of statistics and believexs that it gives us the ability to remove human bias and opinion to discern between what is truly important—and significant—from those things that are not. She loves statistics so much that she earned both her undergrad and graduate degrees on the subject.

Eston Martz’s picture
Eston Martz
For Eston Martz, analyzing data is an extremely powerful tool that helps us understand the world—which is why statistics is central to quality improvement methods such as lean and Six Sigma. While working as a writer, Martz began to appreciate the beauty in a robust, thorough analysis and wanted to learn more. To the astonishment of his friends, he started a master’s degree in applied statistics. Since joining Minitab, Martz has learned that a lot of people feel the same way about statistics as he used to. That’s why he writes for Minitab’s blog: “I’ve overcome the fear of statistics and acquired a real passion for it,” says Martz. “And if I can learn to understand and apply statistics, so can you.”

Front 51

Sampling Distribution of the Mean

Back 51

Sampling Distribution of the Mean
As might be expected, inference with continuous variables is more complicated than with dichotomous variables. Fortunately however, the general principles are the same. Again, we will use a sampling distribution to index the probability that the observed outcome is due to chance.
Empirical (capable of being verified or disproved by observation or experiment)
The sampling distribution of the mean is a probability distribution of the possible values of the mean that would occur if we were to draw all possible samples of a fixed size from a given population.
To get a better feel for this notion, let’s consider an empirical example (or one that could be actually performed). Let us choose 10 samples of size 4 from a population of size 20.
Population
Distribution 10 observed
Sample
Distributions Empirical Sampling
Distribution
of the Mean
6, 2, 9, 5,
0, 1, 3, 2,
1, 1, 5, 2,
7, 7, 7, 8,
1, 1, 3, 7 1, 5, 9, 0 3.75 (& sX=4.11)
0, 3, 1, 5 2.25
5, 8, 3, 0 4.00
1, 5, 0, 7 3.25
7, 6, 1, 3 4.25
3, 2, 1, 7 3.25
2, 0, 3, 5 2.50
1, 2, 1, 1 1.25
2, 7, 1, 7 4.25
9, 7, 6, 2 6.00
Ns=4

Notes:



There are three types of distributions illustrated above: population, sample, and sampling.
Empirical sampling distributions are only used to help students understand the concept. They are not true sampling distributions, since all possible samples are not chosen.

Theoretical
Are called theoretical because all possible samples (an infinite number) should be drawn. Since this is impossible, the characteristics (i.e., the mean & standard deviation) of the distribution are determined mathematically.
It turns out that (note that and are synonyms).
The standard deviation of the distribution of sample means is called the standard error of the mean (or more simply, the standard error). It measures variability in the distribution of sample means or, in other words, sampling error (the amount of error we can expect due to using a sample mean to estimate a population mean). Perhaps it is easier to think of sampling error as "chance" like we did at the beginning of the semester.

One would expect the size of the standard error to be related to the sample size, and it is.
When population values are known:

Thus, as the sample size gets bigger, sampling error gets smaller.
When population values are estimated from sample values:

This formula requires sx to be an unbiased estimator of sx
Computational formula for the standard error estimated from sample values:

Example (using the population distribution from the empirical sampling distribution above):

If we didn’t know the population values, we could use the SX from the first sample.

As you can see, only estimates and it does so poorly in this case (because of the small sample size).
Sampling Distributions & Normality
The techniques that we are discussing require that the sampling distribution (in this case the distribution of sample means) be normal in shape. This will be the case if either of the following two conditions are met.
The population distribution of raw scores is normal.
It is difficult to actually know this, but fortunately, many variables are.
The sampling distribution will approach normality as the sample size is increased.
This occurs even though the population distribution may not be normal in shape. Note, though, that the more skewed the population distribution, the larger the N (sample size) needed for the sampling distribution of the mean to be normal.
II. 1 Sample Z - Parameters Known
Rationale
Now that we have an understanding of sampling distributions of a continuous variable, we can go on to test a hypothesis. Recall that any normally distributed variable can be transformed into a standard normal distribution (i.e., z scores). We also saw that area under the curve implies probability. Thus, if the sampling distribution of the mean is normal we can establish the probability of obtaining a particular sample mean.

