COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Instructor: Dr. John J. Kerbs, Associate Professor Joint Ph.D. in Social Work and Sociology Chapter 4: Variability © DO NOT CITE, QUOTE, REPRODUCE, OR DISSEMINATE WITHOUT WRITTEN PERMISSION FROM THE AUTHOR: Dr. John J. Kerbs can be ed for permission at

Variability The goal for variability is to obtain a measure of how spread out the scores are in a distribution. The goal for variability is to obtain a measure of how spread out the scores are in a distribution. Variability indicates if scores are clustered together or spread out over a large distance Variability indicates if scores are clustered together or spread out over a large distance Variability is usually defined in terms of distance Variability is usually defined in terms of distance Example: Distance between two scores or distance between a score and a mean (average) Example: Distance between two scores or distance between a score and a mean (average) A measure of variability usually accompanies a measure of central tendency as basic descriptive statistics for a set of scores. A measure of variability usually accompanies a measure of central tendency as basic descriptive statistics for a set of scores.

Central Tendency and Variability Central tendency describes the central point of the distribution, and variability describes how the scores are scattered around that central point. Central tendency describes the central point of the distribution, and variability describes how the scores are scattered around that central point. Together, central tendency and variability are the two primary values that are used to describe a distribution of scores. Together, central tendency and variability are the two primary values that are used to describe a distribution of scores.

Variability Variability serves both as a descriptive measure and as an important component of most inferential statistics. Variability serves both as a descriptive measure and as an important component of most inferential statistics. As a descriptive statistic, variability measures the degree to which the scores are spread out or clustered together in a distribution. As a descriptive statistic, variability measures the degree to which the scores are spread out or clustered together in a distribution. In the context of inferential statistics, variability provides a measure of how accurately any individual score or sample represents the entire population. In the context of inferential statistics, variability provides a measure of how accurately any individual score or sample represents the entire population.

Variability (Continued) When the population variability is small, all of the scores are clustered close together and any individual score or sample will necessarily provide a good representation of the entire set. When the population variability is small, all of the scores are clustered close together and any individual score or sample will necessarily provide a good representation of the entire set. On the other hand, when variability is large and scores are widely spread, it is easy for one or two extreme scores to give a distorted picture of the general population. On the other hand, when variability is large and scores are widely spread, it is easy for one or two extreme scores to give a distorted picture of the general population.

Variability (Example)

Measuring Variability Variability can be measured with Variability can be measured with 1. The range 1. The range 2. The variance 2. The variance 3. The standard deviation 3. The standard deviation In all cases, variability is determined by measuring distance. In all cases, variability is determined by measuring distance.

The Range The range is the total distance covered by the distribution, from the highest score to the lowest score (using the upper and lower real limits of the range). The range is the total distance covered by the distribution, from the highest score to the lowest score (using the upper and lower real limits of the range). RANGE = URL for X max – LRL for X min RANGE = URL for X max – LRL for X min If scores span values 1 to 5, then the range is 5.5 to 0.5 = 5 Points If scores span values 1 to 5, then the range is 5.5 to 0.5 = 5 Points When scores are measured in whole numbers, the range is also a measure of the number of discrete measurement categories: When scores are measured in whole numbers, the range is also a measure of the number of discrete measurement categories: 1, 2, 3, 4, 5 = 5-pt range 1, 2, 3, 4, 5 = 5-pt range Many computer programs use alternative definition as calculated by upper minus lower value: 5 – 1 = 4 Many computer programs use alternative definition as calculated by upper minus lower value: 5 – 1 = 4

The Range (Continued) Problems with the use of range as a measure of variability in a distribution Problems with the use of range as a measure of variability in a distribution The range does not consider all scores in a distribution, which makes it less than optimal for describing an entire distribution The range does not consider all scores in a distribution, which makes it less than optimal for describing an entire distribution Thus, the range is considered a crude and unreliable measure of variability. Thus, the range is considered a crude and unreliable measure of variability. Although the textbook does not care about which definition you use to measure the range, you need to know how you can calculate all three using the 1) range formula, 2) whole number approach for discrete categories, and 3) computer approach (see prior slide for details) Although the textbook does not care about which definition you use to measure the range, you need to know how you can calculate all three using the 1) range formula, 2) whole number approach for discrete categories, and 3) computer approach (see prior slide for details)

