#### Transcript less - Cengage Learning

CHAPTER 4 Measures of Dispersion Chapter Outline Introduction The Index of Qualitative Variation (IQV) The Range (R) and Interquartile Range (Q) Computing the Range and Interquartile Range The Standard Deviation Chapter Outline Computing the Standard Deviation: An Additional Example Computing the Standard Deviation for Grouped Data Interpreting the Standard Deviation Interpreting Statistics: The Central Tendency and Dispersion of Income in the United States In This Presentation Measures of dispersion. You will learn Basic Concepts How to compute and interpret the Range (R) and the standard deviation (s) These other measures of dispersion are covered in the text: IQV Q s2 The Concept of Dispersion Dispersion = variety, diversity, amount of variation between scores. The greater the dispersion of a variable, the greater the range of scores and the greater the differences between scores. The Concept of Dispersion: Examples Typically, a large city will have more diversity than a small town. Some states (California, New York) are more racially diverse than others (Maine, Iowa). Some students are more consistent than others. The Concept of Dispersion The taller curve has less dispersion. The flatter curve has more dispersion. The Range Range (R) = High Score – Low Score Quick and easy indication of variability. Can be used with ordinal or intervalratio variables. Why can’t the range be used with variables measured at the nominal level? Standard Deviation The most important and widely used measure of dispersion. Should be used with interval-ratio variables but is often used with ordinal-level variables. Standard Deviation Formulas for variance and standard deviation: Standard Deviation To solve: Subtract mean from each score. Square the deviations. Sum the squared deviations. Divide the sum of the squared deviations by N. Find the square root of the result. Interpreting Dispersion Low score=0, Mode=12, High score=20 Measures of dispersion: R=20–0=20, s=2.9 Years of Education (Full Sample) 900 800 700 600 500 400 300 200 100 0 Interpreting Dispersion What would happen to the dispersion of this variable if we focused only on people with college-educated parents? We would expect people with highly educated parents to average more education and show less dispersion. Interpreting Dispersion Low score=10, Mode=16, High Score=20 Measures of dispersion: R=20-10=10, s=2.2 Years of Education (Both Parents w BA) 45 40 35 30 25 20 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Interpreting Dispersion Entire sample: Mean = 13.3 Range = 20 s = 2.9 Respondents with college-educated parents: Mean = 16.0 R = 10 s =2.2 Interpreting Dispersion As expected, the smaller, more homogeneous and privileged group: Averaged more years of education (16.0 vs. 13.3) And was less variable (s = 2.2 vs. 2.9; R = 10 vs. 20) Measures of Dispersion Higher for more diverse groups (e.g., large samples, populations). Decrease as diversity or variety decreases (are lower for more homogeneous groups and smaller samples). The lowest value possible for R and s is 0 (no dispersion).