Formal Example - [Minitab]
Let us look at an example from your book. Animal studies suggest that the anticholinergic drug physostigmine improves memory. This could have some clinical applications in humans (e.g., senility, Alzheimer’s disease). Studies with humans typically report that we remember an average of seven of 15 words given an 80-minute retention interval. These studies also suggest a standard deviation for the population of two.
Research Question
Does physostigmine improve memory in humans?

Hypotheses
In Symbols In Words
HO m=7 Physostigmine has no effect on memory.
HA m¹7 Physostigmine has an effect on memory.
Assumptions
Population of non-drugged folks has m=7 and s=2 (i.e., the null).
Sample is randomly selected.
Population of non-drugged folks is normal.
Reason is so that the sampling distribution of the mean will be normal. Although a large sample size would also produce a normally shaped sampling distribution, we will rarely use large samples.

Decision Rules
We will use the standard normal curve (Z scores) to obtain the probabilities. Our alpha level is .05 with a two-tailed test. When we look in a Z table, we see that the critical value of Z is 1.96 (Zcrit).

Thus, the shaded area is the critical region. If our observed z value falls into this area, we will reject the null hypothesis. More formally:
If Zobs £ -1.96 or Zobs ³ 1.96, then reject HO.
If Zobs > -1.96 and Zobs < 1.96, then do not reject HO.
Computation
The computations have two goals corresponding to the descriptive and inferential statistics. Suppose we obtain the following scores for a sample of 20 subjects:

9
8
8
9
9
7
7
8
8
10
8
10
8
10
7
9
8
8
7
9

The first step is to describe the data. The most important descriptive statistic in this case is the mean or average number of words remembered by the 20 subjects receiving the drug. The calculation reveals a mean of (åX/N=167/20=) 8.35, which is greater the the mean of the population of 7.

The second step of the computation is to perform an inferential test to determine whether this difference between means is worth paying attention to (in other words, is the improvement in memory due to sampling error or to the drug?).
Remember:

More generally:

Thus, the appropriate formula would be:

And substituting the values in for the standard error gives:

Decision
Since 3.02 (Zobs) > 1.96 (Zcrit) we reject HO and assert the alternative. Now we must go beyond this simple decision of rejecting the null or not to what it all means. In other words, we need to make a conclusion based on our decision and the particular results observed. In this case, we would conclude that the physostigmine improves memory. Notice that we have actually gone beyond the alternative hypothesis by specifying that the effect has a direction (memory was improved). We do this because the mean words remembered for the drugged group was higher than for the population.
III. Errors & the Power of a Test
As can be seen, hypothesis testing is just educated guessing. Moreover, guesses (educated or not) are sometimes wrong. Consider the possible decisions we can make:
Possibilities: Actual Situation
HO is True HO is False
Decision Reject HO Type I Error Correct Decision II
Do not reject HO Correct Decision I Type II Error
Let us now consider each decision in more detail.
A Type I Error is the false rejection of a true null. It has a probability of alpha (a). In other words, this error occurs as a result of the fact that we have to somehow separate probable from improbable.

Correct Decision I occurs when we fail to reject a true null. It has a probability of 1-a. From a scientist's perspective this is a "boring" result.


A Type II Error is the false retention of a false null. It has a probability equal to beta (b).


Correct Decision II occurs when we reject a false null. The whole purpose of the experiment is to provide the occasion for this type of decision. In other words, we performed the statistical test because we expect the sample to differ. This decision has a probability of 1-b. This probability is also known as the power of the statistical test. In other words, the ability of a test to find a difference when there really is one, is power.


Factors Influencing Power:

Alpha (a). Alpha and beta are inversely related. In other words, as one increases, the other decreases (i.e., a ´ b = K). Thus, all other things being equal, using an alpha of .05 will result in a more powerful test than using an alpha of .01.

Sample Size (N). The bigger the sample (i.e., the more work we do), the more powerful the test.

Type of Test. Metric tests (as compared to nonparametric tests that we discuss later in the semester) are generally more powerful due to assumptions that are more restrictive.

