Any degrees of freedom, to full precision, and the gap from the normal.
The t table, without the rounding
Ten observations. The 95% critical value is 2.262 — 15.4% above z.
t = 2.2620 on 9.0000 degrees of freedom
Two-tailed p = 0.050013
One-tailed: 0.025006 in the upper tail, 0.974994 in the lower.
P(T ≤ t)
0.97499358
P(T ≥ t)
0.02500642
Two-tailed p
0.05001285
Same tail under z
0.01184870
the normal, for comparison
The normal would put 0.02369740 in the two tails; the t puts 0.05001285. The t has heavier tails because it accounts for the standard deviation being estimated rather than known, and the gap grows as the degrees of freedom fall. Using z when you should use t makes every p-value look smaller than it is.
Variance is df/(df − 2) = 1.285714, which exceeds 1 and approaches it from above. That is the excess spread over a standard normal, and it is where the wider confidence intervals come from. Below df = 3 the fourth moment is infinite too, so kurtosis is undefined even where the variance is not.
What this tool shows
“Use z above 30 degrees of freedom” is off by 4.199%. At df = 30 the two-sided 95% critical value is 2.042272 against z’s 1.959964, so an interval built on z is 4% too narrow. The gap closes slowly: df must reach 4,746 before t comes within 0.0005 of 1.95996.
One-tailed and two-tailed probabilities at any t and any degrees of freedom
Critical values at any confidence level, not just the six a table prints
The corresponding normal value, and the percentage gap between them
Non-integer degrees of freedom, which Welch’s test produces and tables cannot hold
The density curve with your t or your critical values marked
Which moments exist: none at df = 1, infinite variance at df = 2
Any df, any level The gap from z Welch’s fractional df Moments stated
Updated 12 September 2026 · Works in any browser, no installation
The t distribution is what a standardised mean follows when the standard deviation is estimated from the same sample. That extra uncertainty makes its tails heavier than the normal’s, which is why confidence intervals built on it are wider — and the effect is large at small samples and never quite disappears.
At a glance
Formula shown
With ν degrees of freedom the density is proportional to (1 + t²/ν)^(−(ν+1)/2), and the CDF comes from the regularised incomplete beta function I_x(ν/2, 1/2) at x = ν/(ν + t²) rather than from a table. As ν → ∞ the density converges to the standard normal. The mean exists only for ν > 1 and the variance, ν/(ν − 2), only for ν > 2.
Scenario support
Confidence intervals for a mean when sigma is unknown, one-sample and two-sample t tests, paired tests, regression coefficient tests, and Welch’s unequal-variance test — which produces fractional degrees of freedom that no printed table contains.
Educational estimate
Planning support from the values you enter — not professional advice.
The rule that says switch to z at 30
Every introductory course teaches it, and it was a concession to the physical size of a printed table rather than a claim about when the difference stops mattering.
At df = 30 the two-sided 95% value is 2.042272. Against z’s 1.959964 that is 4.199% larger. An interval built on z is 4% too narrow and a p-value computed from z is too small — both in the direction that finds things.
At df = 120, where tables usually stop, it is still 1.019% out. At df = 500, 0.243%. The convergence is real and it is slow, because the gap falls roughly as 1/df.
Reaching three decimal places takes df = 4,746. That is where t first comes within 0.0005 of 1.95996, which is what it takes for the printed value to be 1.960. Nobody has that many degrees of freedom in a t test.
The old tables ended at 30 and then jumped to infinity because rows cost paper. The rule that emerged from that layout got taught as a statistical fact, and it outlived the constraint by several decades.
There is no reason to approximate now. The exact value costs an incomplete beta evaluation. Use t whenever sigma was estimated, at every df, and the question of when to switch does not arise.
At df = 1 there is no mean, and at df = 2 no variance
The variance of a t distribution is df/(df − 2). That formula is quoted constantly and its domain condition is dropped almost as often.
It only holds for df > 2. At df = 2 it divides by zero, and the truth is that the variance is infinite — the integral diverges. A calculator that prints a finite number there is evaluating a formula outside its domain.
