Measures of Dispersion

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Measures of Dispersion

• Learning Objectives
• Explain what is meant by variability
• Describe, know when to use, interpret and
  calculate range, variance, and standard deviation

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More Statistical Notation

• indicates the sum of squared Xs.
• Square ea score (22 22)
• Find sum of squared Xs 448
• indicates the squared sum of X.
• (22)2

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Measures of Variability

• describe the extent to which scores in a
  distribution differ from each other.

A B C
0 8 6
2 7 6
6 6 6
10 5 6
12 4 6
X6 X6 X6
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A Chart Showing the Distance Between the
Locations of Scores in Three Distributions
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Variability

• Provides a quantitative measure of the degree to
  which scores in a distribution are spread out or
  clustered together
• Figure 4.1

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Kurtosis

• Kurtosis based on size of a distributions tail.
• Leptokurtic thin or skinny dist
• Platykurtic flat
• Mesokurtic same kurtosis (normal distribution)

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Three Variations of the Normal Curve
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The Range, Semi-Interquartile Range, Variance,
and Standard Deviation
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The Range

• indicates the distance between the two most
  extreme scores in a distribution
• Crude measurement
• Used w/ nominal or ordinal data
• Range?difference btwn upper real limit of max
  score and lower real limit of min score
• Range highest score lowest score

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The Interquartile Range

• Covered by the middle 50 of the distribution
• Interquartile range Q3-Q1
• Semi-Interquartile Range
• Half of the interquartile range

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Variance and Standard Deviation

• Variance standard deviation communicate how
  different the scores in a distribution are from
  each other
• We use the mean as our reference point since it
  is at the center of the distribution and
  calculate how spread out the scores are around
  the mean

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The Population Variance and the Population
Standard Deviation
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Population Variance

• The population variance is the true or actual
  variance of the population of scores.

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Population Standard Deviation

• The population standard deviation is the true or
  actual standard deviation of the population of
  scores.

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Describing the Sample Variance and the Sample
Standard Deviation
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Sample Variance

• The sample variance is the average of the squared
  deviations of scores around the sample mean

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Sample Variance

• Variance is average of squared deviations
  (usually large) squared units
• Difficult to interpret
• Communicates relative variability

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Standard Deviation

• Measure of Var. that communicates the average
  deviation
• Square root of variance

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Sample Standard Deviation

• The sample standard deviation is the square root
  of the average squared deviation of scores around
  the sample mean.

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The Standard Deviation

• indicates
• average deviation from mean,
• consistency in scores,
• how far scores are spread out around mean
• larger the value of SD, the more the scores are
  spread out around mean, and the wider the
  distribution

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Normal Distribution and the Standard Deviation
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Normal Distribution and the Standard Deviation

• Approximately 34 of the scores in a perfect
  normal distribution are between the mean and the
  score that is one standard deviation from the
  mean.

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The Estimated Population Variance and the
Estimated Population Standard Deviation
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Estimating the Population Variance and Standard
Deviation

• The sample variance is a biased
  estimator of the population variance.
• The sample standard deviation is a
  biased estimator of the population standard
  deviation.

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Estimated Population Variance

• By dividing the numerator of the sample variance
  by N - 1, we have an unbiased estimator of the
  population variance.

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Estimated Population Standard Deviation

• By dividing the numerator of the sample standard
  deviation by N - 1, we have an unbiased estimator
  of the population standard deviation.

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Unbiased Estimators

• is an unbiased estimator of
• is an unbiased estimator of
• The quantity N - 1 is called the degrees of
  freedom
• Number of scores in a sample that are free to
  vary so that they reflect variability in pop

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Uses of , , and

• Use the sample variance and the sample
  standard deviation to describe the
  variability of a sample.
• Use the estimated population variance and
  the estimated population standard deviation
  for inferential purposes when you need to
  estimate the variability in the population.

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Organizational Chart of Descriptive and
Inferential Measures of Variability
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Always..

• Determine level of measurement
• Examine type of distribution
• Calculate mean
• Calculate variability

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American Psychological Association (5th ed)

• Mean
• M
• Standard Deviation
• SD

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Example

• Using the following data set, find
• The range,
• The semi-interquartile range,
• The sample variance and standard deviation,
• The estimated population variance standard
  deviation

14 14 13 15 11 15
13 10 12 13 14 13
14 15 17 14 14 15
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Example Range

• The range is the largest value minus the smallest
  value.

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ExampleSample Variance
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ExampleSample Standard Deviation
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ExampleEstimated Population Variance
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ExampleEstimated Population Standard Deviation