Variability. Generally speaking, variability in the sample and/or population results in a less powerful test.

Test Directionality. One-tailed tests have the potential to be more powerful than two-tailed tests.

Robustness of the Effect. Six beers are more likely to influence reaction time than one beer.
IV. 1 Sample t - Sigma Unknown
In the 1 Sample Z example, both the mean (m) and standard deviation (s) of the population were given. However, these parameters are rarely known. In this section, we will consider how the test is performed when s is unknown.
Rationale
As we noted earlier, can be used to estimate . One complication of doing this is that the shape of the theoretical distribution of sample means will depend on the sample size. Thus, this sampling distribution is actually a family of distributions and is called Student’s t. To better understand the t distributions, we need to consider a new way of thinking of sample size.
The Degrees of Freedom (df) for a statistic refer to the number of calculations in its computation that are free to vary. For example, the df for the variance of a sample (Sx2) is N-1.

In other words, since the sum of the deviations equals zero, N-1 of the deviations are free to vary. That is, given N-1 of the deviations, we can easily determine the final deviation because it is not free to vary. In the example below where N=5, the unknown value must be 2.
c
-2
-1
0
1
?
åc=0
With the 1 sample t test, the df for t equals the df for Sx which is N-1. And Student’s t is a family of distributions differing in their kurtosis (or peakedness).

Note that when df are infinite (i.e., the sample size is very large), the t distribution will equal the z distribution.
As for the formula, remember the z test:

The formula for the t is similar.

Like the z test, the critical values of t are obtained from a table. To determine the critical value of t from the table, you will need to know a, the df, and whether you are using a one- or two-tailed test. You must be conservative when using these tables. For example, if your df=45 and the table only gives values for a df of 40 and 60, then you must use the critical value given for the df of 40 (or find yourself a better table).
Formal Example - [Minitab]
You are interested in whether the average IQ for a group of "bad kids" (the ones that put a tack on your seat before you sit down) in a school is different from the rest of the kids in the school. The average IQ for the school as a whole is 102 with the standard deviation unavailable.
Research Question
Do "bad kids" have normal intelligence?

Hypotheses
In Symbols In Words
HO m=102 Bad kids have normal IQs.
HA m¹102 Bad kids do not have normal IQs.

Assumptions
Population of IQ has m=102 (i.e., the Null).
Sample is randomly selected.
Population of IQ is normal.
Reason is so that the sampling distribution of the mean will be normal. Although a large sample size would also produce a normally shaped sampling distribution, we will rarely use large samples.

Decision Rules
Using alpha of .05 with a two-tailed test and N=20 (df=N-1=19), we determine from the t table that the critical value is 2.093.


Thus:
If tobs £ -2.093 or tobs ³ 2.093, then reject HO.
If tobs > -2.093 and tobs < 2.093, then do not reject HO.
Computation
The IQs for the 20 bad kids are as follows:

Subj. X X2
1
106
11236
2
120
14400
3
118
13924
4
124
15376
5
111
12321
6
123
15129
7
88
7744
8
116
13456
9
120
14400
10
127
16129
11
97
9409
12
118
13924
13
88
7744
14
91
8281
15
110
12100
16
114
12996
17
109
11881
18
130
16900
19
92
8464
20
108
11664
å 2,210 247,478
N 20
Mean 110.5

Describing the data, we see that the average IQ is 110.5 which is higher than the "normal kids". We also need the standard deviation to be able to estimate the standard error when performing the inferential test. Thus,

Now we can compute the t test:


Decision
Since 2.90 (tobs) > 2.093 (tcrit) we reject HO and assert the alternative. In other words, we conclude that the "bad kids" are smarter than average. Notice that we have actually gone beyond the alternative hypothesis by specifying that the effect has a direction (bad kids are smarter).
V. Interval Estimation
Suppose you are a researcher interested in self-destructiveness. You develop a scale to measure this trait. Example questions might include:
I like to listen to loud music.
I use (or have used) drugs.
I like to drive fast.
Next you obtain a random sample of 25 people and give them the scale. (The difficulty in obtaining a random sample might be noted.) The mean for this sample is 120 and the standard deviation is 10.
One of the things that we may want to know is what is the range of scores expected for the population. If we knew this, we would be easily able to identify the deviant scorer (possibly for a case study).
Let’s say we wanted to know the expected range of scores for 95% of the population. This is termed the 95% Confidence Interval (CI) and is given by:
with df=N-1,
and remembering
Thus, the current example has df=N-1=25-1=24, alpha of .05 (for 95% CI), two-tailed test. From the t table, we determine that the tcrit is 2.064.
Therefore, the upper limit will be

