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Numeric Summaries and Descriptive Statistics

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Numeric Summaries and Descriptive Statistics populations vs. samples we want to describe both samples and populations the latter is a matter of inference ... – PowerPoint PPT presentation

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Title: Numeric Summaries and Descriptive Statistics


1
Numeric Summaries and Descriptive Statistics
2
populations vs. samples
  • we want to describe both samples and populations
  • the latter is a matter of inference

3
outliers
  • minority cases, so different from the majority
    that they merit separate consideration
  • are they errors?
  • are they indicative of a different pattern?
  • think about possible outliers with care, but
    beware of mechanical treatments
  • significance of outliers depends on your research
    interests

4
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5
summaries of distributions
  • graphic vs. numeric
  • graphic may be better for visualization
  • numeric are better for statistical/inferential
    purposes
  • resistance to outliers is usually an advantage in
    either case

6
general characteristics
peakedness
  • kurtosis

leptokurtic? ?platykurtic
7
  • skew (skewness)

8
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9
central tendency
  • measures of central tendency
  • provide a sense of the value expressed by
    multiple cases, over all
  • mean
  • median
  • mode

10
mean
  • center of gravity
  • evenly partitions the sum of all measurement
    among all cases average of all measures

11
mean pro and con
  • crucial for inferential statistics
  • mean is not very resistant to outliers
  • a trimmed mean may be better for descriptive
    purposes

12
mean
R mean(x)
13
trimmed mean
R mean(x, trim.1)
14
median
  • 50th percentile
  • less useful for inferential purposes
  • more resistant to effects of outliers

15
median
16
mode
  • the most numerous category
  • for ratio data, often implies that data have been
    grouped in some way
  • can be more or less created by the grouping
    procedure
  • for theoretical distributionssimply the location
    of the peak on the frequency distribution

17
modal class hamlets
isolated scatters
hamlets
villages
regional centers
regional centers
18
dispersion
  • measures of dispersion
  • summarize degree of clustering of cases, esp.
    with respect to central tendency
  • range
  • variance
  • standard deviation

19
range
R range(x)
20
variance
R var(x)
  • analogous to average deviation of cases from mean
  • in fact, based on sum of squared deviations from
    the meansum-of-squares

21
variance
  • computational form

22
  • note units of variance are squared
  • this makes variance hard to interpret
  • ex. projectile point sample
  • mean 22.6 mm
  • variance 38 mm2
  • what does this mean???

23
standard deviation
  • square root of variance

24
standard deviation
  • units are in same units as base measurements
  • ex. projectile point sample
  • mean 22.6 mm
  • standard deviation 6.2 mm
  • mean /- sd (16.428.8 mm)
  • should give at least some intuitive sense of
    where most of the cases lie, barring major
    effects of outliers

25
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26
trimmed dispersion measures
  • variance and sd are even more sensitive to
    extreme values (outliers) than the mean
  • why??
  • you can calculate a trimmed version of the
    variance simply by eliminating cases from the
    tails, and calculating the variance in the normal
    way

27
trimmed standard deviation
  • trimmed sd is calculated differently
  • sT trimmed standard deviation n number of
    cases in untrimmed batch s2w variance of
    trimmed (winsorized) batch nT number of cases
    in the trimmed batch
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