Welch's t-Statistic

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A Welch's t-Statistic is a test statistic used in a Welch's t-test.

[math]\displaystyle{ t= \frac{(\overline{x}_1-\overline{x}_2)}{s_w} }[/math]
where [math]\displaystyle{ \overline{x}_1,\overline{x}_2 }[/math] are the individual sample mean values and [math]\displaystyle{ D=\mu_1-\mu_2 = 0 }[/math].


References

2017

[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.[1]

2017b

[math]\displaystyle{ t= \frac{\overline{x}_1-\overline{x}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}} }[/math]
where
[math]\displaystyle{ \overline{x}_1 }[/math]= Mean of first sample
[math]\displaystyle{ \overline{x}_1 }[/math] = Mean of second sample
[math]\displaystyle{ n_1 }[/math] = Sample size (i.e., number of observations) of first sample
[math]\displaystyle{ n_1 }[/math] = Sample size (i.e., number of observations) of second sample
[math]\displaystyle{ s_1 }[/math] = Standard deviation of first sample
[math]\displaystyle{ s_1 }[/math] = Standard deviation of second sample
  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.