Difference between revisions of "Standard deviation"

From Conservapedia
Jump to navigation Jump to search
(Redid the page. I will add examples later.)
 
(15 intermediate revisions by 6 users not shown)
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]]

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.
Retrieved from "https://www.conservapedia.com/index.php?title=Standard_deviation&oldid=2152953"