T Statistic Calculator
Compute a one-sample t statistic from summary numbers: enter your sample mean, the hypothesized population mean, the sample standard deviation, and the sample size. You get t, the standard error, degrees of freedom, and an honest read against two-tailed critical values at α = 0.05 and 0.01.
Example: with Sample mean (x̄) 52 · Hypothesized mean (μ₀) 50 · Sample standard deviation (s) 6 · Sample size (n) 25 → t statistic: 1.667.
- Standard error (s / √n)1.2000
- Degrees of freedom (n − 1)24
- Against critical valuesNot significant at α = 0.05, two-tailed (|t| = 1.667 < critical 2.064)
Computed by the calculator below using its default values. Change any input to see your own numbers.
t = (x̄ − μ₀) / (s / √n). Compared against the Student t distribution with n − 1 degrees of freedom, two-tailed. Between table rows this tool uses the next lower df — the conservative choice.
What the t statistic measures
The t statistic counts how many standard errors your sample mean sits from the hypothesized mean. The standard error, s / √n, is what the sample mean typically wobbles by from sample to sample — so t = 1.67 means your mean landed 1.67 typical wobbles above μ₀. Small |t| is consistent with the null hypothesis; large |t| says the gap is bigger than chance sampling easily explains.
It differs from a z score in one honest way: you estimated the standard deviation from the same sample, and that estimate is itself noisy when n is small. The Student t distribution has fatter tails to account for that extra uncertainty, which is why the critical value at df = 4 is 2.776 but shrinks toward the normal 1.960 as the sample grows.
How it’s calculated
t = (x̄ − μ₀) / SE with SE = s / √n and df = n − 1. Default example: (52 − 50) / (6 / √25) = 2 / 1.2 = 1.667 with df = 24. The verdict compares |t| to two-tailed Student t critical values (df 1-30 exact, then 40, 60, 120, and the normal limit 1.960 / 2.576); between rows the next lower df is used, which is conservative.
Assumes a simple random sample from a roughly normal population (or n large enough for the CLT); the verdict is a critical-value comparison, not an exact p-value.
Two-tailed t critical values
| df | α = 0.05 | α = 0.01 |
|---|---|---|
| 5 | 2.571 | 4.032 |
| 10 | 2.228 | 3.169 |
| 15 | 2.131 | 2.947 |
| 20 | 2.086 | 2.845 |
| 25 | 2.060 | 2.787 |
| 30 | 2.042 | 2.750 |
| 60 | 2.000 | 2.660 |
| ∞ (normal z) | 1.960 | 2.576 |
Source: standard Student t distribution table, two-tailed.
Common mistakes
- Dividing by s instead of s / √n — the test asks how far the mean is from μ₀, and means are √n times steadier than single observations.
- Using n instead of n − 1 for degrees of freedom when looking up the critical value.
- Reading a big |t| as a big effect: with huge n, a trivial difference can clear the critical value. Check the raw gap x̄ − μ₀ too.
- Comparing a two-sided question against a one-tailed critical value, which doubles your real false-positive rate.
Frequently asked questions
What is the t statistic formula?
t = (x̄ − μ₀) / (s / √n): the sample mean minus the hypothesized mean, divided by the standard error. Degrees of freedom are n − 1.
What is a significant t statistic?
Compare |t| to the two-tailed critical value for your df. At df = 24, that is 2.064 for α = 0.05 and 2.797 for α = 0.01 — |t| at or above the cutoff rejects the null at that level.
What is the difference between a t statistic and a z score?
Both count standard errors, but a z score assumes the population standard deviation is known, while t uses the sample estimate s. That extra uncertainty gives t fatter tails at small n; by df ≈ 120 the two are nearly identical.
Can the t statistic be negative?
Yes — the sign just says which side of μ₀ your sample mean fell on. Significance for a two-tailed test depends on |t|, so t = −4.0 is exactly as strong as t = +4.0.
Why does this tool not show an exact p-value?
An exact p-value needs the t distribution CDF. Comparing against published critical values at α = 0.05 and 0.01 gives the same accept/reject decision without pretending to more precision; use a p-value calculator if you need the number itself.