t = (x̄ − μ₀) / (s / √n), df = n − 1
| Sample mean, x̄ | 5.062500 |
| Standard error | 0.110093 |
| t statistic | 0.567700 |
| Degrees of freedom | 7.0000 |
| Two-tailed p-value | 0.587976 |
| Left-tail, P(T ≤ t) | 0.706012 |
| Right-tail, P(T ≥ t) | 0.293988 |
Two-tailed p = 0.587976 is not below α = 0.05, so the data do not show a significant difference at that level.
The t test assumes the observations within each sample are independent and that the means are roughly normal, which holds when the data are close to normal or n is moderate to large. Welch's version does not assume equal variances, so it is the safer default for two groups. Use the paired test when each x has a matching y from the same subject or unit. Read the p-value as the chance of a difference at least this large if the hypothesised value were true, not as the chance that the hypothesis is true.
One-sample, Welch and paired t tests with the p-value. A t test checks whether a sample mean, a difference between two means or a mean of paired differences is larger than chance alone would produce, when the population standard deviation is unknown. This calculator runs the one-sample, Welch two-sample and paired versions from raw data or summary statistics, and reports the t statistic, degrees of freedom and one- and two-tailed p-values.
Choose the test before you look at the data. Paired measurements entered as two independent samples throw away the matching and usually lose power. Welch's test is the safer default for two groups because it does not assume equal variances. 100% free, no registration, and complete privacy — everything runs locally in your browser, so your data never touches a server.
t = (x̄ − μ0) / (s / √n), with df = n − 1. Tests whether the mean differs from a fixed value.
t = (x̄1 − x̄2) / √(s1²/n1 + s2²/n2), with Welch–Satterthwaite degrees of freedom, so unequal variances are allowed.
Takes the differences d = x − y, then runs the one-sample test on d with n − 1 degrees of freedom.
The standard error, degrees of freedom and both tails are listed, so the result can be checked against a printed t table.
One-tailed and two-tailed p-values for a z or t test statistic.
Confidence intervals for a mean (z or t) and a proportion.
Least-squares slope, intercept and the slope test.
Z score, percentile and probabilities for a value.
Welch's test does not assume the two groups have the same variance, and it loses little when they do. Student's pooled test assumes equal variances and is only slightly more powerful when that assumption holds. Many statisticians use Welch as the default unless there is a strong reason to assume equal variances.
It subtracts each pair first, so the variability between subjects drops out of the standard error. This is why paired designs often detect smaller effects. The two lists must be the same length and in matching order.
Observations within each group should be independent, and the sample mean should be approximately normal. The mean is close to normal for moderate samples even when the data are not, but strong skew or outliers in a small sample can distort the p-value.
No. The p-value is the probability of a t statistic at least as extreme as the one observed, assuming the hypothesised value is true. It measures how surprising the data would be under that assumption, not the probability that the hypothesis itself is true.
We use cookies for analytics and personalized ads to help keep these tools free. Until you accept, ads stay non-personalized and analytics cookies are off. See our Privacy Policy.