日: 2012年12月28日

Student’s t-test

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

Student t検定

　Student t検定として知られる平均値の検定は下図のフローチャートに従ってどの検定を行うか決めます．

t-test(JP)

　対応のあるt検定は以下のように求まります．

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

　サンプルサイズが異なり分散も異なる独立した2群の平均値を比較する場合の t 統計値は次式で求まります．Welch検定とも言います．

\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}}

　同数かつ等分散の独立した2群の平均値を比べる場合の t 統計値は次式で求まります．

\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}}

　サンプルサイズが異なり等分散の独立した2群の平均値を比較する場合の t 統計値は次式で求まります．

\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}}

参照：Student’s t-test