MEASURES OF DISPERSION I. Introduction. A. Measures of dispersion describe a distribution by providing a measure of how dispersed or spread out the distribution is. They indicate how much variation there is in the distribution of a variable. B. Dispersion can be determined for both categorical and numerical data. II. Measures of dispersion for numerical data. A. The minimum, maximum, and range. 1. The minimum is the lowest score in a numerical distribution. 2. The maximum is the highest score in a numerical distribution. 3. The range is the distance between the minimum and maximum scores and is calculated by subtracting the minimum from the maximum. B. Percentiles, quartiles, and the interquartile range. 1. A percentile represents a value below which a specific percentage of observations fall. 2. To find a percentile, rank the data set from highest to lowest, multiply the desired percentile (expressed as a proportion) by sample size, round to the nearest whole number, then count up from the bottom that number of observations. 3. Observations falling between the minimum and the 25th percentile are in the first quartile (Q1 ). Observations falling between the 25th percentile and the 50th percentile (the median) are in the second quartile (Q2). Observations between the 50th percentile and the 75th percentile are in third quartile (Q3). Observations between 75th percentile and the maximum are in the fourth quartile (Q4). 4. The interquartile range is the distance between the 25th and 75th percentiles. Basically, it excludes the highest and lowest 25% of observations and calculates the range of the middle 50% of observations. C. Standard deviation. 1. Deviations. a. A deviation is the distance between a numerical score and the mean of the distribution. b. To calculate a deviation for a score, you simply subtract the mean from the score. c. Scores that are greater than the mean will have positive deviations. Scores that are lower than the mean will have negative deviations. d. If you add together all the deviations for a distribution of scores, the sum is zero. 2. Variance. a. The variance is the sum of the squared deviations divided by sample size. b. Calculating the variance. c. The variance is always positive. d. Larger variances indicate greater variation. e. The variance is inflated in that it is a squared measure. 3. Standard Deviation a. The standard deviation is simply the square root of the variance. b. Calculating standard deviation. c. Interpreting standard deviation. i. Standard deviation can be seen as index or measure of the amount of variation in a distribution. ii. It is always greater than or equal to 0. iii. The greater it is, the greater the variation. iv. The standard deviation and the mean. III. Measures of dispersion for categorical data. A. Proportions as measures of dispersion. 1. For categorical variables with a limited number of attributes, dispersion can be measured by comparing the relative frequency of each category. 2. If the relative frequency of all categories is similar, there is much dispersion. If there is great inequality in the relative frequencies, there is less dispersion. B. Index of Qualitative Variation 1. The IQV is the ratio of the amount of variation actually observed to the amount of variation possible. 2. It ranges from 0 (no variation) to 1 (maximum variation). 3. It can be calculated for variables measured at any level, but is most commonly reported for categorical (nominal and ordinal) variables. 4. Calculating the IQV.