The Standard Deviation Standard deviation measures the standard (average) distance between a score and the mean. Standard deviation measures the standard (average) distance between a score and the mean. The calculation of standard deviation can be summarized as a four-step process: The calculation of standard deviation can be summarized as a four-step process:

The Standard Deviation (Continued) 1. Compute the deviation (distance from the mean) for each score - - a.k.a., the deviation score = (X – μ) 2. Square each deviation: (X - μ) 2 3. Compute the mean of the squared deviations. For a population, this involves summing the squared deviations (sum of squares, SS) and then dividing by N. The resulting value is called the variance (σ 2 ) or mean square and measures the average squared distance from the mean. For samples, variance (s 2 ) is computed by dividing the sum of the squared deviations (SS) by n - 1, rather than N. The value, n - 1, is know as degrees of freedom (df) and is used so that the sample variance will provide an unbiased estimate of the population variance. 4. Finally, take the square root of the variance to obtain the standard deviation.

Calculating The Standard Deviation

Standard Deviation for Population Two ways to calculate: 1) using definitional formula, and 2) using calculation formula. Two ways to calculate: 1) using definitional formula, and 2) using calculation formula. Use definitional formula for SS when the mean () is a whole number (μ = 8/4 = 2) Use definitional formula for SS when the mean (μ) is a whole number (μ = 8/4 = 2) Variance (σ 2 ) = SS/N = 22/4 = 5.50 Variance (σ 2 ) = SS/N = 22/4 = 5.50 SS = Σ(X - μ) 2 = = 22 SS = Σ(X - μ) 2 = = 22 Standard Deviation (σ) = = Standard Deviation (σ) = = Score (X) Deviation (X - μ) Squared Deviation (X - μ) 2 1(1 - 2) = - 1(-1) 2 = 1 0(0 - 2) = - 2(-2) 2 = 4 6(6 - 2) = +4(+4) 2 = 16 1(1 - 2) = - 1(-1) 2 = 1

Standard Deviation for Population Two ways to calculate: 1) using definitional formula, and 2) using calculation formula. Two ways to calculate: 1) using definitional formula, and 2) using calculation formula. Use computational formula for SS when the mean () is not a whole number Use computational formula for SS when the mean (μ) is not a whole number Example for scores 3, 1, 5, 1 (μ = 10/4 = 2.5) Example for scores 3, 1, 5, 1 (μ = 10/4 = 2.5) SS = Σ X 2 – [(ΣX ) 2 / N ] SS = Σ X 2 – [(ΣX ) 2 / N ] SS = ( ) – [(10) 2 / 4 ] SS = ( ) – [(10) 2 / 4 ] SS = 36 – 25 = 11 SS = 36 – 25 = 11 Variance (σ 2 ) = SS/N = 11/4 = 2.75 Variance (σ 2 ) = SS/N = 11/4 = 2.75 Standard Deviation (σ) = Standard Deviation (σ) =

Standard Deviation for Sample As compared to the calculations for populations, the variance (and thus the standard deviation) for samples is computed by dividing the sum of the squared deviations (SS) by n - 1, rather than N. The value, n - 1, is know as degrees of freedom (df), which determine the number of scores in the sample that are independent and free to vary. DF is used so that the sample variance will provide an unbiased estimate of the population variance. Without correcting the denominator in the calculations for variance with the proper degrees of freedom, the variance will be under-estimated (i.e., biased). Without correcting the denominator in the calculations for variance with the proper degrees of freedom, the variance will be under-estimated (i.e., biased). All of the prior steps for calculating SS still apply in relation to both the use of definitional and computational formulas, except that you will replace μ with M and N with n All of the prior steps for calculating SS still apply in relation to both the use of definitional and computational formulas, except that you will replace μ with M and N with n

Standard Deviation for Sample Sample Variance a.k.a., Estimated Population Variance In the context of inferential statistics, the variance that exists in a set of sample data is often classified as error variance to indicate that the variance represents unexplained or uncontrolled differences between scores. Sample Standard Deviation Sample Standard Deviation a.k.a., Estimated Population Standard Deviation a.k.a., Estimated Population Standard Deviation As “estimates,” there is always the potential for biases. As “estimates,” there is always the potential for biases. A sample statistic is biased if the average value of the statistic either underestimates or overestimates the corresponding population parameter. A sample statistic is biased if the average value of the statistic either underestimates or overestimates the corresponding population parameter. A sample statistic is unbiased if the average value of the statistic equals the population parameter. A sample statistic is unbiased if the average value of the statistic equals the population parameter. NOTE: The average value of the statistic is obtained from all the possible samples for a specific sample size, n. NOTE: The average value of the statistic is obtained from all the possible samples for a specific sample size, n.

