AP Stats T-Test Calculator

Run a one-sample t-test in seconds. Enter your sample mean, the population mean from the null hypothesis, the sample standard deviation, and the sample size. This calculator gives you the t statistic, the degrees of freedom, and guidance on how to interpret the result.

What Is a T-Test?

A t-test checks whether a sample mean is far enough from a claimed value to be surprising. It is used instead of a z-test when the population standard deviation is unknown and you must estimate it from the sample. The test statistic follows a t-distribution, whose shape depends on the degrees of freedom.

How to Use This Calculator

1) Type the sample mean from your data. 2) Type the population mean you are testing against, which is the null hypothesis value. 3) Type the sample standard deviation, s, not the population standard deviation. 4) Type the sample size, n. 5) Press Calculate T-Test and read the t statistic and degrees of freedom, then use a t-table or p-value calculator to finish the decision.

How to Calculate a One-Sample T-Test

The formula is: t = (x-bar - mu) / (s / square root of n), with degrees of freedom = n - 1. The denominator, s / square root of n, is the standard error of the mean. A larger absolute t value means the sample mean is farther from the null value, relative to sampling variation.

Worked example: x-bar = 105, mu = 100, s = 15, n = 25. Standard error = 15 / 5 = 3. t = (105 - 100) / 3 = 1.67. Degrees of freedom = 25 - 1 = 24. A t of 1.67 with df = 24 gives a two-tailed p-value of about 0.108, so you would fail to reject the null at alpha = 0.05.

Tips and Common Mistakes

• Use s, the sample standard deviation, for a t-test. If you know the population standard deviation, use a z-test instead.
• Do not use n in the denominator directly. Divide s by the square root of n to get the standard error.
• Degrees of freedom for a one-sample t-test is n - 1, not n.
• Check conditions before you conclude: random sample, and either a normal population or a large enough sample, plus independence (the 10% condition).
• A t statistic alone does not finish the test. You still need a p-value or critical value to make a decision.

Frequently Asked Questions

When do I use a t-test instead of a z-test?

Use a t-test when the population standard deviation is unknown and you estimate it with the sample standard deviation, s, especially for smaller samples.

What are the degrees of freedom for a one-sample t-test?

Degrees of freedom = n - 1, where n is the sample size.

What does a t statistic of 0 mean?

It means the sample mean exactly equals the null value, so there is no evidence against the null hypothesis from this sample.

Why is the t-distribution wider than the normal distribution?

Because estimating the standard deviation from the sample adds uncertainty, especially when n is small. As n grows, the t-distribution approaches the normal distribution.

Can I use this calculator for a two-sample t-test?

No, this page is for a one-sample t-test only. A two-sample t-test needs two sample means, two standard deviations, and two sample sizes, and it uses a different standard error and degrees of freedom.

What if my sample size is very small?

Then the normality condition matters more. Check a graph of the data for strong skew or outliers, and be cautious about your conclusion.

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