T-Value Calculator

Degrees of freedom (df)

Significance level (α)

Confidence level, two-tailed (%)

Common α

One-tailed critical t

1.8125

All of α = 0.05 sits in one tail.

Two-tailed critical t

±2.2281

α split in half: 0.025 in each tail, for 95% confidence.

t(0.05, 10) = 1.8125

With 10 degrees of freedom, reject H₀ if your t-statistic is above 1.8125 (one-tailed) or outside ±2.2281 (two-tailed).

The t-table row for df = 10

Critical t at each standard significance level, for your degrees of freedom. The highlighted column is the α you entered.
α, one tail	0.1	0.05	0.025	0.01	0.005
α, two tails	0.2	0.1	0.05	0.02	0.01
Critical t	1.3722	1.8125	2.2281	2.7638	3.1693

Which one do you want? Use the one-tailed value when your hypothesis has a direction — "the new version is faster," "the drug lowers blood pressure." Use the two-tailed value when you'd care about a difference in either direction, which is the default for most coursework and most papers. Both are on screen because picking the wrong one is the single most common mistake in this calculation, and a page that makes you choose before it shows you anything can't help you catch it. If you already have a z-score instead, the z-score calculator and the p-value calculator handle the normal-distribution case; if you have a data set and want the interval rather than the cutoff, use the confidence interval calculator.

The t-table in the back of your textbook has 34 rows and stops at df 100. Type your degrees of freedom instead. Both critical values appear as you type — the one-tailed cutoff and the two-tailed ±cutoff, side by side, so there's no tail to pick before you can see anything. Underneath sits the t-table row for your exact df across the five standard columns, with the column you asked for highlighted, which is the part you check against the printed table. Two things here that most t-value pages don't do. Fractional degrees of freedom compute properly: a Welch's t-test that hands you df = 9.47 gets 2.2452, not an error message and not the df = 9 row. And the numbers come from inverting the actual t-distribution with a continued-fraction expansion of the incomplete beta function, not from interpolating between two rows of a stored table — which is exactly the step that goes wrong on fractional df. Nothing you type leaves your browser, and there's nothing to sign up for.

Built by Bob QA by Ben Shipped September 20, 2026

How to use

1
Type your degrees of freedom. For a one-sample t-test with n observations, df = n − 1. For a paired test it's the number of pairs minus one. For Welch's unequal-variance test it's whatever fractional number your software reported — type it in as-is, decimals included.
2
Set the significance level. Use the α field (0.05 is the default) or the confidence-level field (95) — they're two readings of the same number and editing either updates the other instantly. The five common values are one click away as presets.
3
Read both answers. The one-tailed value puts all of α in a single tail; the two-tailed value splits it, α/2 per side, which is why it's always the larger number. They're shown together on purpose: choosing the wrong one is the most common error in this calculation.
4
Check the plain-language line underneath. It spells out the decision rule with your actual numbers in it — what to compare your t-statistic against, and in which direction.
5
Scan the t-table row. It gives the critical t at all five standard significance levels for your df, with your column highlighted, so you can cross-check against a printed table or see how much the cutoff moves if you tighten α.
6
Copy what you need. Each critical value copies as a bare number that pastes cleanly into a spreadsheet; the notation line copies as t(0.05, 10) = 1.8125 for dropping straight into a write-up.

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Frequently asked questions

What is a critical t-value, and what do I do with it?

It's the cutoff. You run a t-test, you get a t-statistic out of your data, and the critical value is the number that statistic has to beat for the result to count as statistically significant at the significance level you chose. If your t-statistic is larger than the critical value, you reject the null hypothesis; if it isn't, you don't. The critical value depends on exactly two things — how much data you have (the degrees of freedom) and how strict you want to be (α). That's why this page only asks for those two numbers: nothing else about your data changes the cutoff.

One-tailed or two-tailed — which value should I use?

Use the one-tailed value when your hypothesis has a direction and you'd only care about a difference one way: "the new version is faster," "the drug lowers blood pressure." Use the two-tailed value when a difference in either direction would be interesting, which covers most coursework, most published research, and any situation where you'd have been surprised either way. The two-tailed value is always larger, because splitting α across two tails leaves less probability in each one, which pushes the cutoff further out. The honest rule: pick your tail before you see your data. Choosing one-tailed after the fact because two-tailed didn't reach significance is the classic way to fool yourself, and it's why this page shows both rather than making you commit up front.

How do I find my degrees of freedom?