And the lower limit will be:

We can now be confident that 95% of people would be expected to score between 115.87 and 124.13.
Contents Index APA Style Guide Dr. P's Place Copyright © 1997-2014 M. Plonsky, Ph.D.
Comments? [e-mail].

Front 52

Continuous Distribution
Chi - Square Distribution

Back 52

Chi - square distribution
The Chi-square distribution is most often for hypothesis tests and in determining confidence intervals.

1) Chi-square test for independence in an "Row x Column" contingency table.

2) Chi-square test to determine if the standard deviation of a population is equal to a specified value.

Unlike the normal distribution, the chi-square distribution is not symmetric. Separate tables exist for the upper and lower tails of the distribution.

This statistical test can be used to examine the hypothesis of independence between two attribute variables and determine if the attribute variables are related.

Chi square

Assumptions
Chi-Square is the underlying distribution for these tests
Attribute data (X data and Y data are attribute)
Observations must be independent
Ideally used when comparing more than two samples otherwise use the 2-Proportions Test (with two samples) or 1-Proportion Test (with one sample).

Formula
The sum of the expected frequencies is always equal to the sum of the observed frequencies

The chi-squared statistic can be used to:

1) Test how well the distribution fits the population

2) Test association of two attribute variables

Chi-Squared

Possible Applications
Test to see if a particular region of the country is an important factor in the number wins for a baseball team
Determine if the number of injuries among a few facilities is statistically different
Determine if a coin or a set of dice is biased or fair.
Goodness of Fit (GOF) Hypothesis Test
The GOF test compares the frequency of occurrence from an observed sample to the expected frequency from the hypothesized distribution.

As in all Hypothesis Tests, craft a statement (without numbers) and use simple terms for the team's understanding and then create the numerical or statistical version of the problem statement.

State the practical problem
State the statistical problem
Develop null and alternative hypotheses
Create table of observed and expected frequencies
Calculate the test statistic
The Degrees of Freedom = (# of Rows - 1) * (# of Columns - 1)

Two methods can be applied to test the hypotheses. The decision to reject the null (and infer the alternative hypothesis) if:

1) Calculate the critical value the chi-squared test statistic and reject the null hypothesis if the calculated value is GREATER THAN the critical value

OR

2) Reject the null hypothesis if the p-value is LESS THAN the alpha-risk. Your team determines the alpha-risk. For a Confidence Level of 95% the alpha-risk = 5% or 0.05.

Using EXCEL to determine the p-value is done by:

P-value = CHIDIST (Chi-Square calculated, degrees of freedom)

OR

P-value = CHITEST(Observed Range, Expected Range)

Observed and Expected Values of Attribute Data
Create the table of Observed values and create the table of Expected values. Creating a table helps visualize the values and ensure each condition is calculated correctly and then the sum of those is equal to actual Chi-Square calculated value.

Calculate Expected values for each condition (fe).

fe = (row total * column total) / grand total.

The Chi-Square calculated value is compared to the Chi-Square critical value depending on the Confidence Level desired (usually 95% which is alpha-risk of 5% or 0.05)

Chi-Squared Table
Chi-Square Table

Chi-Square "Goodness of Fit" for TI-83/84 Calculators


Chi-Square "Independence" Tests for TI 84 Calculator

Front 53

Present Value Calculation

Back 53

What it is:

Present value describes how much a future sum of money is worth today.

