T Test Critical Value Calculator
Before you run a t-test, one number decides it: the critical t-value, the threshold your statistic must beat for the result to count as significant at your chosen alpha. This calculator finds that threshold exactly, for any alpha and any degrees of freedom, no table interpolation and no missing rows. Enter alpha, pick the tail, give df or the sample size, and you get the critical value, the rejection region in words, the equivalent z, and how much wider t runs than z at your settings.
The critical value solves P(T > t) = p by bisection on the exact t-distribution tail (regularized incomplete beta function), so any alpha and any df work, including values no printed table carries. A p-value answers “how surprising is my result”; the critical value answers “how extreme must a result be to count.”
Quick Answer
For a two-tailed t-test at α = 0.05 with 29 degrees of freedom (for example, n = 30), the critical value is ±2.0452: reject the null hypothesis when |t| > 2.0452. The equivalent normal z is 1.9600, so t runs 4.4% wider than z at these settings. The calculator above is pre-filled with this example.
Key takeaways
- The critical value is the threshold your t-statistic must beat; the p-value is how surprising your t-statistic was. Different questions, answered by different tools: this page for the threshold, the t-statistic p-value calculator for the p-value.
- Two-tailed α = 0.05 with df = 29 gives ±2.0452, matching the textbook 2.045 to three decimals.
- Any alpha works here, including values no printed table carries (try 0.03): the value is computed by numerical inversion, never interpolated.
- Enter the sample size n and the calculator sets df = n − 1 for you.
- At small df, t is dramatically wider than z (548.3% wider at df = 1); by df = 100 the penalty is down to 1.2%.
- Quick Answer
- How to use the t critical value calculator
- How the exact critical value is computed
- Worked example: n = 30, α = 0.05
- The small-sample penalty: t vs z
- What competing critical-value calculators get wrong
- Critical value vs p-value: which question are you asking?
- Sample size or degrees of freedom: which to enter
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the t critical value calculator
Four inputs, one threshold. The result stays hidden until you calculate, and the worked example is pre-filled.
- Enter the significance level α (0.05 pre-filled). Any decimal up to 0.5 works, so 0.03 and 0.025 are fine.
- Pick the test type. Two-tailed rejects when |t| beats the value; right-tailed rejects when t is large and positive; left-tailed when t is large and negative.
- Enter degrees of freedom (29 pre-filled), or enter the sample size n and let the calculator set df = n − 1. If both are filled, n wins.
- Click Calculate. The big number is the critical value; below it sit the df used, the rejection region in plain words, the equivalent z, and the small-sample penalty percentage.
Invalid inputs are stopped with a message: a non-numeric alpha (“Please enter a valid significance level.”), alpha at or below zero (“Alpha must be greater than 0.”), alpha above 0.5 (“Alpha must be 0.5 or less.”), a bad df (“Please enter valid degrees of freedom.” / “Degrees of freedom must be at least 1.”), and a sample size below 2 (“Sample size must be at least 2.”). One honest quirk: a fractional df like 2.5 is rounded to the nearest whole number (3) rather than rejected, and a fractional n is truncated by the integer parse, so 3.9 becomes n = 3. The results note says this too.
How the exact critical value is computed
For a right-tail probability p, the critical value is the t that solves P(T > t) = p, where the tail comes from the regularized incomplete beta function. The calculator brackets the root (doubling the search bound until the tail fits inside p) and bisects it to about nine decimals. Two-tailed α uses p = α/2; a one-tailed test uses p = α and negates the result for the left tail.
df from n: df = n − 1 Small-sample penalty: (t* − z*) ÷ z* × 100
The same answers in the tools you may already use: Excel’s =T.INV.2T(0.05,29) returns 2.0452 and =T.INV(0.05,29) returns 1.6991; in R, qt(0.975, 29); in Python, scipy.stats.t.ppf(0.975, 29). This matters because the previous generation of this tool held a 12-row lookup table and linearly interpolated between rows, and any alpha not in the table (say 0.03) silently fell through to a hardcoded 1.96, which is simply wrong for most inputs. The current tool computes the exact value for every input and never falls back.
Worked example: n = 30, α = 0.05
A two-tailed test on 30 observations means df = 29. Enter n = 30 (or df = 29 directly), keep α = 0.05 and two-tailed, and click Calculate.
The classic textbook value for df = 29 is 2.045, matching to three decimals. The stress case the old prose promised still holds: α = 0.01 two-tailed with df = 1 gives ±63.6567, a case where table interpolation fails completely. More real runs, all reproducible in the calculator:
| Inputs | t critical | Rejection region | Equivalent z |
|---|---|---|---|
| α = 0.05, two-tailed, df = 29 | ±2.0452 | |t| > 2.0452 | 1.9600 |
| α = 0.05, right-tailed, df = 10 | 1.8125 | t > 1.8125 | 1.6449 |
| α = 0.05, left-tailed, df = 20 | −1.7247 | t < −1.7247 | 1.6449 |
| α = 0.01, two-tailed, df = 1 | ±63.6567 | |t| > 63.6567 | 2.5758 |
| α = 0.03, two-tailed, df = 14 | ±2.4149 | |t| > 2.4149 | 2.1701 |
The last row is the one printed tables cannot give you: α = 0.03 appears in no standard t-table, and the old lookup version of this tool would have answered 1.96. The exact answer is 2.4149.
