t Distribution Calculator

What Does a t Distribution Calculator Do?

A t distribution calculator turns a t statistic into a probability, or a probability into a critical t value, for the degrees of freedom you choose. Give it df = 15 and t = 2.3 and it tells you the two-tailed p-value is 0.0362. Give it df = 9 and a 95% confidence level and it returns the critical value 2.2622.

The t distribution looks like the bell curve, only with heavier tails. Those fatter tails are the price of estimating the standard deviation from a small sample instead of knowing it. The smaller the sample, the heavier the tails, and the bigger the critical value you need before a result counts as unusual.

This page does both directions in one place: p-value from t, and critical t from a probability. It handles left-tail, right-tail, two-tailed and middle areas, accepts decimal degrees of freedom for Welch tests, and shows the curve with your area shaded so you can check the setup at a glance.

How Do You Use This t Distribution Calculator?

Choose what you want to find, enter the degrees of freedom, then fill in the t value or the probability.

1. Pick the direction. "p-value from t" starts with a t statistic and returns an area. "Critical t from a probability" starts with an area and returns the t value that cuts it off.
2. Enter the degrees of freedom. For a one-sample or paired t test this is n − 1. For a two-sample pooled test it is n₁ + n₂ − 2. A Welch test gives a decimal, and decimals are fine here.
3. Choose the tail. Left tail is the area below t. Right tail is the area above t. Two-tailed is the area in both tails beyond ±|t|. Middle is the area between −|t| and +|t|.
4. Enter t or the area. Areas can be decimals such as 0.025 or percentages such as 95%.

Everything updates as you type. The result cards show the answer and its complement, the working shows which formula was used, and the table at the bottom lists critical values for your degrees of freedom at the usual confidence levels.

What Is the t Distribution?

The t distribution, also called Student's t, is the sampling distribution of a standardized sample mean when the population standard deviation is unknown. William Sealy Gosset described it in 1908 while working at the Guinness brewery, and published under the pen name "Student".

When you draw a sample of n values from a normal population and compute t = (x̄ − μ) / (s / √n), that t follows a t distribution with n − 1 degrees of freedom. The extra spread, compared with a z-score, comes from the fact that s is itself a noisy estimate of σ. Sometimes s lands too small and t gets large, so large values show up more often than a normal curve would predict.

The distribution is symmetric around zero, bell-shaped and has a single parameter, the degrees of freedom ν. Its mean is 0 for ν > 1, its variance is ν / (ν − 2) for ν > 2, and as ν grows it settles into the standard normal curve. At ν = 5 the variance is 1.6667 and the peak height is 0.3796, against 0.3989 for the standard normal. The normal curve calculator shows that reference shape.

What Are Degrees of Freedom?

Degrees of freedom count how many values in your data are free to vary after you have used some of them to estimate something. A sample mean uses up one, which is why the usual df for a single sample is n − 1.

Take five numbers whose mean is fixed at 10. Pick the first four however you like, and the fifth is forced. Only four values are free, so df = 4. The same logic explains df = n − 2 in a pooled two-sample test (two means estimated) and in simple regression (a slope and an intercept).

The Welch test for two samples with unequal spread uses the Welch–Satterthwaite formula, which usually gives a non-integer df. With 12 values at SD 4.1 and 15 values at SD 5.6 it comes to 24.85, slightly under the pooled 25. You can type 24.85 straight into the calculator. The standard deviation calculator and the variance calculator give you the s and s² those formulas need.

How Do You Find a p-Value From a t Statistic?

You compute the area in the tail or tails beyond your t value, using the t distribution with your degrees of freedom. The tail you pick depends on the alternative hypothesis.

• Right-tailed test (H₁: μ > μ₀): p = P(T ≥ t).
• Left-tailed test (H₁: μ < μ₀): p = P(T ≤ t).
• Two-tailed test (H₁: μ ≠ μ₀): p = 2 × P(T ≥ |t|).

Because the curve is symmetric, the two-tailed p-value is exactly double the one-tailed value for the same |t|. With t = 2.3 and df = 15, the right tail is 0.0181 and the two-tailed p-value is 0.0362.