At df = 1 the t IS the Cauchy distribution, and it has no mean either. The integral defining the expectation does not converge, so there is nothing for a sample mean to converge to.
Which breaks the intuition that averaging helps. The mean of n Cauchy draws has exactly the same distribution as one draw. Averaging a thousand of them buys nothing. The law of large numbers requires a mean to exist, and here one does not.
The tails are the reason. At df = 1 the 95% two-sided critical value is 12.706 and the 99.5% point is above 63 — against 1.96 and 2.58 for a normal. Those are not outliers; that is the distribution.
Kurtosis needs df > 4, and skewness df > 3. Each moment has its own threshold, and below it the quantity is not large but undefined. The tool says which exist at the df you entered rather than printing a number for all of them — the same care the kurtosis page takes about which definition is in use.
Fractional degrees of freedom, which no table contains
Welch’s t test — the default two-sample test in R and the one that does not assume equal variances — produces degrees of freedom like 17.43. A printed table has no such row.
The Welch-Satterthwaite formula is a weighted combination of the two samples’ variances and sizes, and it lands wherever it lands. It is not rounded to an integer as part of the method; rounding is what people do because their table forces it.
Rounding down is conservative and rounding to nearest is not. Rounding 17.43 to 17 gives a slightly larger critical value, which is safe. Rounding 17.62 to 18 gives a smaller one, which is not. The habit is inconsistent, and the inconsistency runs in the unsafe direction half the time.
This tool takes any positive real df. The incomplete beta function is defined for non-integer parameters, so 17.43 is no harder to evaluate than 17, and the rounding step disappears.
Welch is the better default anyway. The equal-variance pooled test is more powerful only when the variances really are equal, and the pre-test for equality is itself unreliable at the sample sizes where it would matter. Using Welch unconditionally costs very little and removes an assumption.
The same applies to regression output. Robust standard errors and mixed models both produce fractional degrees of freedom, by Satterthwaite or Kenward-Roger. Any of them can be looked up here directly.
Why it is called Student’s t
The name is a pseudonym, and the reason for the pseudonym is also the reason the distribution exists.
William Sealy Gosset worked at Guinness in Dublin. He published the 1908 paper as “Student” because the brewery treated statistical methods as a trade secret and did not allow employees to publish under their own names.
The problem was small samples. Brewing chemistry gave him four or five measurements at a time, not hundreds, and the normal approximation that worked for astronomers with long observation runs did not work for him.
The insight was that estimating sigma costs something. Replacing a known standard deviation with a sample estimate adds variability that the normal distribution does not account for, and at small n that extra variability dominates.
Fisher supplied the proof and the modern form. Gosset derived the distribution partly by simulation — drawing cards from a physical sample of criminal measurements — and Fisher later derived it properly and introduced the t notation.
The practical lesson has not changed. If you estimated the spread from the same data you are testing, use t. The distinction the z test requires — a genuinely known sigma — almost never holds outside a textbook exercise, which is why the t test is the one people actually run.
Reading the tails correctly
More errors come from picking the wrong tail than from anything about the distribution itself, and the tool prints all three numbers to make the choice visible.
A two-tailed p is twice the one-tailed p, for a symmetric distribution. The t is symmetric, so this holds exactly — and it is why a one-tailed test declared after seeing the direction of the effect is a 10% test wearing a 5% label.
Critical values differ between tails too. At df = 9 the two-sided 95% value is 2.262 and the one-sided is 1.833. Using the one-sided value for a two-sided test is a 10% test again.
The confidence level and the significance level are complements. A 95% interval uses the same critical value as a two-sided 5% test. If they disagree in your output, one of them was computed one-tailed.
The sign of t carries the direction and nothing else. The distribution is symmetric, so P(T ≤ −2.262) equals P(T ≥ 2.262). A negative t is not a smaller effect; it is the same effect the other way.
And a p-value from the t is still a p-value. It is the probability of data at least this extreme if the null is true, not the probability the null is true. The p-value page covers what it does and does not say.
What has to be true for the t to apply
The distribution is exact under specific conditions, and approximately right under much weaker ones. Both statements matter.
