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 © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

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. © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

Notice that both distributions have the same mean, yet they are shaped differently. © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

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. IQV = K ( Σf 2 ) (K-1) Where: K= the number of categories N= the total number of cases in the distribution Σf 2 = the sum of all squared frequencies or percentages © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

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. © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

IQV Example 1 © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

IQV Example 1, Continued © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

Racial Diversity in the U.S., 2008 © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

IQV: Example 2 – Category Sensitive Comparing Racial Diversity at the Univ. of California – Berkley, 1984 & 2009 Racial & Ethnic Groups: IQV for 1984 =.70 (61% White, 24% Asian, 6% Latino, 5% Black, 4% Other) IQV for 2009 =.89 (30% White, 40% Asian, 12% Latino, 4% Black, 14% Other) What happens to the IQV if we divide the Asian category into Chinese Americans and Other Asians? © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

IQV: Example 2 – Category Sensitive, Continued IQV = K ( Σ f 2 )/100 2 (K-1) =6(10,000-2,058)/10,000 (5) =47,652/50,000 =.95 Or 95% of the maximum possible differences. We can then conclude that the larger the number of categories, the larger the variance. © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

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. © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

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  Q3 = 75th percentile (75% below and 25% are above this number)  Q1 = 25th percentile (25% below and 75% are above this number) © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

Finding Q 1 and Q 3 1.Sort values from high to low (or low to high). 2.Identify the first quartile (25 th Percentile) (N)(0.25)=51(0.25)=12.75 state (between Illinois and Utah), so /2= Identify the third quartile (75 th Percentile) (N)(0.75)=51(0.75)=38.25 state (between Washington and Maine), so /2= IQR=Q 1 -Q 3 = = The interquartile range of the percentage increase in the elderly population is percentage points. © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

The Difference Between the Range and IQR Shows greater variability These values fall together closely Yet the ranges are equal! Importance of the IQR © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

The Box Plot © 2011 SAGE Publication sFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e 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.)

Box Plot Example © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

Variance Variance – A measure of variation for interval-ratio variables; it is the average of the squared deviations from the mean © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

Standard Deviation Standard Deviation – A measure of variation for interval-ratio variables; it is equal to the square root of the variance. © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

Finding the Mean © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

Finding the Standard Deviation © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e

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. © 2011 SAGE PublicationsFrankfort-Nachmias and Leon-Guerrero, Statistics for a Diverse Society, 6e