Descriptive Statistics: Overview

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Descriptive Statistics Overview
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Central tendency

• Seeks to provide a single value that best
  represents a distribution

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Central tendency
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Central tendency
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Central tendency
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Central tendency

• Seeks to provide a single value that best
  represents a distribution
• Typical measures are
• mode
• median
• mean

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Mode

• the most frequently occurring score value
• corresponds to the highest point on the frequency
  distribution

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Mode

• The mode is not sensitive to extreme scores.

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Mode

• a distribution may have more than one mode

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Mode

• there may be no unique mode, as in the case of a
  rectangular distribution

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Median

• the score value that cuts the distribution in
  half (the middle score)
• 50th percentile

For N 15 the median is the eighth score 37
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Median
For N 16 the median is the average of the
eighth and ninth scores 37.5
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Mean

• this is what people usually have in mind when
  they say average
• the sum of the scores divided by the number of
  scores

Changing the value of a single score may not
affect the mode or median, but it will affect the
mean.
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Mean
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In many cases the mean is the preferred measure
of central tendency, both as a description of the
data and as an estimate of the parameter.
X7.07
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Mean
The mean is sensitive to extreme scores and is
appropriate for more symmetrical distributions.
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Symmetry

• a symmetrical distribution exhibits no skewness
• in a symmetrical distribution the Mean Median
  Mode

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Skewed distributions

• Skewness refers to the asymmetry of the
  distribution

• A positively skewed distribution is asymmetrical
  and points in the positive direction.

Mode 70,000 Median 88,700 Mean 93,600

• mode lt median lt mean

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Skewed distributions

• A negatively skewed distribution

• mode gt median gt mean

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Measures of central tendency
-
Mode quick easy to compute useful for nominal data poor sampling stability
Median not affected by extreme scores somewhat poor sampling stability
Mean sampling stability related to variance inappropriate for discrete data affected by skewed distributions
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Distributions

• Center mode, median, mean
• Shape symmetrical, skewed
• Spread

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Measures of Spread

• the dispersion of scores from the center
• a distribution of scores is highly variable if
  the scores differ wildly from one another
• Three statistics to measure variability
• range
• interquartile range
• variance

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Range

• largest score minus the smallest score
• these two
• have same range (80)
• but spreads look different
• says nothing about how scores vary around the
  center
• greatly affected by extreme scores (defined by
  them)

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Interquartile range

• the distance between the 25th percentile and the
  75th percentile
• Q3-Q1 70 - 30 40
• Q3-Q1 52.5 - 47.5 5
• effectively ignores the top and bottom quarters,
  so extreme scores are not influential
• dismisses 50 of the distribution

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Homework

• Page 275 1, 2, 3, 6 - 8