How to Do a T-Test by Hand: Two-Sample Worked Example

Written by Dackohn; last reviewed November 2026.

To do a two-sample t-test by hand: find the standard error of the difference, divide the difference in means by it to get t, then compare t with the t-distribution. For means of 24.1 and 20.5, SDs of 3.2 and 4.1 and 20 in each group, t = 3.10 and p = .004.

The data

A tutoring group (M = 24.1, SD = 3.2, n = 20) and a control group (M = 20.5, SD = 4.1, n = 20). Did tutoring make a difference?

Step 1: The standard error of the difference

Divide each variance by its sample size and add: 3.2² / 20 = 0.512 and 4.1² / 20 = 0.8405. Then SE = √(0.512 + 0.8405) = 1.163.

Step 2: The t statistic

t = (24.1 − 20.5) / 1.163 = 3.6 / 1.163 = 3.10. The difference is about three standard errors from zero.

Step 3: Degrees of freedom

Welch’s test, the safer default, uses the Welch–Satterthwaite formula:

df = (0.512 + 0.8405)² / (0.512² / 19 + 0.8405² / 19) = 35.88

Student’s test simply uses n₁ + n₂ − 2 = 38, but it assumes equal variances. See Welch vs Student.

Step 4: The p-value

With a t-table, round df down to 35 or use the nearest row below. The two-sided critical value at α = .05 is about 2.03, and 3.10 exceeds it, so the result is significant. Software gives the exact value: p = .004. In Excel: =T.DIST.2T(3.0955,35.88).

Step 5: Confidence interval

Difference ± t_crit × SE = 3.6 ± 2.028 × 1.163, which is 1.24 to 5.96. It excludes 0, matching the test.

Step 6: Effect size

The pooled SD is √((19 × 3.2² + 19 × 4.1²) / 38) = 3.68, so Cohen’s d = 3.6 / 3.68 = 0.98, a large effect.

Writing it up

“A Welch’s t-test showed significantly higher test scores for the tutoring group (M = 24.10, SD = 3.20) than for the control group (M = 20.50, SD = 4.10), t(35.88) = 3.10, p = .004, d = 0.98.”

Common mistakes

Using the SD instead of the SE in the denominator.
Treating paired data as independent. See paired vs independent samples.
Forgetting the square root when combining the two variances.
Rounding too early, which can shift the p-value.

Try it yourself

T-Test From Means and SDs →Check this example, or run your own from summary statistics. T-Test Calculator →Run the test from raw data.

Frequently asked questions

What is the formula for a two-sample t-test?

t = (mean 1 - mean 2) / square root of (s1 squared / n1 + s2 squared / n2). This is the Welch version.

Do I need raw data for a t-test?

No. The means, SDs and sample sizes are enough for an independent-samples test.

Why does the pooled test give the same t here?

With equal group sizes, the pooled and Welch standard errors are identical. Only the degrees of freedom differ.

Sources & further reading: NIST/SEMATECH e-Handbook of Statistical Methods · OpenStax Introductory Statistics. How we compute & validate: methodology. Written & reviewed by Dackohn; last reviewed August 2026.