The one-sample t-statistic, used for hypothesis testing when the population standard deviation is unknown and estimated from the sample itself.
How it works
The difference between the sample mean and the hypothesized population mean is divided by the standard error (sample standard deviation divided by the square root of the sample size).
What this does not include
This does not include converting the t-statistic to a p-value — that additionally requires the t-distribution’s degrees of freedom, which varies the critical-value table by sample size, and is out of scope here to avoid presenting an approximated figure without the full distribution table backing it.
How to use this calculator
- Enter the sample mean, hypothesized population mean, sample standard deviation, and sample size.
A worked example
Sample mean 105, population mean 100, sample std dev 15, sample size 25: t = (105−100)/(15/√25) = 1.6667, with 24 degrees of freedom.
Sample mean 98, population mean 100, std dev 10, sample size 16: t = −0.8 — a negative t-statistic, since the sample mean fell below the population mean.
What the variables mean
| Variable | Meaning |
|---|---|
| Sample mean | Average of the observed sample |
| Population mean | The hypothesized or known population average |
| Sample std dev | Standard deviation of the sample |
| Sample size | Number of observations |
Edge cases worth knowing
A negative t-statistic isn’t an error — it just means the sample mean fell below the population mean rather than above it, the mirror image of a positive result.
A sample standard deviation of zero makes the t-statistic undefined — every observation being identical leaves no variability to divide by, so the calculator declines to show a result.
Frequently asked questions
How is a t-test different from a z-test?
A t-test is used when the population standard deviation is unknown and must be estimated from the sample; a z-test assumes the population standard deviation is already known.
What does “degrees of freedom” mean here?
It’s the sample size minus one, reflecting that one piece of information is “used up” estimating the sample mean before the standard deviation can be calculated.
What’s a practical use for a t-statistic?
Testing whether a sample’s mean is significantly different from a claimed or expected population value, common in scientific and quality-control research.
