Difference between revisions of "Standard deviation"

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(Redid the page. I will add examples later.)
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The '''standard deviation''' of a set of values is a measure of how widely the values differ from each other.  Specifically, the '''standard deviation''' is the square root of the average of the squares of the differences between the data values and their [[mean]].  If the distribution of the values is [[normal distribution|normal]] then about 68% of the values will fall within one standard deviation of the mean.
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{{math-h}}
  
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Examples help illustrate this concept.  Learning the median height of basketball players tell us that half are above that height and half are below.  Learning the '''standard deviation''' of their heights tells us how varied their heights are.
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'''Standard deviation''' is a measure for how much a set of values ''varies''. It allows for one to find how likely it is for a specific value to be obtained by doing a [[Z-test]].
  
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Mathematically, the standard deviation of a [[random variable]] ''X'' is:
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==Standard Deviation==
  
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:<math>\sigma = \sqrt{\operatorname{E}((X-\operatorname{E}(X))^2)} </math> <math>= \sqrt{\operatorname{E}(X^2) - (\operatorname{E}(X))^2}</math>
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The standard deviation of a set of values is a measure of how widely the values differ from each other.  Specifically, standard deviation follows the equation:
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:<math>\sigma(x) = \sqrt {\sum(x - \bar x) \over n - 1}</math>
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This is the square root of the variance, which is:
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:<math>Var(x) = {\sum (x - \bar x) \over n - 1}</math>
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Where:
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*<math>\bar x</math> is the arithmetic mean of all values of x
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*<math>\sum</math> is the [[summation]] function
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*<math>n</math> is the number of <math>x</math> values
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If the distribution of the values is [[normal distribution|normal]] then it follows the [[Empirical rule]], which states that:
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:{{main|Empirical rule}}
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*68% of the values will fall within 1<math>\sigma</math> of the mean.
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*95% of all values will fall within 2<math>\sigma</math> of the mean.
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*99.7% of all values will fall within 3<math>\sigma</math> of the mean.
  
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where the [[expected value]] of ''X'' is E(''X'').
 
 
[[category:statistics]]
 
[[category:statistics]]

Revision as of 15:09, January 4, 2009

This article/section deals with mathematical concepts appropriate for late high school or early college.

Standard deviation is a measure for how much a set of values varies. It allows for one to find how likely it is for a specific value to be obtained by doing a Z-test.

Standard Deviation

The standard deviation of a set of values is a measure of how widely the values differ from each other. Specifically, standard deviation follows the equation:

This is the square root of the variance, which is:

Where:

  • is the arithmetic mean of all values of x
  • is the summation function
  • is the number of values


If the distribution of the values is normal then it follows the Empirical rule, which states that:

For a more detailed treatment, see Empirical rule.
  • 68% of the values will fall within 1 of the mean.
  • 95% of all values will fall within 2 of the mean.
  • 99.7% of all values will fall within 3 of the mean.
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