It depends on the test. One-sample t-test: df = n − 1, where n is how many observations you have. Paired t-test: df = (number of pairs) − 1. Two-sample t-test assuming equal variances: df = n₁ + n₂ − 2. Two-sample with unequal variances (Welch's, which is what most software runs by default): df comes out of the Welch–Satterthwaite formula and is usually not a whole number — something like 17.34. Regression: df = n − k − 1, where k is the number of predictors. The intuition behind the subtraction is that each parameter you estimate from the data (like the sample mean) uses up one piece of independent information, and what's left is how much freedom the residuals still have.

Why does this accept a fractional df like 9.47 when most calculators don't?

Because Welch's t-test produces fractional degrees of freedom, and Welch's is the default two-sample test in R, SciPy, and most modern statistics software. Calculators that only accept integers are built on a stored t-table plus interpolation, and interpolation is precisely the step that breaks on fractional input — you end up with a number between two table rows rather than the value the distribution actually has. This page inverts the t-distribution's cumulative distribution function directly, using a continued-fraction expansion of the regularized incomplete beta function. Fractional df isn't a special case for that method; it's just another input. So df = 9.47 returns 2.2452, which is the real answer, not a blend of the df = 9 and df = 10 rows.

What's the difference between α and the confidence level?

They're the same number seen from opposite sides. α is the probability you're willing to accept of calling something significant when it isn't — a false positive. The confidence level is what's left over: 100 × (1 − α). α = 0.05 is a 95% confidence level, α = 0.01 is 99%, α = 0.10 is 90%. The pairing is the two-tailed reading, which is the convention when people say "a 95% confidence interval." Both fields are on this page and linked, so you can type whichever one your assignment or your paper uses and read the other off without doing the arithmetic.

When should I use t instead of z?

Use t when you're estimating the population standard deviation from your own sample — which is almost always, unless you've been handed the true σ. The t-distribution has fatter tails than the normal distribution precisely to account for the extra uncertainty in that estimate, and the fewer observations you have, the fatter they get. At df = 1 the two-tailed 95% cutoff is 12.7062; at df = 10 it's 2.2281; at df = 30 it's 2.0423; by df = 100,000 it's 1.9600, which is the z value to four decimals. That convergence is why old textbooks tell you to "use z for n > 30" — it's a printing-cost shortcut from the era of paper tables, not a rule. If you're computing rather than looking up, use t and let the degrees of freedom do the work.

Why is the t-value for df = 1 so enormous?

Because with one degree of freedom you know almost nothing about the spread of your data, and the distribution reflects that honestly. The df = 1 case is the Cauchy distribution, whose tails are so heavy it has no defined mean. The two-tailed 95% critical value is 12.7062, meaning your t-statistic would have to be nearly thirteen times its standard error before you could call the result significant. That's not a glitch in the calculator; it's the correct price of having two data points. The cutoff drops fast as you add observations: df = 2 is 4.3027, df = 5 is 2.5706, df = 20 is 2.0860.

My software gave me a p-value. Do I still need a critical value?

Strictly, no — comparing p against α gives you the same decision as comparing your t-statistic against the critical value. They're two routes to one conclusion. Critical values still earn their place in three situations: coursework that asks you to report the cutoff, write-ups that use the t(df) = value notation, and any time you want to know how close a result was to the line rather than just which side of it landed on. Seeing that your t-statistic was 2.19 against a cutoff of 2.2281 tells you something that "p = 0.053" states more abstractly.

Does the sign of the critical value matter?

Only in how you state the comparison. The t-distribution is symmetric around zero, so the lower critical value is just the negative of the upper one — that's why the two-tailed result is written as ±2.2281 rather than as two separate numbers. For a two-tailed test you reject when the absolute value of your t-statistic exceeds the cutoff. For a one-tailed test in the negative direction ("the new version is slower"), you reject when your t-statistic falls below the negative of the one-tailed value. Same number, flipped sign.

How accurate are these numbers?

They match a standard printed t-table to four decimal places. The values come from inverting the t-distribution's CDF by bisection on a continued-fraction evaluation of the incomplete beta function, which reaches machine precision; above 10,000 degrees of freedom the calculation switches to a Cornish–Fisher expansion around the normal quantile, which agrees with the exact inversion to about twelve decimal places at that size. Every value shown here was checked two independent ways and against the printed table before the page shipped. Four decimals is where it stops because that's what the tables give you and what any write-up will ask for.

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