James _03_ Statistics Ch. 3

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Mean

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"average", computed by adding all values of the variable in the data set and dividing by the number of observations. It is not resistant!

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Population Mean

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Is computed using all the individuals in a population. This is also a parameter; µ - pronounced "mew"

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Sample Mean

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pronounced "x-bar"; (bar over an x for equation) is computed using sample data. This is a statistic

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Median

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the value that lies in the middle of the data when arranged in ascending order. We use "M" to represent this in equation. It is resistant!

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Steps of finding Median in Data Set:

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1.) Arrange data in ascending order
2.) Determine the number of observations, n
3.) Determine the observation in the middle of the data set
a.) If # of observations is odd, the the median is the data value exactly in the middle of the data set. This lays in the n+1/2 position
b.) If the number of observations is even, the median is the mean of the two middle observations in the data set. That is, the median is the mean of the observation that lie in the n/2 position and the n/2 + 1 position

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Resistant

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A numerical survey of data is said to be _________ if extreme values (very large or small) relative to the data do not affect its value substantially.

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Mode

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the most frequent observation of the variable that occurs in the data set.

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No Mode

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If no observations occur more than once in a data set

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Bimodal

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If a data set has two data values that occur with the highest frequency, the data set is ____________.

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Multimodal

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If a data set has three or more data values that occur with the highest frequency, the data set is _______________.

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Dispersion

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The degree to which the data are spread out

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Range (R)

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The difference between the largest and smallest data value

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Range (R) Equation

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Range = R = largest data value - smallest data value

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Deviation about the mean

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( xi - µ )
The sum of all deviations about the mean must equal zero
**pg's 132, 133

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Population Standard Deviation

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Is the square root of the sum of squared deviations about the population mean divided b the number of observations in the population, N. (*PSD divided by N*)

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Sample Standard Deviation (s)

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s, of a variable is the square root of the sum of squared deviations about the sample mean divided by n-1, where n is the sample size

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Degrees of Freedom

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we call n-1 the _________ ___ ________ because the first n-1 observations have freedom to be whatever value they wish, but the nth value hs no freedom. I must be whatever value forces the sum of the deviations about the mean to equal zero.

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Determining Standard Deviation Using Technology

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Interpretations of Standard Deviation (to know)

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(to know) If we are comparing two populations, then the larger the standard deviation, the more dispersion the distribution has, provided that the variable of interest from the two populations has the same unit of measure.
(IT SHOWS DISPERSION)

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Variance

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The _______ of a variable is the square of the standard deviation.

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Population Variance

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Sample Variance

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Front 23

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Empirical Rule (when distribution is bell shaped)

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Front 24

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Z-score

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Represents the distance that a data value is from the mean in terms of standard deviations. It is unitless, it has a mean of 0 and a standard deviation of 1

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Quartiles

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divide the data sets into fourths, or four equal parts (ie. each 25%)

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Interquartile Range, IQR

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is the range of the middle 50% of the observations in a data set. That is, the _____ is the difference between the third and first quartiles. The more spread a set of data has, the higher the _____ will be.

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Describe the Distribution (must know!)

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Front 28

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Outliers

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Extreme observations in a data set

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Fences

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Serve as cutoff points for determining outliers

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Checking for Outliers by using Quartilels

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1.) Determine the first and third quartiles of the data
2.) Compute the IQR
3.) Determine the fences.
Lower fence = Q1 - 1.5(IQR)
Upper fence = Q3 - 1.5(IQR)
4.) If a data value is less than the lower fence or greater than the upper fence, it is considered and outlier

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Five-Number Summary

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MINIMUM Q1 M Q3 MAXIMUM

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Boxplot

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Another graph that can be created using the five-number summary

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Steps for Drawing a Box Plot

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1.) Find the upper and lower fences
2.) Draw horizontal # line with a scale that will accommodate our graph. Draw vertical lines at Q1, M, and Q3. Enclose vertical lines in a box
3.) Label the lower & upper fences ([ and ])
4.) Draw whiskers = the smallest and larges data value inside the fence (horizontal lines to these points)
5.) Identify outliers (denote by asterisk *)

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Determining the Mean and Median (Calculator!)

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1.) Enter the raw data in L1 by pressing STAT and selecting 1:Edit.
2.) Press STAT, highlight CALC menu, and select 1:1-Var Stats
3.) with 1-Var Stats appearing on the HOME screen, press then 2nd then 1 to insert L1 on the HOME screen. Press ENTER

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Determining the Mean and Median (Calculator!)

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1.) Enter the class midpoint in L1 and the frequency or relative frequency in L2 by pressing STAT and selecting 1:Edit.
2.) Press STAT, highlight the CALC menu, and select 1:1-Var Stats
3.) With 1-Var Stats appearing on the HOME screen, press 2nd then 1 to insert L1 on the HOME screen. Then press the comma and press 2nd 2 to insert L2 on the HOME screen. So the HOME screen should have the following: 1-Var Stats L1, L2
Press ENTER to obtain the mean and standard deviation.