Unpooled Standard Deviation

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

[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]


References

2017

[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]

  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.