The small-sample penalty: t vs z
With few degrees of freedom the t-distribution has far heavier tails than the normal, so the critical value sits much further out than the familiar z of 1.96. The calculator quantifies that as a percentage on every run, and the mini t-table under the results shows the convergence: as df grows, t melts into z.
| df (α = 0.05 two-tailed) | t critical | How much wider than z |
|---|---|---|
| 1 | 12.7062 | 548.3% wider |
| 5 | 2.5706 | 31.2% wider |
| 10 | 2.2281 | 13.7% wider |
| 30 | 2.0423 | 4.2% wider |
| 100 | 1.9840 | 1.2% wider |
This is the honest reason the textbook rule of thumb says “use z when n is large”: at df = 100 the difference is 1.2%, at df = 1 it is a factor of six. Using 1.96 for a small-sample t-test understates the threshold and inflates false positives, which is exactly what the penalty readout is there to prevent.
What competing critical-value calculators get wrong
The ranking pages for this keyword fall into familiar camps:
- Exact calculators (SageCalculator’s critical-value tool): the closest rival: one- and two-tailed, any df, with a worked n = 15 example (df = 14, ±2.1448, which this calculator reproduces). Its gaps: no equivalent-z comparison, no small-sample penalty readout, no sample-size input, no mini-table.
- Static t-tables (MeraCalculator, StatsCalculators, EasyUnitConverter): correct numbers for the rows they print, but fixed alpha columns and fixed df rows. An alpha of 0.03 or a df of 137 sends you to interpolation, and interpolation on the t-tail is exactly where the error concentrates.
- Two-tailed-only tools (SageCalculator’s two-tailed page): add a z/t choice, but only answer the two-tailed question. A directional hypothesis needs the one-tailed boundary, which is a different number (1.8125 vs 2.2281 at df = 10).
- Tutorial pages (Sorumatik, Pearson’s Excel guide): good teaching of T.INV.2T and the SciPy/R equivalents, but no live tool: you read the formula, then open a spreadsheet anyway.
- The gap this page fills: exact inversion for arbitrary alpha and df with no fallback value, the rejection region stated in words (|t| > 2.0452, not just the number), the z comparison with the penalty percentage, and the sample-size shortcut, all on one page.
Critical value vs p-value: which question are you asking?
The two numbers answer different questions and get confused constantly. The critical value is the planning number: before you collect data, it tells you how extreme a result must be to count as significant. The p-value is the post-hoc number: after you have a t-statistic, it tells you how surprising that statistic is under the null. Look the critical value up once, then compare every t from your analysis against it; compute a p-value fresh for each t you observe. When you need to run the test itself from sample data first, start at the t-statistic calculator.
Sample size or degrees of freedom: which to enter
For a one-sample t-test, df = n − 1, so n = 30 gives df = 29. Enter whichever you have: the n field converts automatically and takes precedence if both are filled. For two-sample tests the df formula depends on the variant (pooled: n1 + n2 − 2; Welch’s: the approximation), so enter the df your software reports rather than forcing it through n − 1. The paired t-test is a one-sample test on the differences, so df = number of pairs − 1. If you are unsure which formula your design calls for, the degrees of freedom calculator works it out from your sample sizes.
Frequently asked questions
The t-statistic threshold beyond which you reject the null hypothesis at your chosen alpha. For a two-tailed test at alpha 0.05 with 10 degrees of freedom it is plus or minus 2.2281: any t further from zero than that is significant.
The critical value is the bar your statistic must clear, fixed before you see the data. The p-value measures how surprising your actual statistic is, computed after. Use this calculator for the bar; use the t-statistic p-value calculator for the p-value.
Yes. Enter n and the calculator sets df = n − 1 automatically. For a one-sample or paired test that is the whole conversion; for two-sample tests, enter the df your software reports instead.
At alpha 0.05 two-tailed with 10 degrees of freedom, t is 13.7 percent wider than z. The tool reports the exact penalty for your settings on every run, and the mini table shows the two converging as df grows.
Enter it anyway. The calculator inverts the t-distribution numerically for any alpha up to 0.5, so 0.03 gives the exact ±2.4149 at df = 14 instead of a table interpolation or a wrong fallback value.
A left-tailed test rejects for large negative t, so the boundary sits on the negative side: −1.7247 at alpha 0.05 with df = 20. Reject when your t is below that number (t < −1.7247).
For a two-tailed test, compare the absolute value: reject when |t| exceeds the critical value. For one-tailed tests, compare the signed t against the signed boundary: t > 1.8125 for right-tailed, t < −1.8125 for left-tailed (df = 10, alpha 0.05).
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References
- NIST/SEMATECH. e-Handbook of Statistical Methods: Two-Sample t-Test for Equal Means (t-test procedure, critical values, and the reject/fail-to-reject decision rule). Accessed October 24, 2026.
- R Project for Statistical Computing. The Student t Distribution: pt, qt, dt, rt (the qt quantile function is the reference implementation of the critical-value inversion this calculator is checked against). Accessed October 24, 2026.
- SciPy Community. scipy.stats.t: Student’s t continuous random variable (the ppf percent-point function computes critical values; isf gives one-tailed boundaries directly). Accessed October 24, 2026.
- Microsoft Support. T.INV.2T function (Excel’s two-tailed inverse t function, including the 2×probability rule for one-tailed values: T.INV.2T(0.05,10) = 2.28139, T.INV.2T(0.10,10) = 1.81246). Accessed October 24, 2026.
- Microsoft Support. T.DIST.2T function (Excel’s two-tailed t-distribution function, with the df ≥ 1 requirement). Accessed October 24, 2026.
- OpenStax, Rice University. Introductory Statistics (hypothesis testing, critical values vs p-values, and the t-distribution’s convergence to the normal). Accessed October 24, 2026.