About T-Test Calculator
This t-test calculator is free, needs no sign-up and runs in your browser, so your data is never uploaded. It covers the three t-tests people need: a one-sample t-test against a hypothesised mean, a paired t-test for before and after measurements, and an independent two-sample t-test in both the Student (equal variances) and Welch (unequal variances) forms.
Enter raw data or just the mean, standard deviation and size of each group. The result shows the t statistic, the degrees of freedom, the p-value, the critical value, the standard error, a confidence interval and Cohen's d with a plain label for its size. A bell curve shades the rejection region and marks your t value, and a step-by-step panel writes every formula with your numbers filled in.
For two samples the page also compares Welch and Student side by side and gives a recommendation. Welch is the safer default because it stays valid when the group variances or sizes differ, and it gives almost the same answer as Student when they do not. Choose a two-tailed, left-tailed or right-tailed alternative and a significance level from 0.10 to 0.001.
How to use T-Test Calculator
- Choose the test: one-sample, paired or two-sample.
- Choose raw data or summary statistics. For raw data, paste each list; for paired data, keep the two lists in the same order and the same length.
- For a one-sample test type the hypothesised mean. For two samples pick Welch or Student.
- Pick the alternative hypothesis (two-tailed, right or left) and the significance level.
- Read the t statistic, p-value, interval and Cohen's d, then open the working to see every formula.
Features
- One-sample, paired and independent two-sample t-tests.
- Welch (unequal variances) and Student (pooled) two-sample tests, compared side by side with a recommendation.
- Raw data or summary statistics (mean, standard deviation, n).
- Cohen's d with a size label, plus the standard error and a confidence interval.
- Left, right or two-tailed tests, with the critical value for your significance level.
- A bell curve with the rejection region shaded and the t statistic marked.
- Step-by-step KaTeX working and a plain-English conclusion.
The formulas
Each test divides a difference of means by its standard error. The one-sample and paired tests use n minus 1 degrees of freedom. The Student test pools the two variances and uses n1 + n2 minus 2. The Welch test keeps the variances apart and estimates the degrees of freedom with the Welch-Satterthwaite formula.
one-sample t = (x̄ − μ0) / (s / √n), df = n − 1
paired t = d̄ / (s_d / √n), df = n − 1
Student t = (x̄1 − x̄2) / √(s_p² (1/n1 + 1/n2)), df = n1 + n2 − 2
Welch t = (x̄1 − x̄2) / √(s1²/n1 + s2²/n2)
Cohen's d d = (x̄1 − x̄2) / s_pWorked examples
Sleep data, drug 1 and drug 2 (10 patients each), Welch gives t = -1.861, df = 17.78, p = 0.079, d = -0.832
Student (1908) data as in R. Treated as two independent groups the difference is not significant at 0.05.
Same sleep data, paired t-test gives t = -4.062, df = 9, p = 0.003
The same patients took both drugs, so pairing removes the person-to-person variation and the difference becomes clear.
NIST car mileage: US 249 cars, mean 20.14458, SD 6.41470; Japanese 79 cars, mean 30.48101, SD 6.10771 (Student) gives t = -12.6206, df = 326, critical values ±1.9673
Matches the worked example in the NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.3.
Cohen's d, rough benchmarks
| Cohen's d (size) | Label | Overlap of the two groups |
|---|---|---|
| below 0.2 | negligible | more than 92% |
| 0.2 to 0.5 | small | about 80% to 92% |
| 0.5 to 0.8 | medium | about 69% to 80% |
| 0.8 and above | large | under 69% |
Good to know
Welch or Student?
The Student test assumes both groups have the same variance. If that is false and the group sizes differ, its p-value can be too small or too large. The Welch test does not make that assumption and loses very little power when the variances are in fact equal, so most statisticians use it by default. Testing for equal variances first and then picking a test is not recommended, because the two-step procedure distorts the error rates.
Assumptions and limits
All t-tests assume independent observations and roughly normal data, or a large enough sample for the mean to be roughly normal. With small samples and heavy skew or outliers, a rank-based test such as Mann-Whitney is safer, and this calculator does not run one. The paired test assumes the pairs are independent of each other and that the differences are roughly normal.
Privacy: how it works
- The test is calculated by code in your own browser tab. Your data is not sent to Ulkaa or to any other server.
- There is no account and nothing is stored. Closing the tab clears what you entered.
Tips
- Use the paired test when the same people or items are measured twice. Using the two-sample test on paired data throws away the pairing and usually loses power.
- Decide one-tailed or two-tailed before you see the data. A two-tailed test is the safe default.
- Report the effect size and the interval with the p-value. A tiny p-value with a negligible Cohen's d is statistically significant but may not matter in practice.
- If your lists have different lengths, that is fine for the two-sample test, but the paired test needs equal lengths.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your data are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your data are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
Which t-test should I use?
Use a one-sample t-test to compare one mean with a known value, a paired t-test for two measurements on the same subjects, and a two-sample t-test for two separate groups. For two separate groups choose Welch unless you have a strong reason to assume equal variances.
What is the difference between Student and Welch?
Student pools the two variances and assumes they are equal. Welch keeps them apart and adjusts the degrees of freedom. When the variances and group sizes are similar they agree closely, and when they differ Welch is the reliable one.
What is Cohen's d?
It is the difference between the means divided by the pooled standard deviation, so it is in standard deviation units. Around 0.2 is small, 0.5 medium and 0.8 large, by convention. For a paired or one-sample test the calculator divides by the standard deviation of the differences or of the sample.
What does the p-value tell me?
It is the probability of a t statistic at least this extreme if the null hypothesis were true. A value below your significance level, usually 0.05, means you reject the null hypothesis. It does not give the size of the effect, which is why the calculator also reports Cohen's d and an interval.
How many observations do I need?
The calculator needs at least 2 values in each group. For reliable results you want more, and the sample size calculator on this site can estimate how many you need to detect a given effect.
Can I test with only summary statistics?
Yes. Switch to summary statistics and type the mean, standard deviation and sample size of each group. For a paired test, type the mean, standard deviation and count of the differences.
Does it round the degrees of freedom for Welch?
No. It uses the fractional degrees of freedom from the Welch-Satterthwaite formula, as R and most statistics packages do. Some textbooks round down to a whole number, which gives a slightly larger p-value.
Popular T-Test Calculator pages
- One-Sample T-Test CalculatorCompare the mean of one sample with a value you expect and see the t statistic, p-value and working.
- Paired T-Test CalculatorPaste two matched lists and get the paired t statistic, p-value and working.
- Two-Sample T-Test CalculatorCompare the means of two independent groups with the classic pooled Student t-test.
- Welch's T-Test CalculatorCompare two means without assuming equal variances, with Welch-Satterthwaite degrees of freedom.
- Cohen's d CalculatorGet Cohen's d for two groups from summary statistics, with the matching t-test and a size label.