CHAPTER 4 Measures of Dispersion

In This Presentation  Measures of dispersion.  You will learn Basic Concepts How to compute and interpret the Range (R) and the standard deviation (s)

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: Interval/ratio variables  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 interval- ratio 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 in a distribution of scores Square the deviations (this eliminates negative numbers). Sum the squared deviations. Divide the sum of the squared deviations by N: this is the Variance 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

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

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).