Chapter 4 Variability

Variability In statistics, our goal is to measure the amount of variability for a particular set of scores, a distribution. In statistics, our goal is to measure the amount of variability for a particular set of scores, a distribution. If all the scores are the same no variability If all the scores are the same no variability If small difference, variability is small If small difference, variability is small If large difference, variability is large If large difference, variability is large

Variability Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together. Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together. Goal: to describe how spread out the scores are in a distribution Goal: to describe how spread out the scores are in a distribution

Figure 4.1 Population distributions of heights and weights Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Variability (cont.) Variability will serve two purposes Variability will serve two purposes Describe the distribution Describe the distribution Close together Close together Spread out over a large distance Spread out over a large distance Measure how well an individual score (or group of scores) represents the entire distribution Measure how well an individual score (or group of scores) represents the entire distribution

Variability (cont.) Variability provides information about how much error to expect when you are using a sample to represent a population. Variability provides information about how much error to expect when you are using a sample to represent a population. Three measures of variability Three measures of variability Range Range Interquartile range Interquartile range Standard deviation Standard deviation

Range The range is the difference between the upper real limit of the largest (maximum) X value and the lower real limit of the smallest (minimum) X value. The range is the difference between the upper real limit of the largest (maximum) X value and the lower real limit of the smallest (minimum) X value. Range is the most obvious way to describe how spread out the scores are. Range is the most obvious way to describe how spread out the scores are.

Range (cont.) Problem: Completely determined by the two extreme values and ignores the other scores in the distribution. Problem: Completely determined by the two extreme values and ignores the other scores in the distribution. It often does not give an accurate description of the variability for the entire distribution. It often does not give an accurate description of the variability for the entire distribution. Considered a crude and unreliable measure of variability Considered a crude and unreliable measure of variability

Interquartile Range and Semi-Interquartile Range Divide the distribution into four equal parts Divide the distribution into four equal parts Q1, Q2, Q3 Q1, Q2, Q3 The interquartile range is defined as the distance between the first quartile and the third quartile The interquartile range is defined as the distance between the first quartile and the third quartile

Interquartile Range Q1 Q2 Q3 25% Semi-interquartile Range

Figure 4.2 The interquartile range Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Interquartile Range (cont.) When the interquartile range is used to describe variability, it commonly is transformed into the semi-interquartile range. When the interquartile range is used to describe variability, it commonly is transformed into the semi-interquartile range. Semi-interquartile range is one-half of the interquartile range Semi-interquartile range is one-half of the interquartile range

Interquartile Range (cont.) Because the semi-interquartile range is derived from the middle 50% of a distribution, it is less likely to be influenced by extreme scores and therefore gives a better and more stable measure of variability than the range. Because the semi-interquartile range is derived from the middle 50% of a distribution, it is less likely to be influenced by extreme scores and therefore gives a better and more stable measure of variability than the range.

Interquartile Range (cont.) Does not take into account distances between individual scores Does not take into account distances between individual scores Does not give a complete picture of how scattered or clustered the scores are. Does not give a complete picture of how scattered or clustered the scores are.

Standard Deviation Most commonly used Most commonly used Most important measure of variability Most important measure of variability Standard deviation uses the mean of the distribution as a reference point and measures variability by considering the distance between each score and the mean. Standard deviation uses the mean of the distribution as a reference point and measures variability by considering the distance between each score and the mean.

Standard Deviation (cont.) Are the scores clustered or scattered? Are the scores clustered or scattered? Deviation is the average distance and direction from the mean. Deviation is the average distance and direction from the mean.

