T Distribution Calculator

Pick what you want: a probability from a t value, the probability between two t values, a critical t from a probability, or a confidence interval from sample data. Enter the degrees of freedom and the other values, then press Calculate. For t = 2.1 and df = 18 the two-tailed p-value is 0.05009.

Enter t and the degrees of freedom, and choose the tail that matches your test. For t = 2.1 and df = 18 the right-tail p-value is 0.025045, the left-tail is 0.974955 and the two-tailed p-value is 0.05009. The calculator shows all three, with the one you chose first.

A two-tailed p-value is P(|T| ≥ |t|), which is twice the smaller tail. For t = 2.1 and df = 18 the right tail is 0.025045, so the two-tailed p-value is 0.05009. Pick the two-tailed option and the calculator gives it directly, with the incomplete beta function behind it.

Choose the critical value option, enter alpha, the degrees of freedom and the tail. For alpha = 0.05, two-tailed, df = 10 the critical value is 2.2281, so you reject when |t| exceeds it. For alpha = 0.01, right-tailed, df = 24 it is 2.4922.

It goes from a probability back to a t value, the opposite of finding a p-value. Enter an area and df and it returns the t that cuts off that area. A left-tail area of 0.975 with df = 10 gives t = 2.2281, which is the familiar 2.2281 used for a 95% two-sided interval.

It is 2.2281. For a 95% two-sided interval, alpha is 0.05, so each tail holds 0.025 and the critical value is t(0.975, 10). The NIST table lists 2.228 for df = 10 at the 0.975 column, and the calculator agrees to all four decimals.

Find the row for your degrees of freedom and the column for your tail area, for example 0.025 for a two-sided 95% interval. The entry is the critical t. For df = 10 and 0.025 the table gives 2.228. A table only holds a few areas and some df rows, while this calculator gives any value.

The density is f(t) = Gamma((df + 1) / 2) / (√(df x pi) x Gamma(df / 2)) × (1 + t² / df)^(-(df + 1) / 2). The cumulative probabilities use the regularized incomplete beta function I_x(df / 2, 1 / 2) with x = df / (df + t²). The calculator evaluates both numerically.

Find t* for the confidence level with df = n − 1, compute the standard error s / √(n), and take the sample mean plus or minus t* x SE. With mean 72, SD 8.5, n = 16 and 95%, t* = 2.1314, SE = 2.125 and the interval is 67.47 to 76.53.

Many Casio graphing calculators have a distribution menu reached through the statistics menu, with t density, cumulative t and inverse t options. You enter the t value or the area and the degrees of freedom. Menus differ by model, so check your manual. This page gives the same numbers without a menu.

Press 2nd, DISTR. Use tcdf(lower, upper, df) for probabilities, for example tcdf(2.1, 1E99, 18) for the right tail, which is 0.025045. Use invT(area, df) for the t value from a left-tail area, for example invT(0.975, 10) gives 2.2281. The calculator returns the same figures.

Use T.DIST(t, df, TRUE) for the left tail, T.DIST.RT(t, df) for the right tail and T.DIST.2T(t, df) for the two-tailed value, with t non-negative. T.INV(area, df) and T.INV.2T(alpha, df) give critical values. T.DIST.2T(2.1, 18) returns 0.05009, the same as this page.

Degrees of freedom control the shape of the t curve. For a one-sample t test or interval they equal the sample size minus one, because one value of freedom is used up in estimating the mean. Few degrees of freedom give heavy tails and a wide curve. Many give a curve close to the normal.

For one sample or paired data, df = n − 1. For two independent samples with equal variances, df = n1 + n2 − 2. For unequal variances, the Welch formula gives a non-integer df, which this calculator accepts. For regression with k predictors it is n − k - 1. Check your test's rules.

Yes. Welch's t test often gives a fractional df such as 17.4, and this calculator accepts any positive value up to 1,000,000. For instance t = 2.1 at df = 0.7 gives a two-tailed value of 0.357878, though very small df values are rarely meaningful in practice.

The t distribution has heavier tails, so extreme values are more likely, especially with small samples. P(|T| > 1.96) is 0.1073 at df = 5 against 0.05 for the normal. As df grows the two curves match: at df = 1000 the figure is 0.0503.

Match the tail to your alternative hypothesis. Use left when you test whether the mean is lower, right when it is higher, and two when you test for any difference. Confidence intervals use two tails. When you do not know the direction in advance, the two-tailed choice is the safer default.

It is a symmetric bell shape centred on zero, like the normal curve but with lower peaks and heavier tails. At t = 0 the density is 0.3183 for df = 1, 0.3796 for df = 5 and 0.3956 for df = 30, approaching the normal peak 0.3989. The diagrams on this page show the curves.

It uses the regularized incomplete beta function with a continued fraction and solves for critical values by bisection to near double precision. We checked it against all 600 values of the NIST critical-value table and against high-precision mathematics software, and it agrees to the 3 decimals NIST prints and far beyond.

Yes. The solution view shows the model, the conversion of t to the beta argument x, the two-tailed and one-tailed probabilities and the density for the probability mode. In the critical value and interval modes it shows how alpha was split, the value found and a check. A toggle hides the steps.

A t-score, t-value or t statistic measures how far a sample mean is from a claimed value, counted in standard errors: t = (sample mean - claimed mean) / (s / √(n)). A large absolute value is evidence against the claim. Enter it with the degrees of freedom to get a p-value, for example t = 2.1 and df = 18 gives 0.05009.

It approaches the standard normal distribution. The 97.5th percentile falls from 12.706 at df = 1 to 2.228 at df = 10, 2.042 at df = 30 and 1.962 at df = 1000, heading for the normal value of 1.960.