Sociology 220

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Sociology 220

• Spring 2007
• Prof. J. Brines
• May 2, 2007
• MGH 389

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Measures of spread in a distribution

• How spread out are the values for a set of
  cases?

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The Range

• Range Upper value-Lower value 1
• Consider N12 test scores
• 8 11 15 19 23 29 30 32 32 38 41 46
• The range for the test scores is
• 46-81 39.

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Features of the Range

• Strengths easy to calculate
• Weaknesses can poorly represent the amount of
  overall variation in the data.

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Example of the problem

• Consider the following two sets of data re.
  number of first cousins a person has. Seven
  people contribute to each set of data (N7).
• Group 1 0 4 4 4 4 5 5 Range6
• Group 2 0 1 2 2 3 4 5 Range6

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• In the preceding example, using the range to
  represent variation suggests that for both
  groups, there is equally wide variation in number
  of cousins. But thats not the case!
• Given this flaw, as a measure of spread we
  usually prefer

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The Variance

• __
• Formula ?2 ? ((Yi Y )2)
• N
• __
• (Yi Y) difference between the value for the
  ith case and the mean.
• __
• (Yi - Y)2 squared difference between case i and
  the mean.
• __
• ? (Yi Y)2 Squared difference between case 1
  and the mean squared difference between case 2
  and the mean squared difference between last
  case and the mean

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• __ __ __
• Example Yi Y Yi Y (Yi Y)2
• 10 35.58 -25.58 654.34
• 21 35.58 -14.58 212.58
• 25 -10.58 111.94
• 27 - 8.58 73.62
• 32 - 3.58 12.82
• 32 - 3.58 12.82
• 35 - .58 .34
• 37 1.42 2.01
• 42 6.42 41.22
• 49 13.42 180.10
• 56 35.58 20.42 416.98
• 61 35.58 25.42 646.18
• ? -- -- 0.00 2364.95

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• N 12
• __
• ?2 ? ((Yi Y)2) 2364.95
• N 12
• 197.08
• ? square root (197.08) 14.04

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• ? or the standard deviation represents the
  average deviation of individual values (Yi) from
  the mean for Y.