Measures of Descriptive Statistics

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Measures of Descriptive Statistics
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Class Objectives

• Objectives Upon completion of this lesson the
  student will be able to
• Describe the different measures of descriptive
  statistics central tendency, variability,
  symmetry and kurtosis.
• Identify the statistic appropriate for each
  measurement scale.
• Present the various descriptive statistics in
  narrative and table formats.

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Measures of Central Tendency

• Purpose is to report a single score or category
  that best describes a set of observations.
• Determined by the scale of measurement.
• Three common measures are
• Mean
• Median
• Mode
• These measures can be used to
• compare one group to another,
• identify some behavior that is unknown, or
• compare a group with a standard.

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Mode

• The category or score that occurs most
  frequently.
• Easy to obtain.
• Not very stable.
• A group of scores may have more than one mode.
• Most likely but may not be the majority.

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Mode Nominal Scale

• Mode

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Mode Ordinal Scale

• C
• B
• A
• B
• C
• A
• A

B D F B C B A

• A
• B
• A
• D
• D
• A
• A

• Mode A

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Mode Interval Scale

• 75
• 85
• 90
• 85
• 70
• 95
• 100

85 65 57 80 75 85 90

• 98
• 85
• 92
• 67
• 66
• 90
• 99

• Mode 85

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Median

• The score below which 50 of the scores fall.
• For raw scores, the median is the middle score
• If there is an even number of scores, the median
  is the midpoint between the two middle scores
• Responds to the number of scores above or below
  it but not how far the scores lie from it.
• Is less sensitive to a few extreme scores.

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Median Ordinal Scale

• C
• B
• A
• B
• C
• A
• A

B D F B C B A

• A
• B
• A
• D
• D
• A
• A

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Median Ordinal Scale

• A A A A A A A A B B B B B B C C C D D D F
• 21 scores
• Look for 11th score

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Median Ordinal Scale

• C
• B
• A
• B
• C
• A
• A

B D F B C B A

• A
• B
• A
• D
• D
• A
• A

• Median B

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Median Interval Scale

• 75
• 85
• 90
• 85
• 70
• 95
• 100

85 65 57 80 75 85 90

• 98
• 85
• 92
• 67
• 66
• 90
• 99

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Median Interval Scale

• 57 65 66 67 70 75 75 80 85 85 85 85 85
  90 90 90 92 95 98 99 100
• 21 scores
• Look for 11th score

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Median Interval Scale

• 75
• 85
• 90
• 85
• 70
• 95
• 100

85 65 57 80 75 85 90

• 98
• 85
• 92
• 67
• 66
• 90
• 99

• Median 85

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Mean

• The arithmetic average of all scores.
• Calculated by summing all scores and dividing by
  the number of scores.
• The mean is responsive to every score, including
  extreme scores.

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Mean Interval Scale

• 75
• 85
• 90
• 85
• 70
• 95
• 100

85 65 57 80 75 85 90

• 98
• 85
• 92
• 67
• 66
• 90
• 99

• Mean 82.5

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Summary Interval Scale

• 75
• 85
• 90
• 85
• 70
• 95
• 100

85 65 57 80 75 85 90

• 98
• 85
• 92
• 67
• 66
• 90
• 99

• Mode 85
• Median 85
• Mean 82.5

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Descriptive Statistics
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Measures of Variability

• Determine some indication of the spread or
  dispersion of the group.
• Measures include
• Range
• Variance
• Standard deviation

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Range

• Exclusive range the difference between the
  largest and smallest score
• For example
• 1, 3, 4, 6,7, 9
• Range 9 - 1 8

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Semi-Interquartile Range

• Q3 - Q1
• 2
• Q3 75th percentile
• Q1 25th percentile

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Variance
Mean 76.7
Sum of Squared Deviation 853.4
Deviation score - mean
Squared Deviation deviation deviation
Variance (sum of squared deviation)/no of scores
Variance 121.92
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Standard Deviation

• Most widely used measure of variability
• Calculated by taking the square root of the
  variance
• Expressed in terms of one standard deviation
  above above the mean
• Value becomes apparent as we understand the
  relationship between standard deviation and the
  normal curve

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Standard Deviation
Mean 76.7
Sum of Squared Deviation 853.4
Variance 121.92
Standard deviation square root of variance
Standard deviation 11.93
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Descriptive Statistics
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Symmetry

• Shape of the distribution
• Measured in terms of the degree to which the
  distribution is asymmetrical.
• A normal curve is symmetrical.
• Any degree of variance from a normal curve is
  skewness, the measure for the shape of
  distribution

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Symmetry

• Range of 3 (positively skewed) to 3 (negatively
  skewed)
• Positively skewed indicated that scores are
  shifted to the left
• Negatively skewed contains scores shifted to the
  right.
• The value of the skewness indicates the degree to
  which the scores have shifted away from symmetry.

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Descriptive Statistics
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Kurtosis

• Measure of how peaked or how flat the curve is
• Concerns the steepness of the curve in the
  vicinity of the mode
• A positive value indicates a curve more peaked
  (leptokurtic) than the normal distribution
• A negative value indicates a curve more flat
  (platykurtic) than the normal distribution

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Descriptive Statistics
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Reporting Descriptive Statistics

• APA Style Tables
• Narrative examples

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Normal Curve
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Positively Skewed
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Negatively Skewed