Average,%20Median,%20and%20Standard%20Deviation

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Average, Median, and Standard Deviation
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Center and Spread

• The average and the median are measures of the
  center.
• The standard deviation is a measure of the spread.

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Center and Spread
The centers of the two histograms are the same,
but the histogram on the right is more spread out.
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Average, Mean, Median

• The average of a list of numbers equals their
  sum, divided by how many there are.
• The average is sometimes called the (arithmetic)
  mean.

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How Many Are Above Average?
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What Happens to the Mean?
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The Median

• The median of a list is the value with half the
  entries to the left, and half the entries to the
  right.
• Find the median of 9, 1, 2, 2, 0
• Order the list 0, 1, 2, 2, 9
• Find the median of 7, 2, 1, 5
• Order the list 1, 2, 5, 7.
• The average of 2 and 5 is 3.5

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Average vs. Median Income

• For persons age 25 and over in the U.S. would the
  average or the median be higher for income? For
  years of schooling completed?

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The Tail of a Histogram
Average is bigger than the median
Average is about the same as the median
Average is smaller than the median
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Standard Deviation
The standard deviation tries to quantify the
spread.
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The root-mean-square

• The average of the list 0, 5, -8, 7, -3 is 0.2.
• The positive numbers wipe out the negative
  ones.
• We could neglect the signs. The average of 0, 5,
  8, 7, 3 is 4.6
• Instead, we use the root-mean-square.

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The root-mean-square

• Square all the entries. This will get rid of all
  the signs.
• Take the mean (average) of the squares.
• Take the square root of the mean.

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Example

• Find the root-mean-square of 0,
  5, -8, 7, -3.
• Square 0, 25, 64, 49, 9.
• Mean (0 25 64 49 9) 5 29.4
• Root

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Computing the Standard Deviation

• Find the standard deviation of 20, 10, 15, 15.
• Find the average 15.
• Find the deviations from the average 5, -5,
  0, 0.
• Find the root-mean-average

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Which list has the larger SD?

• 50, 40, 60, 30, 70, 25, 75
• 50, 40, 60, 30, 70, 25, 75, 50, 50, 50

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68 95 - 99

• In many cases, roughly 68 percent of the entries
  on a list are within 1 SD of average roughly 95
  within 2 SDs 99 percent are within 3 SDs of the
  average.

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Example

• Average height for boys age 11 is 146 cm and the
  SD was 8 cm.
• How much below average is a boy 170 cm tall? 148
  cm tall?
• How tall is a boy 1.5 SD below average?
• Four boys are 150 cm, 130 cm, 165 cm, 140 cm. Who
  is unusually short, about average, unusually tall?