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Lesson 14 of 22 · 14 min

Two means: independent vs paired samples

Before comparing two means, decide whether the samples are independent (pooled t-test on the difference in means) or related (paired comparisons test on the mean of the differences).

In short

Independent samples from normal populations with unknown but equal variances: pooled t-test with $n_1 + n_2 - 2$ df.
The pooled variance $s_p^2$ is a df-weighted average of the two sample variances.
Dependent (paired) samples, such as two assets over the same dates or the same firms before and after: compute each pair's difference and run a paired comparisons test with $n - 1$ df.
On related data the paired test is more powerful, because differencing strips out variation the samples share.
Usual null: no difference ( $\mu_1 - \mu_2 = 0$ or $\mu_d = 0$ ). One-sided versions work just as for a single mean.

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