How it works/Example:

The formula for present value is:

PV = CF/(1+r)n

Where:
CF = cash flow in future period
r = the periodic rate of return or interest (also called the discount rate or the required rate of return)
n = number of periods

Let's look at an example. Assume that you would like to put money in an account today to make sure your child has enough money in 10 years to buy a car. If you would like to give your child $10,000 in 10 years, and you know you can get 5% interest per year from a savings account during that time, how much should you put in the account now? The present value formula tells us:

PV = $10,000/ (1 + .05)10 = $6,139.13

Thus, $6,139.13 will be worth $10,000 in 10 years if you can earn 5% each year. In other words, the present value of $10,000 in this scenario is $6,139.13.

It is important to note that the three most influential components of present value are time, expected rate of return, and the size of the future cash flow. To account for inflation in the calculation, investors should use the real interest rate (nominal interest rate - inflation rate). If given enough time, small changes in these components can have significant effects.

Why it Matters:

The concept of present value is one of the most fundamental and pervasive in the world of finance. It is the basis for stock pricing, bond pricing, financial modeling, banking, insurance, pension fund valuation, and even lottery payouts. It accounts for the fact that money we receive today can be invested today to earn a return. In other words, present value accounts for the time value of money.

In the stock world, calculating present value can be a complex, inexact process that incorporates assumptions regarding short and long-term growth rates, capital expenditures, return requirements, and many other factors. Naturally, such variables are impossible to predict with perfect precision. Regardless, present value provides an estimate of what we should spend today (e.g., what price we should pay) to have an investment worth a certain amount of money at a specific point in the future -- this is the basic premise of the math behind most stock- and bond-pricing models.

Present value is one of the most important concepts in finance. Luckily, it's easy to calculate once you know a few tricks. Click here to learn How to Calculate Present Value Using Excel or a Financial Calculator.

Front 54

Standard Deviation
Simple Example of Calculating Standard Deviation

Back 54

Standard Deviation
Simple Example of Calculating Standard Deviation

Let's say we wanted to calculate the standard deviation for the amounts of gold coins pirates on a pirate ship have.

There are 100 pirates on the ship. In statistical terms this means we have a population of 100. If we know the amount of gold coins each of the 100 pirates have, we use the standard deviation equation for an entire population:













What if we don't know the amount of gold coins each of the 100 pirates have? For example, we only had enough time to ask 5 pirates how many gold coins they have. In statistical terms this means we have a sample size of 5 and in this case we use the standard deviation equation for a sample of a population:













The rest of this example will be done in the case where we have a sample size of 5 pirates, therefore we will be using the standard deviation equation for a sample of a population.

Here are the amounts of gold coins the 5 pirates have:

4, 2, 5, 8, 6.

Now, let's calculate the standard deviation:

1. Calculate the mean:









2. Calculate for each value in the sample:











3. Calculate :







4. Calculate the standard deviation:







The standard deviation for the amounts of gold coins the pirates have is 2.24 gold coins.

See also:

Front 55

What is a X2 Distribution? What does is look like? When is it used? Chi square

Back 55

[2/14, 6:25 AM] Darrah: Rajan, What is a X2 Distribution? What does is look like? When is it used?
[2/14, 6:33 AM] ‪[phone]‬: Chi square....it is used in two conditions mainly....comparison of standard deviation and to know existence of relationship between two discrete variables. Also can be use to know parent distribution for a sample.

Front 56

T Distribution Table with a One Tail, alpha (a)? How do use the table for right or left tail test? For using T distribution table :

Back 56

[2/14, 1:01 PM] Darrah: T Distribution Table with a One Tail, alpha (a)? How do use the table for right or left tail test?
[2/14, 10:25 PM] [phone]: For using T distribution table :

1. Select the right test out of three... Is it two tail test, right tailed or left tailed... Reading the problem statement carefully will give you the clues... If it is written.. Not equal.... Then alternative hypothesis can be in either direction... Then two tailed test has to be done.

If it is written greater than then select right tailed test otherwise select left tailed test.

2. For two tailed test given Alpha value is halved and accordingly t table is looked...

3. In one sided t test alpha value remains same and accordingly t test is looked..

4. Normally drawing normal curve and marking critical value on it... avoids confusion.. Which area of Curve being taken into account.