One-Sample T-Test Calculator

A one-sample t-test calculator tests whether a sample mean differs from a known or claimed value. Enter the sample mean, standard deviation and size — or paste a raw data list — plus the hypothesised mean, and it returns the t statistic, degrees of freedom, p-value, critical t, Cohen’s d and the 95% confidence interval, with a shaded t curve.

How to use the one-sample t-test calculator

Use Summary stats when you already know the sample mean, standard deviation and size. Use Raw data when you have the individual measurements — the calculator works out the mean and sample standard deviation (s) for you.

1. Choose a mode: Summary stats or Raw data.
2. Enter the sample mean, sample SD and sample size n — or paste the values in Raw data mode, separated by commas, spaces or line breaks.
3. Enter the hypothesised mean (μ₀), the value you are testing against, then choose the significance level (α) and the tail: two-tailed, or one-tailed for μ > μ₀ or μ < μ₀.
4. Read the headline t statistic, then check the results grid for the p-value, critical t, degrees of freedom, Cohen’s d, 95% confidence interval and standard error. The calculation ledger shows every step, and the chart shades the rejection region.

Try it now: with x̄ = 105, s = 15, n = 25 and μ₀ = 100, the calculator gives t = 1.667, df = 24 and a two-tailed p-value of 0.1086 — not significant at the 5% level. Paste the raw list 98, 102, 105, 110, 95 with μ₀ = 100 and it gives t = 0.761 with df = 4.

When should you use a one-sample t-test?

Use a one-sample t-test when you have one sample and want to know whether its mean is compatible with a known, claimed or target value — for example, whether packs of coffee really average 250 g, or whether a class mean differs from the national average. It is the right test when:

• The population standard deviation (σ) is unknown and must be estimated from the sample.
• The sample is small to moderate — the t-test was designed for exactly these cases.
• The data are approximately normal, or the sample is large enough that the central limit theorem applies.
• Observations are independent and measured on a continuous scale.

Because σ is unknown, the test uses the sample SD (s) and the t-distribution instead of the normal curve. The same t-distribution powers the t-distribution calculator, where you can look up any area, p-value or critical t for any degrees of freedom.

One-sample t-test vs z-test: which one do you need?

Both tests compare a sample mean with a value, but they assume different things about the population. The t-test is for unknown σ; the z-test is for known σ.

Common t critical values at α = 0.05 (two-tailed) are df = 4 → 2.776, df = 10 → 2.228, df = 24 → 2.064 and df = 30 → 2.042. Notice how the cut-off falls towards 1.960 as the sample grows.

If σ is known, a critical z-value calculator gives the cut-off you need. If you are testing a proportion rather than a mean, use the one-proportion z-test calculator instead. If you need to combine two samples’ spread first, a pooled standard deviation calculator does that step, and if you are starting from a single raw score, first learn how to calculate a z-score.

How do I read the p-value, Cohen’s d and confidence interval together?

Three numbers, one story. Read them as a group rather than in isolation:

• P-value: the probability of getting a sample mean at least as far from μ₀ as yours, if the null hypothesis were true. If p < α, reject H₀; if p > α, do not. Learn more about what a p-value is and what it cannot tell you.
• Cohen’s d: the size of the difference in standard deviation units, d = (x̄ − μ₀) / s. Roughly, 0.2 is small, 0.5 is medium and 0.8 is large. A tiny p-value with a tiny d can mean a real but trivial difference.
• 95% confidence interval: the range of plausible values for the true mean, x̄ ± t* · (s / √n). If the interval excludes μ₀, the two-tailed test at 5% rejects — the two must agree. Its width is set by the margin of error vs confidence interval trade-off, and you can build one directly with a confidence interval calculator for a mean and standard deviation.

Example: in the first worked example below, p = 0.1086 (not significant), d = 0.333 (a small effect) and the 95% CI [98.81, 111.19] contains 100 — all three agree that the evidence against 100 is weak. In the second, p = 0.0073, d = 0.587 (a medium effect) and the CI [102.61, 114.99] excludes 100 — all three point the other way.

Worked examples

Example 1 — not significant (summary stats)

A bakery claims its loaves average 100 g of filling. A sample of 25 loaves has x̄ = 105, s = 15. Testing H₀: μ = 100 at α = 0.05, two-tailed: SE = 15 / √25 = 3.00, so t = (105 − 100) / 3.00 = 1.667 with df = 24. The p-value is 0.1086 and the critical t is 2.064, so |1.667| < 2.064 and we fail to reject H₀. Cohen’s d = 0.333 (small) and the 95% CI is [98.81, 111.19], which contains 100.

Example 2 — significant (summary stats)

With the same s = 15 and n = 25 but a sample mean of x̄ = 108.8, the standard error is still 3.00, so t = (108.8 − 100) / 3.00 = 2.933 with df = 24. The p-value is 0.0073, below 0.05, and 2.933 exceeds the critical t of 2.064, so we reject H₀. Cohen’s d = 0.587 (a medium effect) and the 95% CI is [102.61, 114.99], which excludes 100.

Example 3 — raw data

Five measurements — 98, 102, 105, 110, 95 — tested against μ₀ = 100, two-tailed. The sample mean is 102 and the sample SD is s = 5.874, so SE = 5.874 / √5 = 2.627 and t = (102 − 100) / 2.627 = 0.761 with df = 4. The p-value is 0.4887 and the critical t is 2.776, so we fail to reject H₀. Cohen’s d = 0.341 and the 95% CI is [94.71, 109.29].

Frequently asked questions

Source: the critical t values quoted on this page are from the NIST/SEMATECH e-Handbook of Statistical Methods, Critical Values of the Student’s t Distribution. All examples on this page were checked against that table.

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