Rfc1 - Chapter 12: Descriptive Statistics

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Descriptive Statistics

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1) Statistics is a set of procedures for describing, synthesizing, analyzing, and interpreting quantitative data

- The mean is an example of a statistic.

2) One can calculate statistics by hand or can use the assistance of statistical programs

- Excel, SPSS, and many other programs exist. Some programs are also available on the Web to analyze datasets.

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Preparing Data for Analysis

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1) After data is collected, the first step

toward analysis involves converting

behavioral responses into a numerical

system or categorical organization.

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Indices are calculated for a sample

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1) They are referred to as statistics

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Indices are calculated for the entire population

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1) They are referred to as parameters

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Types of Descriptive Statistics: Frequencies

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1) The frequency refers to the number of

times something occurs.

2) Frequencies are often used to describe

categorical data.

- We might want to have frequently counts of how many males and females were in a study or how many participants were in each condition.

3) Frequency counts are not as helpful in describing interval and ratio data.

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Measures of Central Tendency

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1) Measures of central tendency are indices that represent a typical score among a group of scores.

2) Measures of central tendency provide a way to describe a dataset with a single number.

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Measures of Central Tendency: Mean

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1) Appropriate for describing interval or ratio data

2) The mean is the most commonly used

measure of central tendency.

3) The formula for the mean is:

X= ∑X/n

4) To calculate the mean, all the scores

are summed and then divided by the

number of students.

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Measures of Central Tendency: Median

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1) Appropriate for describing ordinal data

2) The median is the midpoint in a

distribution: 50% of the scores are above

the median and 50% of the scores are

below the median

3) To determine the median, all scores are listed in order of value

4) If the total number of scores is odd, the median is the middle score.

5) If the total number of scores is even, the median is halfway between the two middle scores

6) Median values are useful when there is large variance in a distribution.

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Measures of Central Tendency: Mode

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1) Appropriate for describing nominal data

2) he mode is the most frequently

occurring score in a distribution.

3) The mode is established by looking at a

set of scores or at a graph of scores and

determining which score occurs most

frequently.

4) The mode is of limited value.

Some distributions have more than one mode

(e.g., bi-modal, or multi- modal distributions)

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Measures of Central Tendency: Deciding among measures of central tendency

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1) Generally the mean is most preferred.

2) The mean takes all scores into account.

3) The mean, however, is greatly influenced by

extreme scores.

4) When there are extreme scores present in a

distribution, the median is a better measure

of central tendency.

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Measures of Variability

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1) Measures of variability provide an

index of the degree of spread in a

distribution of scores.

2) Measures of variability are critical to

examine and report because some

distributions may be very different but

yet still have the same mean or

median

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Measures of Variability: Range

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1) The difference between the

highest and lowest score.

2) The range is not a stable measure.

3) The range is quickly determined.

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Measures of Variability: Quartile Deviation

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1) One half the difference between the upper quartile and the lower quartile in a

distribution.

2) By subtracting the cutoff point for the

lower quartile from the cutoff point for

the upper quartile and then dividing by

two we obtain a measure of variability.

3) A small number indicates little variability

and illustrates that the scores are close

together.

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Measures of Variability: Variance

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1) The amount of spread among scores. If the variance is small the scores are close together. If the variance is large the scores are spread

out.

2) Calculation of the variance shows how far

each score is from the mean.

3) The formula for the variance is:

∑(X–X)^2/n

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Measures of Variability: Standard Deviation

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1) The score root of the variance.

2) The standard deviation is used with

interval and ratio data.

3) The standard deviation is the most

commonly used measure of variability.

4) If the mean and the standard deviation

are known, the distribution can be

described fairly well.

5) SD represents the standard deviation of a

sample and the symbol (i.e., the Greek

lower case sigma) represents the

standard deviation of the population.

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The Normal Curve

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1) If a variable is normally distributed then

several things are true about the

distribution of the variable.

2) Fifty percent of the scores are above the

mean and 50% are below the mean.

3) The mean, median, and mode have the

same value.

4) Most scores are near the mean.

5) 34.13% of the scores fall between the

mean and one standard deviation above

the mean and 34.13% of scores fall below

the mean and one standard deviation

below the mean.

6) That is, 68.26% of the scores fall within

one standard deviation of the mean.

7) More than 99% of the scores fall within

three standard deviations above and below

the mean.

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The Normal Curve: Skewed Distributions

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1) When a distribution is not normally distributed, it is said to be skewed.

2) A skewed distribution is not symmetrical.

- The mean, median, and mode are not the same value

- The farther apart the mean and the median, the more skewed the distribution.

3) A negatively skewed distribution has extreme scores at the lower end of the distribution.

- Mean

4) A positively skewed distribution has extreme scores at the higher end of the distribution

- Mean>Median>Mode

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Measures of Relative Position

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1) Measures of relative position indicate

where a score falls in the distribution

relative to all the other scores.

2) Measures of relative position indicate

how well an individual has scored in

comparison to others in the distribution.

3) Measures of relative position express

different scores on a common scale.

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Measures of Relative Position: Percentile Ranks

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1) Percentile ranks indicate the percentage of

scores that fall at or below a given score.

2) Percentile ranks are appropriate for ordinal

data and are also used for interval data.

3) A percentile rank of 50 indicates the median

score.

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Measures of Relative Position: Standard Scores

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1) uses standard deviation units to express how far an individual student’s test score is from

the mean.

-i.e., a standard score reports how many standard deviations a given score is from the mean of a distribution

2) Standard scores allow scores from different tests to be compared on a common scale and therefore allow for calculations on those data

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Z Score

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1) Is the most basic and most often used standard score

2) Directly tied to the standard deviation

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Score that represents the mean

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1) Has a z-score of 0

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A score at 1 standard deviation above

the mean has a z-score of 1.

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A score at 1 standard deviation above

the mean has a z-score of 1.

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Convert a raw score to a z-score

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1) To convert a raw score to a z-score

we use the following formula where

X is the raw score.

Z=X-X/ SD

2) The characteristics of the normal distribution can be used to approximate where a score falls

based upon a standard score.

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T- Score

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1) Standard score sometimes used instead of a z-score

2) Calculated by multilying a z-score by 10 and adding 50

3) T-Scores transform the scores such that there are not negative values

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Measures of Relationships

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1) Measures of relationship indicate the

degree to which two sets of scores are

related.

2) There are several statistical measures

of relationship.

The nature of a given dataset dictates

which measure is used

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Measures of Relationship: Pearson r

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1) Is statistic used to calculate

relationship for interval or ratio data.

2) Pearson r takes into account every score.

3) Pearson r is the most stable measure of

relationship

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Measures of Relationship: Spearman rho

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1) Is used to calculate relationship with ordinal data.

2) There are other tests of relationship for

ordinal data (e.g., Gamma, Kendall’s tau).

3) Spearman rho is most popular.

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Graphing Data

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1) Is a critical step in analyzing any datset

2) All variables should be graphed

- Graphing helps researchers to know their data.

- Statistical packages make graphing data an easy task

- Graphing helps to identify any errors in a dataset.