# Unpooled Standard Deviation

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

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

## References

### 2017

$t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }}\,$
where $\overline{X}_1$, $s_1^2$ and $N_1$ 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 $\nu$  associated with this variance estimate is approximated using the Welch–Satterthwaite equation:
$\nu \quad \approx \quad {{\left( \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \; \right)^2 } \over { \quad {s_1^4 \over N_1^2 \nu_1} \; + \; {s_2^4 \over N_2^2 \nu_2 } \quad }}$
Here $\nu_1 = N_1-1$, the degrees of freedom associated with the first variance estimate. $\nu_2 = N_2-1$, 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]

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.