Standard Deviation for Sample Consider the following sample (n = 7, M = 5) Consider the following sample (n = 7, M = 5) The sample mean (M) = 35/7 = 5 The sample mean (M) = 35/7 = 5 Because the mean is a whole number, you can use the definitional formula for the sum of squares (SS) Because the mean is a whole number, you can use the definitional formula for the sum of squares (SS) Score (X) Deviation (X - M) Squared Deviation (X - M) 2 1(1-5) = - 4(-4) 2 = 16 6(6-5) = + 1(+1) 2 = 1 4(4-5) = - 1(-1) 2 = 1 3(3-5) = - 2(-2) 2 = 4 8(8-5) = + 3(+3) 2 = 9 7(7-5) = + 2(+2) 2 = 4 6(6-5) = + 1(+1) 2 = 1

Standard Deviation for Sample Consider the following sample (n = 7, M = 5) Consider the following sample (n = 7, M = 5) SS sample = = 36 SS sample = = 36 Variance (s 2 ) = SS/(n-1) = 36/(7-1) = 36/6 = 6 Variance (s 2 ) = SS/(n-1) = 36/(7-1) = 36/6 = 6 Standard Deviation (s) = Standard Deviation (s) = Score (X) Deviation (X - M) Squared Deviation (X - M) 2 1(1-5) = - 4(-4) 2 = 16 6(6-5) = + 1(+1) 2 = 1 4(4-5) = - 1(-1) 2 = 1 3(3-5) = - 2(-2) 2 = 4 8(8-5) = + 3(+3) 2 = 9 7(7-5) = + 2(+2) 2 = 4 6(6-5) = + 1(+1) 2 = 1

Standard Deviation for Sample with Computational Formula Consider the following sample (n = 7, M = 5) Consider the following sample (n = 7, M = 5) Σ X = 35 Σ X = 35 (Σ X) 2 = 1225 (Σ X) 2 = 1225 Σ X 2 = 211 Σ X 2 = 211 SS sample = Σ X 2 - [(Σ X) 2 /n] = 211- (1225/7) = = 36 SS sample = Σ X 2 - [(Σ X) 2 /n] = 211- (1225/7) = = 36 Variance (s 2 ) = SS/(n-1) = 36/(7-1) = 36/6 = 6 Variance (s 2 ) = SS/(n-1) = 36/(7-1) = 36/6 = 6 Standard Deviation (s) = Standard Deviation (s) = Score (X) Score Squared (X 2 ) Do NOT confuse the SS denominator with the denominator of the sample variance (s 2 )

Properties of Standard Deviation If a constant is added to every score in a distribution, the standard deviation will not be changed. If a constant is added to every score in a distribution, the standard deviation will not be changed. If you visualize the scores in a frequency distribution histogram, then adding a constant will move each score so that the entire distribution is shifted to a new location. If you visualize the scores in a frequency distribution histogram, then adding a constant will move each score so that the entire distribution is shifted to a new location. The center of the distribution (the mean) changes, but the standard deviation remains the same. The center of the distribution (the mean) changes, but the standard deviation remains the same. If each score is multiplied by a constant, the standard deviation will be changed - - i.e., the standard deviation will be multiplied by the same constant. If each score is multiplied by a constant, the standard deviation will be changed - - i.e., the standard deviation will be multiplied by the same constant. Multiplying by a constant will multiply the distance between scores, and because the standard deviation is a measure of distance, it will also be multiplied. Multiplying by a constant will multiply the distance between scores, and because the standard deviation is a measure of distance, it will also be multiplied.

The Mean and Standard Deviation as Descriptive Statistics If you are given numerical values for the mean and the standard deviation, you should be able to construct a visual image (or a sketch) of the distribution of scores. If you are given numerical values for the mean and the standard deviation, you should be able to construct a visual image (or a sketch) of the distribution of scores. As a general rule, about 70% of the scores will be within one standard deviation of the mean, and about 95% of the scores will be within a distance of two standard deviations of the mean. As a general rule, about 70% of the scores will be within one standard deviation of the mean, and about 95% of the scores will be within a distance of two standard deviations of the mean.