Applied Math Notes

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Applied Math Notes

• 4-2 Measures of Dispersion
• Objectives Find the range, interquartile range,
  variance and standard deviation of sets of data.

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What are measures of dispersion?

• Measures of dispersion are statistics used to
  describe the spread of a set of data.
• Two measures of dispersion are the range and the
  interquartile range.
• The range of a set of data is the difference
  between the greatest value and the least value.
• Range max min
• The interquartile range (IQR) of a set of data is
  a measure of the spread of the middle 50 of the
  data.
• The interquartile range is the difference between
  the upper and lower quartiles.
• IQR Q3 Q1

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What data display would be helpful in computing
the range and the IQR?

• Since range is max min
• And since IQR is Q3 Q1
• Then the data display that would helpful in
  finding the range and IQR is a
• Box-and-whisker plot
• Without making a box-and-whisker plot, what else
  could one do to compute the range and IQR?
• Run 1-variable statistics with the TI-83/84

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Example 1 Finding range and IQR

• Find the range and IQR of the data sets below
• Set 1 27, 11, 20, 19, 25, 19, 35, 17, 20, 28,
  19, 11, 21, 26, 19, 28, 23
• Set 2 19, 17, 20, 18, 21, 21, 21, 25, 20, 21,
  26, 23, 18, 24
• Solution Enter the data from set 1 into L1 and
  set 2 into L2
• Run 1-variable stats on each list
• Set 1 Range max min 35 11 24
• IQR Q3 Q1 26.5 19 7.5
• Set 2 Range max min 26 17 9
• IQR Q3 Q1 23 19 4

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Two more measures of dispersion

• The range is based on extreme values of a data.
• The IQR is based on the quartiles of a data set.
• Variance and standard deviation are based on the
  mean.

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Standard Deviation

• The standard deviation of a set of data describes
  the typical difference, or deviation, between the
  mean and a data value.
• The standard deviation is the positive square
  root of the variance.

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Some nasty-looking formulas
The variance s2 (read as sigma squared) of a
set of data x1, x2, xn is
The standard deviation s (read as sigma) of a
set of data x1, x2, xn is
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Lucky for you

• The TI-83/84 computes one of these with
  1-variable stats
• Recall
• sx is the standard deviation (of the population)
• You can square it to get the variance

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Example 2 Finding Variance and Standard Deviation

• Compute the variance and standard deviation of
  the following data set
• 23, 37, 33, 39, 29, 31

Standard deviation
Ask me how to recall stats variables
So the standard deviation is 5.26
To find the variance, simply square the standard
deviation 5.262 27.67
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Practice

• See Smart Board examples