Measures of Dispersion

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Measures of Dispersion

• Variance and Standard Deviation

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Basic Assumptions about Distributions

• We should be able to plot the number of times a
  specific value occurs on a graph using a line
  chart or histogram (interval/ratio data)
• Some distributions will be normal or bell-shaped.
• Some distributions will be bi-modal or will have
  data points distributed irregularly.
• Some distributions will be skewed to the right or
  skewed to the left.
• Theoretically, samples taken from one population,
  should over time, approximate a normal
  distribution.
• We should have a normal distribution if we are to
  use inferential statistics.

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Other reasons to use Measures of Dispersion

• To see if variables taken from two or more
  samples are similar to one another.
• To see if a variable taken from a sample is
  similar to the same variable taken from a
  population in other words is our sample
  representative of people in the population at
  least on that one variable.

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Variation in Two Samples
Sample 1 Sample 2
1 2
2 3
3 3
4 5
4 7
5 9
6 9
7 10
Mo 4 Mo, 3, 9
Mdn 4 Mdn 6
Mean 4 Mean 6
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Sample 2
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Normal Distributions are Bell-shaped and have the
same number of measures on either side of the
mean. Note According to Montcalm Royse only
unimodal distributions can be normal
distributions.
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Normal Distributions

• 50 of all scores are on either side of the mean.
  
• The distribution is symmetrical same number of
  scores fall above and below the mean.
• The mean is the midpoint of the distribution.
• Mean median mode
• The entire area under the bell-shaped curve
  100.

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A standard deviation is

• The degree to which each of the scores in a
  distribution vary from the mean. (x mean)
• Calculated by squaring the deviation of each
  score from the mean.
• Based on first calculating a statistic called the
  variance.

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Formulas are

• Variance Sum of each deviation squared divided
  by (n -1) where n is the number of values in the
  distribution.
• Standard Deviation the square root of the sum
  of squares divided by (n 1).

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Using Sample 1 as an example
1 (1-4) -3 9  Mean 4  
2 (2-4) -2 4    
3 (3-4) -1 1    
4 (4-4) 0 0 Variance S.D.
4 (4-4) 0 0 28/(8-1) Sq Root 4
5 (5-4) 1 1 4 2
6 (6 - 4) 2 4    
7 (7 4) 3 9    
Total 0 28    
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Another variance/SD example
1 -5.00 25.00  Mean 6  
2 -4.00 16.00    
4 -2.00 4.00    
8 2.00 4.00    
10 4.00 16.00 Variance 90/(6-1) SD sq root of 18
11 5.00 25.00 18.00 4.24
Total 0.00 90.00    
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Other Important Terms in This Chapter

• Mean squares the average of squared deviations
  from the mean in a set of numbers. (Same as
  variance)
• Interquartile range points in a set of numbers
  that occur between 75 of the scores and 25 of
  the scores that is, where the middle 50 of all
  scores lie (use cumulative percentages)
• Box plot gives graphic information about
  minimum, maximum, and quartile scores in a
  distribution.

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Box Plot
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Interquartile Range
Test Scores Frequency Percent Cumulative Percent
100 3 25 100
90 3 25 75
80 3 25 50
70 1 8.3 25
60 2 16.7 16.7
Total 12 100.0
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This information is important to our discussion
of normal distributions
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Central Limit Theorem (we will discuss this in
two weeks) specifies that

• 50 of all scores in a normal distribution are on
  either side of the mean.
• 68.25 of all scores are one standard deviation
  from the mean.
• 95.44 of all scores are two standard deviations
  from the mean.
• 99.74 of all scores in a normal distribution are
  within 3 standard deviations of the mean.

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Therefore, we will be able to

• Predict what scores are contained within one,
  two, or three standard deviations from the mean
  in a normal distribution.
• Compare the distribution of scores in samples.
• Compare the distribution of scores from
  populations to samples.

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To calculate measures of central tendency and
dispersion in SPSS

• Select descriptive statistics
• Select descriptives
• Select your variables
• Select options (mean, sd, etc.)

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SPSS output