Exactly: the observations are independent and normally distributed. Then the standardised mean follows a t exactly, at any n, with no approximation anywhere.
Approximately: the observations are independent and n is not tiny. The central limit theorem makes the sample mean roughly normal even when the data is not, so the t test is famously robust to non-normality.
Skewness is what actually breaks it, not heavy tails. A symmetric heavy-tailed distribution is handled reasonably at moderate n. A strongly skewed one biases the t statistic, because the sample mean and the sample standard deviation are correlated under skew — and that bias does not shrink as fast as the non-normality does.
Independence is the assumption with no defence. Clustered, repeated or time-ordered data inflates significance badly, and no amount of sample size fixes it. Robustness to non-normality gets all the attention; this is the one that invalidates results.
Outliers matter more than the distribution’s shape. One extreme value moves both the mean and the standard deviation, and a residual check or a rank-based alternative like Mann-Whitney is the right response — not a normality test, which at small n cannot detect the deviation and at large n flags one too small to care about.
Sources and methodology
References for the distribution and its computation.
Method. Probabilities come from the regularised incomplete beta function rather than from an interpolated table, so any positive real degrees of freedom is exact — including the fractional values Welch’s test and mixed models produce, which no printed table contains. Critical values are obtained by inverting the same function rather than by root-finding on a series. The corresponding normal value is shown alongside with the percentage gap, because the rule about switching to z at df = 30 is a claim about that gap and is usually made without quoting it. The suite pins the standard table values at df = 1, 2, 30 and 120, confirms the gap from z is 4.199% at df = 30 and 1.019% at df = 120, and checks the identity that ties this distribution to the F distribution: F(1, v) equals t(v) squared at every df and level tested. That engine is verified on every change against 59 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
t-testOne-sample, two-sample and paired t-tests defaulting to Welch, with Student's pooled version printed beside it — and a warning when the two disagree on the verdict.
Normal DistributionProbabilities under a normal curve in all four directions with the region shaded — and the empirical rule given exactly, because two standard deviations is 95.45% and the 95% everyone quotes sits at 1.96σ.
F-DistributionUpper and lower critical values at any degrees of freedom — including the lower tail printed tables omitted, because F(a,b;p) = 1/F(b,a;1−p) — plus F(1,v) against t(v)² computed both ways, which agree exactly.
Confidence IntervalIntervals for a mean or a proportion using t at every sample size and Wilson rather than the textbook Wald formula — with both methods shown, because Wald returns [0,0] at zero successes.
Critical ValueCritical values for z, t, chi-square and F at any alpha and any degrees of freedom — with one- and two-tailed values shown together, because reading the wrong column of a printed table is the classic error.
p-valueA p-value from a t or z statistic, one- or two-tailed — with a panel that holds an effect fixed and grows the sample, so you can watch significance appear from nothing but n.
An educational tool. The t distribution is exact only for independent, normally distributed observations; it is robust to non-normality at moderate sample sizes but not to dependence, and strongly skewed data biases the statistic in a way sample size corrects only slowly.
Published a t-distribution calculator that prints the percentage gap from the normal, because the rule everyone learns — switch to z above 30 degrees of freedom — is a claim about that gap and is always made without quoting it. At df = 30 the two-sided 95% value is 2.042272 against z's 1.959964, which is 4.199% larger; at df = 120 it is still 1.019% larger, and df must reach 4,746 before the gap falls below 0.0005.
Takes any positive real degrees of freedom, including the fractional values Welch's unequal-variance test and mixed models produce and that no printed table contains. The incomplete beta function does not require integer parameters, so 17.43 is no harder than 17 and the rounding step — which is inconsistent and unsafe half the time — disappears.
States which moments exist rather than evaluating df/(df - 2) outside its domain. At df = 1 the t IS the Cauchy distribution: no mean, no variance, a 95% critical value of 12.706 and a 99.5% point above 63. At df = 2 the mean exists and the variance is infinite.
Verified against the standard table values at df = 1, 2, 30 and 120, and against the identity F(1, v) = t(v) squared, which ties it to the F distribution and shows a two-group ANOVA and a two-sided t test are the same test.
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