One-sample t statistic: what it means and how to use it
Compare a sample mean with a null-hypothesis mean.
One-sample t statistic is one of 2 hypothesis tests formulas in the statistics section of this library, and it is used at ap · university level.
Why one-sample t statistic works
The top measures how far the sample mean sits from the claimed value, and the bottom measures how far it would typically wander by chance alone. Dividing converts a gap in the original units into a count of standard errors, which is a scale on which unusual has the same meaning in every problem.
What each symbol means
$\mu_0$ is null value and $n-1$ is the degrees of freedom.
One-sample t statistic: when it holds
Use independent data and verify the t-procedure’s shape or sample-size conditions.
When it stops applying
It assumes independent observations and a population shape close enough to symmetric for the sample size at hand. The most common breach is paired data: measuring the same people before and after and then ignoring the pairing wastes the design, because the correct approach is to run this statistic on the differences instead.
One-sample t statistic: a worked example
$\bar x=52,\mu_0=50,s=8,n=16$ gives $t=1$.
The mistake to avoid
What people do: Leaving s alone on the bottom without dividing it by the square root of n.
Why it goes wrong: With a mean of 52 against a claim of 50, a spread of 8 and 16 observations, the correct statistic is 1. Skipping that square root gives 0.25 and hides every effect the sample size was supposed to reveal.
Do this instead: Write the denominator as s divided by the square root of n as a single quantity, and compute that number before dividing.
One-sample t statistic: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\mu_0$ is null value and $n-1$ is the degrees of freedom.
- Check the conditions before substituting. Use independent data and verify the t-procedure’s shape or sample-size conditions.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Hypothesis tests
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where one-sample t statistic comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Standard Deviation lesson — the lesson behind this formula: measure typical distance from the mean.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about one-sample t statistic
Is a t value of 1 large enough to be significant?
No. With 15 degrees of freedom, a two-sided p-value for t equal to 1 is about 0.333, meaning a gap this size or larger happens in a third of samples when the claim is true.
How do I turn t into a p-value?
Look up the tail area beyond your value on a t distribution with n minus 1 degrees of freedom, then double it for a two-sided test. Calculators and software do this with a single command.
What is the practical difference between this and a z statistic?
This one uses an estimated spread, so its distribution has heavier tails and needs degrees of freedom. Once the sample passes roughly 30 the two give nearly identical answers.
Does a very large t prove the claimed value is false?
No. It says the data would be surprising if the claim were true, which is weaker. A large value can also come from a broken assumption, such as dependent observations.