Chapter 5 – 1 Chapter 5: Measures of Variability The Importance of Measuring Variability IQV (Index of Qualitative Variation) The Range IQR (Inter-Quartile Range) Variance Standard Deviation Considerations for choosing a measure of variation

Chapter 5 – 2 The Importance of Measuring Variability Central tendency - Numbers that describe what is typical or average (central) in a distribution Measures of Variability - Numbers that describe diversity or variability in the distribution. These two types of measures together help us to sum up a distribution of scores without looking at each and every score. Measures of central tendency tell you about typical (or central) scores. Measures of variation reveal how far from the typical or central score that the distribution tends to vary.

Chapter 5 – 3 Notice that both distributions have the same mean, yet they are shaped differently

Chapter 5 – 4 Example What’s the mean, median, and mode for them? A: B: C: ABC

Chapter 5 – 5 Index of Qualitative Variation IQV – A measure of variability for nominal variables. It is based on the ratio of the total number of differences in the distribution to the maximum number of possible differences within the same distribution. Where K= the number of categories Σp* 2 = the sum of all squared percentages Σp 2= the sum of all squared proportions

Chapter 5 – 6 Understanding the Index of Qualitative Variation The IQV is a single number that expresses the diversity of a distribution. The IQV ranges from 0 to 1 An IQV of 0 would indicate that the distribution has NO diversity at all. An IQV of 1 would indicate that the distribution is maximally diverse.

Chapter 5 – 7 IQV in Real Life: Diversity in the U.S.

Chapter 5 – 8 IQV-Example LANGUAGE HOME Population 5 years and over.6,161,4607,486,797 English only5,931,4356,909,648 Spanish105,963378,942 Other Indo-European languages90,979119,961 Asian / Pacific Island languages33,08378,246

Chapter 5 – 9 The Range Range = highest score - lowest score Range – A measure of variation in interval-ratio variables. It is the difference between the highest (maximum) and the lowest (minimum) scores in the distribution.

Chapter 5 – 10 Inter-Quartile Range Inter-Quartile Range (IQR) – A measure of variation for interval-ratio data. It indicates the width of the middle 50 percent of the distribution and is defined as the difference between the lower and upper quartiles (Q1 and Q3.) IQR = Q3 – Q1

Chapter 5 – 11 The difference between the Range and IQR Shows greater variability These values fall together closely Yet the ranges are equal! Importance of the IQR

Chapter 5 – 12 The Box Plot The Box Plot is a graphic device that visually presents the following elements: the range, the IQR, the median, the quartiles, the minimum (lowest value,) and the maximum (highest value.)

Chapter 5 – 13 Variance Variance – A measure of variation for interval-ratio variables; it is the average of the squared deviations from the mean

Chapter 5 – 14 Standard Deviation Standard Deviation – A measure of variation for interval-ratio variables; it is equal to the square root of the variance.

Chapter 5 – 15 Find the Mean and the Standard Deviation

Chapter 5 – 16 Find the Mean and the Standard Deviation Total Variance SD18.49

Chapter 5 – 17 Considerations for Choosing a Measure of Variability For nominal variables, you can only use IQV (Index of Qualitative Variation.) For ordinal variables, you can calculate the IQV or the IQR (Inter-Quartile Range.) Though, the IQR provides more information about the variable. For interval-ratio variables, you can use IQV, IQR, or variance/standard deviation. The standard deviation (also variance) provides the most information, since it uses all of the values in the distribution in its calculation.