Measures of Dispersion

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Measures of DispersionThe Standard Normal
Distribution

• 9/12/06

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The Semi-Interquartile Range (SIR)

• A measure of dispersion obtained by finding the
  difference between the 75th and 25th percentiles
  and dividing by 2.
• Shortcomings
• Does not allow for precise interpretation of a
  score within a distribution
• Not used for inferential statistics.

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Calculate the SIR

• 6, 7, 8, 9, 9, 9, 10, 11, 12
• Remember the steps for finding quartiles
• First, order the scores from least to greatest.
• Second, Add 1 to the sample size.
• Third, Multiply sample size by percentile to find
  location.
• Q1 (10 1) .25
• Q2 (10 1) .50
• Q3 (10 1) .75
• If the value obtained is a fraction take the
  average of the two adjacent X values.

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Variance (second moment about the mean)

• The Variance, s2, represents the amount of
  variability of the data relative to their mean
• As shown below, the variance is the average of
  the squared deviations of the observations about
  their mean

• The Variance, s2, is the sample variance, and is
  used to estimate the actual population variance,
  s 2

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Standard Deviation

• Considered the most useful index of variability.
• It is a single number that represents the spread
  of a distribution.
• If a distribution is normal, then the mean plus
  or minus 3 SD will encompass about 99 of all
  scores in the distribution.

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Definitional vs. Computational

• Definitional
• An equation that defines a measure
• Computational
• An equation that simplifies the calculation of
  the measure

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Calculate the variance using the computational
and definitional formulas.

• 6, 7, 8, 9, 9, 9, 10, 11, 12

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Calculating the Standard Deviation
9

• Interpreting the standard deviation
• Remember
• Fifty Percent of All Scores in a Normal Curve
  Fall on Each Side of the Mean

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Probabilities Under the Normal Curve
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With our previous scores

• What score is one standard deviation above the
  mean?
• Two standard deviations?
• Three standard deviations?
• What score is one standard deviation below the
  mean?
• Two standard deviations?
• Three standard deviations?

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Interpreting the standard deviation

• We can compare the standard deviations of
  different samples to determine which has the
  greatest dispersion.
• Example
• A spelling test given to third-grader children
• 10, 12, 12, 12, 13, 13, 14
• xbar 12.28 s 1.25
• The same test given to second- through
  fourth-grade children.
• 2, 8, 9, 11, 15, 17, 20
• xbar 11.71 s 6.10

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The shape of distributions

• Skew
• A statistic that describes the degree of skew for
  a distribution.
• 0 no skew
• or - .50 is sufficiently symmetrical

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Kurtosis

• Mesokurtic (normal)
• Around 3.00
• Platykurtic (flat)
• Less than 3.00
• Leptokurtic (peaked)
• Greater than 3.00

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From our previous scores

• Calculate the skew
• 6, 7, 8, 9, 9, 9, 10, 11, 12
• xbar 9.00
• mdn 9.00
• s 1.87

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• Calculate Kurtosis
• 6, 7, 8, 9, 9, 9, 10, 11, 12
• Q3 10.5
• Q1 7.5
• P10 6
• P90 12

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The Standard Normal Distribution

• Z-scores
• A descriptive statistic that represents the
  distance between an observed score and the mean
  relative to the standard deviation

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Standard Normal Distribution

• Z-scores
• Convert and distribution to
• Have a mean 0
• Have standard deviation 1
• However, if the parent distribution is not normal
  the calculated z-scores will not be normally
  distributed.

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Why do we calculate z-scores?

• To compare two different measures
• e.g., Math score to reading score, weight to
  height.
• Area under the curve
• Can be used to calculate what proportion of
  scores are between different scores or to
  calculate what proportion of scores are greater
  than or less than a particular score.

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Class practice

• 6, 7, 8, 9, 9, 9, 10, 11, 12
• Calculate z-scores for 8, 10, 11.
• What percentage of scores are greater than 10?
• What percentage are less than 8?
• What percentage are between 8 and 10?

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Z-scores to raw scores

• If we want to know what the raw score of a score
  at a specific tile is we calculate the raw using
  this formula.

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Transformation scores

• We can transform scores to have a mean and
  standard deviation of our choice.
• Why might we want to do this?

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With our scores

• We want
• Mean 100
• s 15
• Transform
• 8 10.