# Unpooled Standard Deviation

An Unpooled Standard Deviation is a linear combination between standard deviations of samples with unequal population variances.

**AKA:**Separate Standard Deviation, Non-Pooled Standard Deviations.**Context:**- It used in the calculation of Welch's t-statistic.
- It can be estimated, for two independent samples with the individual sample standard deviations, as

- [math]s_w=\sqrt{\frac{s_1}{n_1}+\frac{s_2}{n_2}}[/math]

- where [math]n_1, n_2[/math] are the respective sample sizes. This assumes that degrees of freedom is given by:
- [math]DF=\frac{(s_1/n_1+s_2/n_2)^2}{\frac{1}{n_1-1}(s_1/n_1)^2+\frac{1}{n_2-1}(s_1/n_2)^2}[/math]

**Counter-Example(s):****See:**Standard Deviation, Sample Standard Deviation, Point Estimate, Sample Variance.

## References

### 2017

- (Wikipedia, 2017) ⇒ http://en.wikipedia.org/wiki/Welch's_t-test#Calculations
- Welch's
*t*-test defines the statistic*t*by the following formula:

- Welch's

- [math] t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }}\,[/math]

- where [math]\overline{X}_1[/math], [math]s_1^2[/math] and [math]N_1[/math] are the 1st sample mean, sample variance and sample size, respectively. Unlike in Student's
*t*-test, the denominator is*not*based on a pooled variance estimate. - The degrees of freedom [math]\nu[/math] associated with this variance estimate is approximated using the Welch–Satterthwaite equation:
- [math] \nu \quad \approx \quad {{\left( \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \; \right)^2 } \over { \quad {s_1^4 \over N_1^2 \nu_1} \; + \; {s_2^4 \over N_2^2 \nu_2 } \quad }} [/math]

- Here [math]\nu_1 = N_1-1[/math], the degrees of freedom associated with the first variance estimate. [math]\nu_2 = N_2-1[/math], the degrees of freedom associated with the 2nd variance estimate.
- Welch's
*t*-test can also be calculated for ranked data and might then be named Welch's*U*-test.^{[1]}

- ↑ Fagerland, M. W.; Sandvik, L. (2009). "Performance of five two-sample location tests for skewed distributions with unequal variances".
*Contemporary Clinical Trials***30**: 490–496. doi:10.1016/j.cct.2009.06.007.