Standard Deviation

1
Standard Deviation

• Although the 5 number summary is very useful for
  describing a data set, it is not the most widely
  used. The most common measures are the mean for
  the center and the standard deviation to measure
  spread. Standard deviation measures how far the
  observations are from their mean.

2
Variance

• The variance of a set of observations is the
  average of the squares of the deviations of each
  observation from the mean.
• The standard deviation (s) is simply the square
  root of the variance. Your TI-83 calls this Sx.

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Variance
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Calculating Variance

• Seven men took part in a study of metabolic
  rates. Here are the calories burned in 24 hours
  by the men
• 1792 1666 1362 1614
• 1460 1867 1439
• Find s.
• s189.24

5
Why Square It?

• The sum of the deviations from the mean will
  always be 0. This is why we square and square
  root when finding s.

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Why n-1?

• Now the question comes up as to why we divide by
  n-1 instead of n. Because the sum of the
  deviations is always 0, the last deviation can be
  found once we know the first n-1 deviations.
  Since only n-1 of the squared deviations can vary
  freely, we average by dividing the total by n-1.
  The number n-1 is called the degrees of freedom
  of the variance or standard deviation.

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Comments

• 1. s measures spread about the mean - so, use s
  only when using xbar.
• 2. s0 if there is no spread (ie, all
  observations are the same). Else, sgt0. A big s
  value implies the data are spread out.
• 3. s is nonresistant.

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Finally

• The five number summary is usually better than
  the mean and standard deviation for describing a
  skewed distribution.
• Use x-bar and s for reasonably symmetric
  distributions.
• Remember, always graph the data!