Standard Deviation (cont.) Goal of standard deviation is to measure the standard, or typical, distance from the mean. Goal of standard deviation is to measure the standard, or typical, distance from the mean. Deviation is the distance and direction from the mean Deviation is the distance and direction from the mean deviation score = X - 

Standard Deviation (cont.) Step 1 Step 1 Determine the deviation or distance from the mean for each individual score. Determine the deviation or distance from the mean for each individual score. If  X = 53 deviation score = X –   = +3 = +3

Standard Deviation (cont.) If  X = 45 deviation score = X –   = -5 = -5

Standard Deviation (cont.) Step 2: Calculate the mean of the deviation scores Step 2: Calculate the mean of the deviation scores Add the derivation scores Add the derivation scores Divide by N Divide by N

Standard Deviation (cont.) X X –   X –  Deviation scores must add up to zero

Standard Deviation (cont.) Step 3: Square each deviation score. Step 3: Square each deviation score. Why? The average of the deviation scores will not work as a measure of variability. Why? The average of the deviation scores will not work as a measure of variability. Why? They always add up to zero Why? They always add up to zero

Standard Deviation (cont.) Step 3 cont.: Step 3 cont.: Using the squared values, you can now compute the mean squared deviation Using the squared values, you can now compute the mean squared deviation This is called variance This is called variance Variance = mean squared deviation Variance = mean squared deviation

Standard Deviation (cont.) By squaring the deviation scores: By squaring the deviation scores: You get rid of the + and – You get rid of the + and – You get a measure of variability based on squared distances You get a measure of variability based on squared distances This is useful for some inferential statistics This is useful for some inferential statistics Note: This distance is not the best descriptive measure for variability Note: This distance is not the best descriptive measure for variability

Standard Deviation (cont.) Step 4: Make a correction for squaring the distances by getting the square root. Step 4: Make a correction for squaring the distances by getting the square root. Standard deviation = variance Standard deviation = variance

Sum of Squared Deviations (SS) Variance = mean squared deviation = SS Variance = mean squared deviation = SSN Definitional Formula SS =  X –  

Sum of Squared Deviations (SS) Definitional Formula X X –   X –   = 8 11  =  X–  ) 2 =  X –  ) 2

Computational Formula Computational Formula SS =  X 2 – (  X    N

Computational Formula for SS X X2X2X2X  X = 8  X 2 = 38  X 2 – (  X) 2 SS =  X 2 – (  X) 2 N = 38 – (8) 2 = 38 – (8) 2 4 = 38 – 64 = 38 – 64 4 = 38 – 16 = 38 – 16 = 22 = 22

Definitional vs. Computational? Definitional is most direct way of calculating the sum of squares Definitional is most direct way of calculating the sum of squares However if you have numbers with decimals, it can become cumbersome However if you have numbers with decimals, it can become cumbersome Computation is most commonly used Computation is most commonly used

Formulas Variance = SS Variance = SS N Standard deviation = variance = SS Standard deviation = variance = SS N

Formulas (cont.) Variance and standard deviation are parameters of a population and will be identified with a Greek letter –  or  sigma Variance and standard deviation are parameters of a population and will be identified with a Greek letter –  or  sigma Population standard deviation =  SS N Population variance =    SS N

Figure 4.4 Graphic presentation of the mean and standard deviation Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Figure 4.5 Variability of a sample selected from a population Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Figure 4.6 Largest and smallest distance from the mean Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Example (pg. 94) X X –   X –    X =  611n= =  X–  ) 2 = SS =  X –  ) 2 = SS

Degrees of Freedom Degrees of freedom, use for sample variance Degrees of freedom, use for sample variance where n is the number of scores in the sample. where n is the number of scores in the sample. With a sample of n scores, the first n-1 scores are free to vary With a sample of n scores, the first n-1 scores are free to vary but the final score is restricted. but the final score is restricted. As a result, the sample is said to have n-1 degrees of freedom As a result, the sample is said to have n-1 degrees of freedom

Degrees of Freedom Degrees of freedom, or df, for sample variance are defined as Degrees of freedom, or df, for sample variance are defined as df = n – 1 df = n – 1 where n is the number of scores in the sample.

Table 4.2 Reporting the mean and standard deviation in APA format Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning