Standard deviation - Revision history

Conservative at 03:25, August 21, 2025

2025-08-21T03:25:08Z

2025-08-21T03:24:34Z

2016-07-13T19:40:34Z

top: clean up & uniformity

2009-11-14T17:15:52Z

Reverted edits by RyanMatthews (Talk) to last version by Qwestor

2009-11-14T16:45:11Z

Additions

2009-02-27T23:09:20Z

2009-01-19T15:34:25Z

2009-01-19T15:06:42Z

2009-01-19T14:56:01Z

2009-01-19T14:54:00Z

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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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 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.
+<br />
 [[Category:Probability and Statistics]]
 [[Category:Mathematics]]

← Older revision		Revision as of 03:24, August 21, 2025
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	{{math-h}}		{{math-h}}
−
	'''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		'''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

@@ Line 1: / Line 1: @@
 {{math-h}}
-'''Standard deviation''' is a measure in [[Statistics|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
+'''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
 :<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>
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-[[category:Probability and Statistics]]
+[[Category:Probability and Statistics]]
 [[Category:Mathematics]]

← Older revision		Revision as of 17:15, November 14, 2009
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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.		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.
−
−	It may be important to consider the standard deviation is but ''a'' measure of dispersion, and not the only one. For instance, a [[Cauchy distribution]] has an undefined standard deviation (or mean), yet a glance at its [[Probability density function \| pdf]] suggests that it is not "infinitely" dispersed.

← Older revision		Revision as of 16:45, November 14, 2009
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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.		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.
		+
		+	It may be important to consider the standard deviation is but ''a'' measure of dispersion, and not the only one. For instance, a [[Cauchy distribution]] has an undefined standard deviation (or mean), yet a glance at its [[Probability density function \| pdf]] suggests that it is not "infinitely" dispersed.

← Older revision		Revision as of 23:09, February 27, 2009
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	(where <math>\bar X = {\sum_n X_n \over N}</math> is the [[sample mean]]).		(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 estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.	+	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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 The formula for standard deviation must not be confused with the formula
-:<math>S_{n} = \sqrt {\sum_n(x - \bar x)^2 \over n - 1}</math>
+:<math>S_{n} =  \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}</math>
 which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''.  As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

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 :<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>
-where the [[expectation|expected value]] of ''X'' is E(''X'').
+where the [[expectation (math)|expected value]] of ''X'' is E(''X'').
 Thus the standard deviation is