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

where [math]\displaystyle{ n_1, n_2 }[/math] are the respective sample sizes. This assumes that degrees of freedom is given by:

[math]\displaystyle{ 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):
  • Pooled Variance.
  • Pooled Standard Deviation.
• See: Standard Deviation, Sample Standard Deviation, Point Estimate, Sample Variance.

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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:

[math]\displaystyle{ 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]\displaystyle{ \overline{X}_1 }[/math], [math]\displaystyle{ s_1^2 }[/math] and [math]\displaystyle{ 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]\displaystyle{ \nu }[/math]  associated with this variance estimate is approximated using the Welch–Satterthwaite equation:

[math]\displaystyle{ \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]\displaystyle{ \nu_1 = N_1-1 }[/math], the degrees of freedom associated with the first variance estimate. [math]\displaystyle{ \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.

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