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Measures of Variability

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REP [REAGAN] 1982. 11.082 = 122.8. 11.08 -15 26.08 -15. DEM [CARTER] ... REP [NIXON] 1970 -20.922 = 437.6 -20.92 -47 26.08 -47. DEM [LBJ] 1966. 22.082 = 487.5 ... – PowerPoint PPT presentation

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Title: Measures of Variability


1
Measures of Variability
  • Understanding Dispersion

2
Summarizing Data
  • Central tendency (mean, mode, median)
  • Range (what is the minimum point and what is the
    maximum point)
  • Variability/dispersion (how close/far away are
    each of the data points from the mean how much
    heterogeneity do we have)

3
Family of Normal Distribution Curves
4
Mean
  • Mean describes Central Tendency, it tells us what
    the average outcome is.
  • We also want to know something about how accurate
    the mean is when making predictions.

5
Means
  • Consider these means for weekly candy bar
    consumption.

X 7, 8, 6, 7, 7, 6, 8, 7 X
(78677687)/8 X 7
X 12, 2, 0, 14, 10, 9, 5, 4 X
(12201410954)/8 X 7
What is the difference ?
6
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7
How do we describe this?
  • Measures of variability
  • Mean Deviation
  • Variance
  • Standard Deviation

8
Mean Deviation
  • We could just calculate the average distance
    between each observation and the mean.
  • We must take the absolute value of the distance,
    otherwise they would just cancel out to zero!
  • Formula (SX-X)
  • N

9
Mean Deviation An Example
Data X 6, 10, 5, 4, 9, 8
X 42 / 6 7
  • Compute X (Average)
  • Compute X X and take the Absolute Value to get
    Absolute Deviations
  • Sum the Absolute Deviations
  • Divide the sum of the absolute deviations by N

12 / 6 2
Total 12
10
What Does it Mean?
  • On Average, each observation is two units away
    from the mean.

11
Is it Really that Easy?
  • No!
  • Absolute values are difficult to manipulate
    algebraically
  • Absolute values cause enormous problems for
    calculus (Discontinuity)
  • We need something else

12
Variance and Standard Deviation
  • Instead of taking the absolute value, we square
    the deviations from the mean. This yields a
    positive value.
  • This will result in measures we call the Variance
    and the Standard Deviation
  • Sample- Population-
  • s Standard Deviation s Standard Deviation
  • s2 Variance s2 Variance

13
Calculating the Variance and/or Standard Deviation
  • Formulae
  • Variance Standard
    Deviation
  • Examples Follow . . .

14
Example
Data X 6, 10, 5, 4, 9, 8 N 6
Mean
Variance
Standard Deviation
Total 42
Total 28
15
GAINS LOSSES BY PRESIDENTS PARTY IN MIDTERM
ELECTIONS
MEAN
VARIANCE
ST. DEV.
16
What Does it Mean?
  • On Average, each observations is 17.7 seats away
    from the mean seat loss (roughly 18 seats).

17
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18
The Seat Loss Example
-80 -62 -44 -26 -8 10 28
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