Difference between revisions of "Standard deviation"

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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]].  In simple terms, the '''standard deviation''' is the distance from the mean within which the vast majority of data exists.
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'''Standard deviation''' is a measure in [[statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>).  It is defined as the square root of the [[variance:Probability and Statistics|variance]] of these values, where variance is defined as
  
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Examples help illustrate this concept.  Learning the average height of basketball players us that half are above that height and half are below.  Learning the '''standard deviation''' of their heights tells us how much opportunity there is for players who are shorter than the average.
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:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>
  
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Mathematically, the standard deviation of a [[random variable]] ''X'' is:
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where the [[expectation (math)|expected value]] of ''X'' is E(''X'').
  
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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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Thus the standard deviation is
  
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where the [[expected value]] of ''X'' is E(''X'').
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:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>
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[[category:statistics]]
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[[category:probability]]
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The formula for standard deviation must not be confused with the formula
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:<math>S_{n} =  \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}</math>
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(where <math>\bar X =  {\sum_n X_n  \over N}</math> is the [[sample mean]]).
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which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this [[statistical estimator]] itself has a variance which, as the formula indicates, decreases as the sample size increases.
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{{math-h}}
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[[Category:Probability and Statistics]]
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[[Category:Mathematics]]

Latest revision as of 03:25, August 21, 2025

Standard deviation is a measure in statistics of the dispersion of a set of values (represented as <math>X</math>). It is defined as the square root of the variance of these values, where variance is defined as

<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the expected value of X is E(X).

Thus the standard deviation is

<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

<math>S_{n} = \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}</math>

(where <math>\bar X = {\sum_n X_n \over N}</math> is the sample mean).

which is the formula for a point estimate of the true standard deviation from a sample size of n. As such this statistical estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

<math>\frac{d}{dx} \sin x=?\,</math> This article/section deals with mathematical concepts appropriate for late high school or early college.
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