Statistics
T Distribution Calculator
Replace the printed t table.
Enter degrees of freedom, then either a t value (to get the tail probabilities), a probability (to get the matching t), or a significance level (to get the critical value). Every mode also shows the density and how the t curve compares with the standard normal.
n - 1 for one sample; Welch df may be a decimal.
Try an example
Result
Result
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- P(T < t)
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- P(T > t)
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- P(|T| > |t|)
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- Density f(t)
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- Standard normal P(Z < t)
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Study path
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What the t distribution calculator solves
Replace the printed t table. Enter degrees of freedom, then either a t value (to get the tail probabilities), a probability (to get the matching t), or a significance level (to get the critical value). Every mode also shows the density and how the t curve compares with the standard normal.
Formula
How to use it
- Enter the degrees of freedom (n - 1 for a one-A smaller group taken from a population, used to estimate facts about the whole group. test).
- Choose a mode: a t value gives the tail probabilities, a probability gives t, and α gives the critical value.
- Read the main answer plus the left, right, and two-tailed areas for the same t.
- Compare with the standard normal column to see how much the heavier t tails matter at your df.
How to read the answer
Tail probabilities are The chance of seeing results this extreme if nothing real were going on.: the two-tailed area is the p-value for a 'not equal' hypothesis. Critical values mark where the rejection region starts. As df grows the t distribution approaches the standard normal, so t*(0.025) drops from 12.71 at df = 1 toward 1.96.
Common mistakes and edge cases
- Using a two-tailed critical value for a one-tailed test (or the reverse); the tail split changes the number.
- Entering n instead of n - 1 for the degrees of freedom.
- Reading 0.975 as the tail area; it is the left-tail area that leaves 0.025 on the right.
- Assuming the t and z values match for small samples; at df = 5 the two-tailed 5% cutoff is 2.571, not 1.96.
Worked examples
Two-tailed p for t = 2.228, df = 10
Result
0.05001 (5.00%)
Left tail of a negative t
Result
0.07718 (7.72%)
Inverse: t with P(T < t) = 0.975, df = 10
Result
2.2281
Critical value, two-tailed α = 0.05, df = 29
Result
±2.0452
Critical value, one-tailed α = 0.01, df = 5
Result
3.3649
Zero t is the median
Result
0.5 (50.00%)
Impossible: zero degrees of freedom
Result
Error
Frequently asked questions
How do I read a t table?+
Rows are degrees of freedom and columns are tail areas. The entry is the t value that leaves that area in the upper tail. This calculator returns the same number for any df and any area, without interpolation.
What are degrees of freedom?+
The number of independent pieces of information used to estimate the spread. For one sample it is n - 1; for a pooled two-sample test it is n₁ + n₂ - 2; Welch's test gives a decimal.
When does the t distribution become the normal distribution?+
Never exactly, but by df ≈ 30 the critical values are within a few percent of z, and by df = 1000 they match to two decimals.
What is the critical t for a 95% confidence interval?+
The two-tailed value with α = 0.05: t*(0.025, df). For df = 10 that is 2.228; for df = 29 it is 2.045.
Can degrees of freedom be a decimal?+
Yes. Welch's two-sample test produces non-integer df, and the incomplete beta formula handles any positive value.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026