Case 1
- Sample mean (x̄)
- 52
- Hypothesised population mean (μ)
- 50
- Sample standard deviation (s)
- 5
- Sample size (n)
- 25
2.000000
Open with these valuesT-Statistic Calculator
2.000000
Result: 2.000000The t-statistic counts how many standard errors your sample mean lies from a hypothesised mean. The divisor is the standard error s ÷ √n, not the standard deviation itself — that square root is what makes a large sample more convincing than a small one. Judge it against a t-table with n − 1 degrees of freedom.
Worked examples
Case 2
- Sample mean (x̄)
- 105
- Hypothesised population mean (μ)
- 100
- Sample standard deviation (s)
- 15
- Sample size (n)
- 30
1.825742
Open with these valuesCase 3
- Sample mean (x̄)
- 2.7
- Hypothesised population mean (μ)
- 3
- Sample standard deviation (s)
- 0.5
- Sample size (n)
- 16
-2.400000
Open with these valuesHow it's calculated
t = (x̄ − μ) ÷ (s ÷ √n)
- StepEnter the mean of your sample and the mean you are testing against.
- StepAdd the sample standard deviation and how many observations it covers.
- ResultRead the t-statistic and compare it with a t-table at n − 1 degrees of freedom.
What this number means
Divide by the standard error, not by s
The divisor is s ÷ √n. A mean of 105 against 100 with s = 15 and n = 30 gives 5 ÷ 2.738613 = 1.825742; dividing by 15 itself would return 0.333333.
The degrees of freedom are n − 1
A one-sample test scores t against n − 1 degrees of freedom, so n = 25 gives 24. The shape of the t-distribution changes with that number, and the critical value changes with it.
Compare the magnitude against a t-table
For a two-tailed test at 24 degrees of freedom the critical value is about 2.06, so a t of 2 falls just short. A larger magnitude is stronger evidence against the null hypothesis.
t rather than z when s is only an estimate
A z-score uses a known population standard deviation, while t uses the sample one in its place. That extra uncertainty gives the t-distribution heavier tails and ties it to the sample size through the degrees of freedom.
Commonly misread
t = (x̄ − μ) ÷ s.
The divisor is the standard error s ÷ √n, not the standard deviation. The two coincide only at n = 1, and this test needs at least two observations.
A t of 2 always clears the significance bar.
That depends on the degrees of freedom. At 24 of them the two-tailed critical value is about 2.06, so a t of exactly 2 falls just short.
A negative t means something went wrong.
It only says the sample mean lies below the hypothesised one: 48 against 50 with s = 5 and n = 25 gives −2. It is the magnitude that goes into the table.
Reference table
| Mean, μ, s, n | Reading | t |
|---|---|---|
| 48, 50, 5, 25 | Two standard errors below μ | -2 |
| 100, 100, 5, 9 | No difference at all | 0 |
| 52, 50, 5, 25 | Two standard errors above μ | 2 |
| 105, 100, 15, 30 | Just under two standard errors above | 1.825742 |
| 47.5, 50, 6, 36 | Two and a half standard errors below | -2.5 |
Questions
How do I calculate a one-sample t-statistic?
Subtract the hypothesised population mean from the sample mean, then divide by the standard error: t = (x̄ − μ) ÷ (s ÷ √n). A sample of 25 with a mean of 52, a hypothesised mean of 50 and a standard deviation of 5 gives (52 − 50) ÷ (5 ÷ √25) = 2 ÷ 1 = 2.
What is a t-statistic?
It is a standardised measure of how far a sample mean sits from a hypothesised value, expressed in units of standard error. It is the test statistic of the one-sample t-test and follows a Student t-distribution when the data are roughly normal. That lets you decide whether an observed difference is likely real or just sampling noise.
Why divide by s ÷ √n instead of by s?
Because the question is about the mean, not about a single observation. The sample mean varies less than the data do, and its own spread is the standard error s ÷ √n. Dropping the square root would understate t whenever n is larger than 1.
What is the difference between a t-statistic and a z-score?
Both count standard units away from a reference value, but a z-score uses the known population standard deviation while a t-statistic uses the sample standard deviation as an estimate. That extra uncertainty is why the t-distribution has heavier tails than the normal curve, and why it depends on the sample size through its degrees of freedom.
What are degrees of freedom in a t-test?
For a one-sample t-test they equal n − 1, one less than the sample size, so n = 25 gives 24. You use that number to look up the critical value in a t-table or to find the p-value. The shape of the t-distribution changes with the degrees of freedom.
How do I know if my t-statistic is significant?
Compare its absolute value with the critical value from a t-table at your significance level and your degrees of freedom. For a two-tailed test with 24 degrees of freedom the critical value is about 2.06, so a t of 2 would fall just short. A larger magnitude means stronger evidence against the null hypothesis.
Sources and last check
Information, not professional advice.