en>Mogism: Cleanup/Typo fixing, typo(s) fixed: a 19th century → a 19th-century using AWB

2013-12-31T19:04:42Z

Cleanup/Typo fixing, typo(s) fixed: a 19th century → a 19th-century using AWB

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Pooled variance is a method for [[estimation theory|estimating]] [[variance]] given several different [[sample (statistics)|samples]] taken in different circumstances where the [[mean]] may vary between samples but the true variance (equivalently, [[accuracy and precision|precision]]) is assumed to remain the same. It is calculated by

:<math>s_p^2=\frac{\sum_{i=1}^k (n_i - 1)s_i^2}{\sum_{i=1}^k(n_i - 1)}</math>

or with simpler notation,

:<math>s_p^2=\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2+\cdots+(n_k - 1)s_k^2}{n_1+n_2+\cdots+n_k - k}</math>

where ''s''p2 is the pooled variance, ''n''''i'' is the [[sample size]] of the ''i'''th sample, ''s''''i''2 is the variance of the ''i''th sample, and ''k'' is the number of samples being combined. ''n'' − 1 is used instead of ''n'' for the same [[Standard_deviation#With_sample_standard_deviation|reason it may be used in estimating variances from samples]] (i.e. [[Bessel's correction]]).

The square-root of a pooled variance estimator is known as a '''pooled standard deviation'''.

== Motivation ==
In [[statistics]], many times, data are collected for a [[statistical independence|dependent variable]], y, over a range of values for the [[statistical independence|independent variable]], x. For example, the observation of fuel consumption might be studied as a function of engine speed while the engine load is held constant. If, in order to achieve a small [[variance]] in y, numerous repeated tests are required at each value of x, the expense of testing may become prohibitive. Reasonable estimates of variance can be determined by using the principle of '''pooled variance''' after repeating each [[Statistical hypothesis testing|test]] at a particular x only a few times.

== Unbiased least square estimate vs. biased maximum likelihood estimate ==

Both

:<math>s_p^2=\frac{\sum_{i=1}^k (n_i - 1)s_i^2}{\sum_{i=1}^k (n_i - 1)}</math>

and

:<math>s_p^2=\frac{\sum_{i=1}^k (n_i - 1)s_i^2}{\sum_{i=1}^k n_i }</math>

are used in different contexts.{{Citation needed|date=November 2010}} The former can give an unbiased <math>s_p^2</math> to estimate <math>\sigma^2</math> when the two groups share an equal population variance. The latter one can give a more efficient <math>s_p^2</math> to estimate <math>\sigma^2</math> biasedly. Note that the quantities <math>s_i^2</math> in the right hand sides of both equations are the unbiased estimates.

==Example==

Consider the following set of data for ''y'' obtained at various levels of the independent variable ''x''.

{| border="1" cellspacing="0" cellpadding="5" align="center"
! ''x''
! ''y''
|-
| 1
| 31, 30, 29
|-
| 2
| 42, 41, 40, 39
|-
| 3
| 31, 28
|-
| 4
| 23, 22, 21, 19, 18
|-
| 5
| 21, 20, 19, 18,17
|}

The number of trials, mean, variance and standard deviation are presented in the next table.
{| border="1" cellspacing="0" cellpadding="5" align="center"
! ''x''
! ''n''
! ''y''mean
! ''S''''y''2
! ''S''
|-
| 1
| 3
| 30.0
| 1.0
| 1.0
|-
| 2
| 4
| 40.5
| 1.67
| 1.29
|-
| 3
| 2
| 29.5
| 4.5
| 2.12
|-
| 4
| 5
| 20.6
| 4.3
| 2.07
|-
| 5
| 5
| 19.0
| 2.5
| 1.58
|}

These statistics represent the variance and [[standard deviation]] for each subset of data at the various levels of ''x''. If we can assume that the same phenomena are generating [[random error]] at every level of ''x'', the above data can be “pooled” to express a single estimate of variance and standard deviation. In a sense, this suggests finding a [[mean]] variance or standard deviation among the five results above. This mean variance is calculated by weighting the individual values with the size of the subset for each level of ''x''. Thus, the pooled variance is defined by

: <math>S_P^2 = \frac{(n_1-1)S_1^2+(n_2-1)S_2^2 + \cdots + (n_k - 1)S_k^2}{(n_1 - 1) + (n_2 - 1) + \cdots +(n_k - 1)}</math>

where ''n''1, ''n''2, . . . ''n''''k'' are the sizes of the data subsets at each level of the variable ''x'', and ''S''12, ''S''22, . . ., ''S''''k''2 are their respective variances.

The pooled variance of the data shown above is therefore:

: <math>S_P^2 = 2.765 \, </math>

==See also==
* Used for calculating [[Effect size#Cohen.27s_d|Cohen's ''d'' (effect size)]]

==References==
*{{cite journal |author=Killeen PR |title=An alternative to null-hypothesis significance tests |journal=Psychol Sci |volume=16 |issue=5 |pages=345–53 |date=May 2005 |pmid=15869691 |pmc=1473027 |doi=10.1111/j.0956-7976.2005.01538.x }}
==External links==
* [http://goldbook.iupac.org/P04758.html IUPAC Gold Book - pooled standard deviation]
* [http://www.isixsigma.com/dictionary/Pooled_Standard_Deviation-295.htm]
* [http://web.psych.utoronto.ca/~psy379/Stats%20PPT.pdf – also referring to Cohen's ''d'' (on page 6)]

{{DEFAULTSORT:Pooled Variance}}
[[Category:Statistical terminology]]
[[Category:Analysis of variance]]

Pert - Revision history

en>Mogism: Cleanup/Typo fixing, typo(s) fixed: a 19th century → a 19th-century using AWB