MEASURES OF DISPERSION: SPREAD AND VARIABILITY

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MEASURES OF DISPERSIONSPREAD AND
VARIABILITY
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DATA SETS FOR PROJECT

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

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• Outline Key Measures
• of Dispersion (Interval Scale)
• Range
• Variance
• Standard Deviation
• Standard Score

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1. Range of Values Xmax Xmin Range
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i
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2. The Meaning of Variance
Formula Variance s2 S (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?
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2. Variance (continued)

• Thus We want to understand variation in values
  of a variablethe so-called dependent variable.
• Question Might the variation in the dependent
  variable be a function (or consequence) of
  variation in another variablethe independent
  variable?
• Key concept Explanation of variance (in the
  dependent variable).

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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

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4. Standard Scores (Z Scores)

• 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 orangesor level of education
• plus annual income, or GDP per capita plus life
  expectancy)

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Reprise Constructing Composite Indicators--Two
Key Problems

• 1. Aggregating Indicators
• Add, multiply.?
• Apples and oranges?
• 2. Weighting Indicators
• Are some indicators more important?
• Weighting cannot be avoided

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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

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Alternative Results
Composite scale 1/Mobility and Class Equal Z1
3 (-2) 1 Composite scale 2/Social
MobilitygtClass Z2 3(2) (-2) 4 Composite
scale 3/Economic ClassgtMobility Z3 3 2(-2)
-1
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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 gt 1, consider using median not mean
• If Sk/standard error gt2, use median not mean

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