Two-Sample T-Test from Means and Standard Deviations
For checking a published result, or when the raw data is not available.
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The formula
t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂) df = (a + b)² ÷ (a²/(n₁ − 1) + b²/(n₂ − 1)), a = s₁²/n₁, b = s₂²/n₂
That is Welch’s method. The pooled method combines the two variances first and has df = n₁ + n₂ − 2.
What to watch out for
Make sure the figures are standard deviations rather than standard errors. A paired test cannot be done from these figures: it needs the standard deviation of the differences.
Frequently Asked Questions
What is a worked example?
Means of 15 and 13, standard deviations of 3 and 4, and groups of 30 and 25 give t = 2.063 on 43.81 degrees of freedom.
Can degrees of freedom be a fraction?
Yes, with Welch’s method. The result stays correct, so there is no need to round them.