T Statistic Calculator

What the t-statistic calculator does

The t-statistic measures how far a sample mean sits from a hypothesized value, counted in standard errors. This calculator works it out from the sample mean, the value you are testing against, the sample size, and the sample standard deviation.

It is the single number a one-sample t-test is built on, the figure you compare against the t-distribution to judge significance. Below is how it works and how to read it.

How to use it

1. Enter the sample mean and the population mean you are testing against.
2. Enter the sample size and the sample standard deviation.
3. Press Calculate for the t-statistic, or Reset to clear it.

How the t-statistic is worked out

It is a signal-to-noise ratio. The signal is how far your sample mean is from the value you are testing, and the noise is the standard error, how much a sample mean of that size naturally varies:

t = (sample mean minus population mean) ÷ (s ÷ √n)

The bottom of that fraction, the sample standard deviation divided by the square root of the sample size, is the standard error of the mean. Dividing the difference by it expresses the gap in standard errors, which is what puts the result on a scale the t-distribution can judge.

Reading the t-statistic

The size tells you how strong the evidence is. A t-statistic near zero means the sample mean is close to the tested value, so there is little to report. A large t-statistic, well away from zero, means the sample mean is many standard errors off, which is hard to explain by chance alone.

The sign shows the direction: positive means the sample mean is above the tested value, negative means below. On its own, though, the t-statistic is only half the story. How large counts as convincing depends on the degrees of freedom, which is why the next step is to pair it with them and read off a p-value.

A worked example

Suppose you are testing against a value of 100, and your sample of 36 has a mean of 105 and a standard deviation of 15.

The standard error is 15 ÷ √36 = 15 ÷ 6 = 2.5, so the t-statistic is (105 minus 100) ÷ 2.5 = 2.0. The sample mean sits two standard errors above the tested value, positive because it is higher, and large enough to be worth checking against the t-distribution.

Turning it into a p-value

The t-statistic by itself does not tell you whether a result is significant. For that you need its degrees of freedom, which for a single sample is the sample size minus one, and the t-distribution.

Take the t-statistic and the degrees of freedom to our p-value calculator, pick the t-score option, and it returns the p-value. A p-value below your chosen level, usually 0.05, means the difference is unlikely to be down to chance.

Questions people ask

What is a t-statistic?

A measure of how many standard errors a sample mean is from a hypothesized value. It is the test statistic behind a one-sample t-test.

What counts as a large t-statistic?

The further from zero, the stronger the evidence, but how large is convincing depends on the degrees of freedom. That is why it is read together with them via a p-value.

What does the sign mean?

A positive t-statistic means the sample mean is above the tested value, and a negative one means below. The size, not the sign, measures the strength of the evidence.

What is the standard error here?

The sample standard deviation divided by the square root of the sample size. It is the natural variability of the sample mean, and the t-statistic measures the difference in units of it.

References

A quick note on where the methods here come from. The t-statistic and the t-distribution are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering the t-statistic and t-tests.

1. NIST/SEMATECH e-Handbook of Statistical Methods (t-tests and the t-distribution). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (the Student's t-distribution). https://openstax.org/details/books/introductory-statistics-2e