Difference between revisions of "Standard deviation"

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 $X$ ). It is defined as the square root of the variance of these values, where variance is defined as

\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

\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.

{\frac {d}{dx}}\sin x=?\,

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

@@ Line 1: / Line 1: @@
-{{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 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]].
+:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>
-==Standard Deviation==
+where the [[expectation (math)|expected value]] of ''X'' is E(''X'').
-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:
+Thus the standard deviation is
-:<math>\sigma(x) = \sqrt {\sum(x - \bar x) \over n - 1}</math>
+:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>
-This is the square root of the variance, which is:
+The formula for standard deviation must not be confused with the formula
-:<math>Var(x) = {\sum (x - \bar x) \over n - 1}</math>
+:<math>S_{n} =  \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}</math>
-Where:
+(where <math>\bar X =  {\sum_n X_n  \over N}</math> is the [[sample mean]]).
-*<math>\bar x</math> is the arithmetic mean of all values of x
-*<math>\sum</math> is the [[summation]] function
-*<math>n</math> is the number of <math>x</math> values
+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}}
-If the distribution of the values is [[normal distribution|normal]] then it follows the [[Empirical rule]], which states that:
-:{{main|Empirical rule}}
-*68% of the values will fall within 1<math>\sigma</math> of the mean.
-*95% of all values will fall within 2<math>\sigma</math> of the mean.
-*99.7% of all values will fall within 3<math>\sigma</math> of the mean.
-[[category:statistics]]
+[[Category:Probability and Statistics]]
+[[Category:Mathematics]]