Student’s t-test

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When you would like to compare two averages between samples, you could use such formulas of Student’s t-test as following chart according to the number and the variances between dependent or independent samples.

t-test(EN)

Paired t-test as below;

\displaystyle t = \frac{\bar X_D - \mu_0}{SD/\sqrt{n}} \cdot\cdot\cdot(1)

Independent two samples t-test with unequal sample sizes and unequal variances, called as Welch’s t-test, as below;

\displaystyle t = \frac{\bar X_1 - \bar X_2}{SD_p} \cdot\cdot\cdot(2) \vspace{0.2in}\\ SD_p = \sqrt{\frac{SD_1^2}{n_1} + \frac{SD_2^2}{n_2}}

Independent two samples t-test with equal sample sizes and unequal variances as below;

\displaystyle t = \frac{\bar X_1 - \bar X_2}{SD_p \sqrt{\frac{2}{n}}} \cdot\cdot\cdot(3) \vspace{0.2in}\\SD_p = \sqrt{\frac{SD_1^2 + SD_2^2}{2}}

Independent two samples t-test with unequal sample sizes and equal variances as below;

\displaystyle t = \frac{\bar X_1 - \bar X_2}{SD_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \cdot\cdot\cdot(4) \vspace{0.2in}\\ SD_p = \sqrt{\frac{(n_1 -1)SD_1^2 + (n_2 - 1)SD_2^2}{n_1 + n_2 - 2}}

Reference:Student’s t-test

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