Enter the degrees of freedom and a t value to get the left-tail, right-tail, two-tailed or between-bounds probability of Student's t distribution, or enter an area to find the critical t value. The curve shows the shaded area next to the standard normal.
Last updated:
Usually n − 1 for one sample. Decimals (Welch df) are fine.
What do you want to find?
A decimal between 0 and 1, e.g. 0.05 (or a percent like 5%).
Enter your values to see the result.
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How to use the T Distribution Calculator
Enter the degrees of freedom, for example n − 1 for a one-sample t-test.
Pick what you want: a left, right or two-tailed probability, the area between two values, or a critical value from a probability.
Type the t value (or the bounds, or the area such as 0.05).
Read the probability or critical t and check the shaded curve.
Use the critical value table below for other significance levels at the same df.
T distribution formulas
For ν degrees of freedom the probability density of Student's t is
Tail areas are computed exactly with the regularized incomplete beta function: P(T > |t|) = ½ · Iν/(ν+t²)(ν/2, ½). The mean is 0 (for ν > 1) and the variance is ν ÷ (ν − 2) for ν > 2, which is always larger than 1, the variance of the standard normal. Critical values are found by solving the CDF for t numerically.
You want
Mode to choose
TI-84
Excel
Area left of t
P(T < t)
tcdf(−1E99, t, df)
T.DIST(t, df, TRUE)
Area right of t
P(T > t)
tcdf(t, 1E99, df)
T.DIST.RT(t, df)
Two-tailed p-value
P(|T| > t)
2 · tcdf(|t|, 1E99, df)
T.DIST.2T(|t|, df)
Area between a and b
P(a < T < b)
tcdf(a, b, df)
T.DIST(b) − T.DIST(a)
Critical t (left area p)
t from a probability, left tail
invT(p, df)
T.INV(p, df)
Critical ±t (two-tailed α)
t from a probability, both tails
invT(1 − α/2, df)
T.INV.2T(α, df)
Two-tailed critical t values at a glance
df
90% (α = 0.10)
95% (α = 0.05)
99% (α = 0.01)
1
6.314
12.706
63.657
2
2.920
4.303
9.925
5
2.015
2.571
4.032
10
1.812
2.228
3.169
15
1.753
2.131
2.947
20
1.725
2.086
2.845
30
1.697
2.042
2.750
60
1.671
2.000
2.660
120
1.658
1.980
2.617
∞ (z)
1.645
1.960
2.576
Tip: the heavier tails at small df are why small-sample confidence intervals are wider. With only 3 observations (df = 2) the 95% multiplier is 4.303, more than twice the normal value of 1.96.
Frequently asked questions
What is the t distribution?
Student's t distribution is a bell-shaped, symmetric distribution centred on 0 that has heavier tails than the normal distribution. It describes the statistic (x̄ − μ) ÷ (s ÷ √n) when the population standard deviation is unknown and estimated from a sample. Its shape depends on the degrees of freedom: the larger the df, the closer it is to the standard normal.
How do I find the degrees of freedom?
For a one-sample or paired t-test and for a confidence interval for one mean, df = n − 1. For a pooled two-sample t-test, df = n₁ + n₂ − 2. Welch's two-sample test uses the Welch–Satterthwaite formula, which usually gives a decimal df; this calculator accepts decimals.
How do I get the critical t value for a 95% confidence interval?
Choose 't from a probability', enter 0.95 and pick 'in the middle'. The calculator puts 2.5% in each tail and returns ±t*. For example, with df = 24 the critical value is about ±2.064, and with df = 10 it is about ±2.228.
What is the difference between one-tailed and two-tailed probabilities?
A one-tailed probability is the area in a single tail, either P(T > t) or P(T < t). A two-tailed probability counts both tails beyond ±|t|, so it is twice the one-tailed area for the same |t|. Use two-tailed for 'not equal' alternative hypotheses.
When does the t distribution look like the normal distribution?
As the degrees of freedom grow, the t distribution approaches the standard normal. Around df = 30 the two-tailed 5% critical value is 2.042 versus 1.960 for z, and by df = 120 it is 1.980. With df = ∞ the values are identical.
Is this the same as tcdf and invT on a TI-84?
Yes. P(a < T < b) matches tcdf(a, b, df), and the inverse mode with a left-tail area matches invT(area, df). In Excel the equivalents are T.DIST, T.DIST.RT, T.DIST.2T, T.INV and T.INV.2T.
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