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When performing these two tests on the very same populations both times:

  • T-test Assuming Equal Variances
  • T-test Not Assuming Equal Variances

Why do both tests generate the same observed t-statistic?

Supposedly, the formulas for calculating the observed t-statistic differ for these two tests. So I'm curious why I get the same observed t-statistic (t Stat) in the output from Microsoft Excel's Data Analysis Add-In.


  • T-test Assuming Equal Variances:

$$ t=\frac{(\overline{x}_{1}-\overline{x}_{2})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_1^2}{n_{1}}+\frac{s_2^2}{n_{2}}}} $$

  • T-test Not Assuming Equal Variances:

$$ t=\frac{(\overline{x}_{1}-\overline{x}_{2})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_1^2(n_{1}-1)+s_2^2(n_{2}-1)}{n_{1}+n_{2}-2}\cdot\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}}} $$


Population data:

Population A    Population B
-1.235671223    4.15960852
-0.761968905    1.310399095
0.941679605     -1.07118374
-1.378594308    -1.140884186
-0.701344758    7.340783069
0.470609188     -0.337045646
-0.455034508    8.506507035
0.98726877      3.09472358
-0.265557251    8.131838266
0.645189392     3.176737606

Observed T-statistics:

t Stat (Assuming Equal Variance): -2.89968750779
t Stat (Not Assuming Equal Variance): -2.89968750779
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Because the sample sizes are the same:

$$ \sqrt{\frac{s^2_1(n-1) + s^2_2(n-1)}{n+n-2}}*\sqrt{\frac{1}{n}+\frac{1}{n}} =$$

$$ \sqrt{\frac{(s^2_1 + s^2_2)(n-1)}{2(n-1)}}*\sqrt{\frac{2}{n}} = $$

$$ \sqrt{\frac{(s^2_1 + s^2_2)}{1}}*\sqrt{\frac{1}{n}} = $$

$$ \sqrt{\frac{(s^2_1 + s^2_2)}{n}} = \sqrt{\frac{s^2_1}{n} + \frac{s^2_2}{n}}$$

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