MEASURES OF DISPERSION:

SPREAD AND VARIABILITY

DATA SETS FOR PROJECT

• NES2000.sav •  States.sav • World.sav

Outline: Key Measures of Dispersion (Interval Scale) 1.  Range

2.  Variance

3.  Standard Deviation

4.  Standard Score

1. Range of Values

Xmax – Xmin = Range

i

i

2. The Meaning of Variance

Formula: Variance = s2 = Σ (Xi – X)2/n [or n – 1]

Definition: Mean squared deviation from the mean

That is: Overall variation or cumulative “spread” of values around the arithmetic mean (or “average”) for a variable

Relevance: A key goal of statistical analysis is to detect underlying patterns within the overall variance. Question: Why are some values below the mean and some values above? Can we find an explanation?

2. Variance (continued)

•  Thus: We want to understand variation in values of a variable—the so-called “dependent” variable.

•  Question: Might the variation in the dependent variable be a function (or consequence) of variation in another variable—the “independent” variable?

•  Key concept: Explanation of variance (in the dependent variable).

3. The Standard Deviation

•  Measure of how individual observations deviate from or vary around the mean of the variable

•  Allows comparison of variation: – Standard deviation is 0 only if no variation – The greater the spread, the greater the standard

deviation of variable – Two variables with similar means but different

standard deviations differ in extent of variation around mean

• Definition: Zi = (Xi – X)/s

• Measures distance from mean in standard deviation units

• Allows comparison across variables

• Useful in construction of composite variables (i.e., adding apples and oranges—or level of education plus annual income, or GDP per capita plus life expectancy)

4. Standard Scores (Z Scores)

•  1. Aggregating Indicators • Add, multiply….? • Apples and oranges?

•  2. Weighting Indicators • Are some indicators “more important”? • Weighting cannot be avoided

Reprise: Constructing Composite Indicators--Two Key Problems

Example: Socio-economic Status

•  Education: Mean = 10 years , sd = 2, observation X = 16, Zx = (16-10)/2 = + 3

•  Parents’ annual income: Mean = $ 50,000, sd = 5,000, observation X = $40,000,

Zx = (40,000-50,000)/5,000 = -2

Composite scale 1/Mobility and Class Equal Z1 = 3 + (-2) = + 1

Composite scale 2/Social Mobility>Class Z2 = 3(2) + (-2) = + 4

Composite scale 3/Economic Class>Mobility Z3 = 3 + 2(-2) = -1

Alternative Results

Postscript: On Skewness

Sk = (X – Mo)/s = 3 (X – Md)/s

Sk = 0 for a symmetrical (normal) curve, positive if skewed to the right, negative if skewed to the left If Sk > 1, consider using median not mean If Sk/standard error >2, use median not mean