Two-sample t statistic
Compare two independent population means without assuming known population spreads.
Two-sample t statistic is one of 2 inference formulas in the statistics section of this library, and it is used at ap · university level.
Why two-sample t statistic works
The top is the observed gap between the groups minus the gap the null hypothesis claims. The bottom is how much that gap would wobble from sampling alone, and because the two groups are independent their variances add, which is why the two terms are summed underneath a single square root.
What each symbol means
$\Delta_0$ is the null difference, usually zero.
Two-sample t statistic: when it holds
Samples should be independent and conditions for t inference should be checked; use Welch degrees of freedom unless pooling is justified.
When it stops applying
It assumes the two samples are independent of each other. For paired measurements, such as the same subject tested twice, this statistic ignores the pairing and usually inflates the denominator, hiding a real difference that the paired analysis on the differences would have detected.
Two-sample t statistic: a worked example
Substitute both sample means, standard deviations, and sizes before comparing with a t distribution.
The mistake to avoid
What people do: Subtracting the two variance terms under the root instead of adding them.
Why it goes wrong: Uncertainty from two separate samples accumulates; it never cancels. Subtracting can even produce a negative quantity under the root, which is a sure sign the set-up is wrong.
Do this instead: Add the two squared standard errors, then take one square root of the total.
Two-sample t statistic: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\Delta_0$ is the null difference, usually zero.
- Check the conditions before substituting. Samples should be independent and conditions for t inference should be checked; use Welch degrees of freedom unless pooling is justified.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Inference
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where two-sample t statistic comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Standard Deviation — the lesson behind this formula: measure typical distance from the mean.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about two-sample t statistic
What degrees of freedom should I use?
The Welch formula, which software computes and which usually gives a non-whole number. For two samples of 10 with spreads of 2 and 3 it comes out to about 15.7, and a conservative hand method uses the smaller sample size minus 1.
Do the two groups need to be the same size?
No. Unequal sizes are handled automatically because each group contributes its own variance divided by its own count, so the larger group simply contributes less noise.
When is it safe to pool the two variances?
Only when there is good reason to believe the two populations have the same spread. The Welch version does not need that assumption and performs well even when the spreads do match, so it is the safer default.
How much does non-normality matter?
Very little for large samples, thanks to the central limit theorem, but a lot for small ones, especially with skewed data. Look at both samples before trusting a t procedure on fewer than about 15 observations each.