Difference between revisions of "Standard deviation"
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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 | |
| − | |||
| − | '''Standard deviation''' is a measure in [[ | ||
:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math> | :<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 | Thus the standard deviation is | ||
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The formula for standard deviation must not be confused with the formula | The formula for standard deviation must not be confused with the formula | ||
| − | :<math>S_{n} = \sqrt {\ | + | :<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. | ||
| + | <br /> | ||
| + | <br /> | ||
| + | {{math-h}} | ||
| − | [[ | + | [[Category:Probability and Statistics]] |
| + | [[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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} ). It is defined as the square root of the variance of these values, where variance is defined as
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2}
where the expected value of X is E(X).
Thus the standard deviation is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}}
The formula for standard deviation must not be confused with the formula
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{n} = \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}}
(where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar X = {\sum_n X_n \over N}} 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.
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \sin x=?\,} | This article/section deals with mathematical concepts appropriate for late high school or early college. |