Compare p to your significance level α. If p ≤ α, reject the null hypothesis. If p > α, you do not have enough evidence to reject it. A p-value is the chance of seeing a result at least this extreme if the null hypothesis were true. It is not the chance that the null hypothesis is true.

Worked Example: One-Sample t Test

A machine is set to fill packs to 50 g. A sample of 12 packs averages 52.3 g with a sample standard deviation of 4.1 g. Is the true mean different from 50?

The test statistic is t = (52.3 − 50) / (4.1 / √12) = 1.9433, with df = 12 − 1 = 11. Enter t = 1.9433 and df = 11, choose two-tailed, and the calculator gives p = 0.0780.

At α = 0.05 the critical values are ±2.2010, and 1.9433 falls inside them, so you fail to reject. The sample suggests the machine runs heavy, but twelve packs with that much scatter are not enough to be sure. If the claim had been one-sided, "heavier than 50 g", the right-tail p-value would be 0.0390 and the same data would be significant at 5%. That is why you choose the tail before you look at the data.

How Do You Find a Critical t Value?

A critical t value is the cut-off that leaves a chosen area in the tail or tails. You use it for confidence intervals and for deciding a test without computing a p-value. The calculator finds it by working the t CDF backwards, the same idea as the inverse normal calculator but with heavier tails.

Switch to "Critical t from a probability", enter the degrees of freedom, and choose how the area is described:

• Two-tailed α: the total area in both tails, such as 0.05. The result is the ±t* pair.
• Confidence level: the middle area, such as 95%. It is the same as a two-tailed α of 0.05.
• Right tail or left tail: the area on one side only.

For df = 9 and a 95% confidence level, the critical value is 2.2622. For a one-sided 5% test with df = 14, it is 1.7613. For df = 20 and a left-tail area of 0.99, it is 2.5280.

Which Critical Values Come Up Most Often?

This table lists two-sided critical values for the three common confidence levels, checked against SciPy. The bottom row is the standard normal, which is what the t values approach as the degrees of freedom grow.

With one degree of freedom, a 95% interval needs 12.71 standard errors on each side. By df = 30 that is down to 2.04, and by df = 120 it is 1.98, almost the familiar 1.96.

Worked Example: A 95% Confidence Interval for a Mean

A sample of 25 readings has a mean of 18.4 and a standard deviation of 3.2. Build a 95% confidence interval for the population mean.

The degrees of freedom are 24, and the critical value for a 95% confidence level is t* = 2.0639. The standard error is 3.2 / √25 = 0.64. The margin of error is 2.0639 × 0.64 = 1.3209, so the interval is 18.4 ± 1.3209, or 17.08 to 19.72.

Using z = 1.96 instead would give a margin of 1.2544 and an interval about 5% too narrow. For small samples, that is the difference between a 95% interval and one that captures the truth closer to 94% of the time. Use t whenever the standard deviation comes from the sample.

Worked Example: Two Samples With Unequal Spread

Group A has 12 values with a standard deviation of 4.1. Group B has 15 values with a standard deviation of 5.6. A Welch test uses the squared standard errors, 4.1²/12 = 1.4008 and 5.6²/15 = 2.0907, and the Welch–Satterthwaite formula gives df = 24.85.

Enter 24.85 as the degrees of freedom and ask for a 95% confidence level. The critical value is 2.0602. The pooled-variance test would use df = 25 and a critical value of 2.0595. The gap is tiny here because the samples are similar in size and spread. It grows when the groups differ a lot.

Decimal degrees of freedom are not a rounding mistake. The t distribution is defined for any positive df, and rounding down to a whole number is the conservative choice if you only have printed tables to work with.

When Should You Use t Instead of z?

Use t when the population standard deviation is unknown and you estimated it from the sample, which is almost always the case with real data. Use z when σ is known, or when the sample is so large that t and z are practically the same.

At df = 10, a t of 1.96 has a two-tailed p-value of 0.0784. The same 1.96 as a z-score gives 0.0500. Treating a small-sample t as a z-score understates the p-value and overstates the evidence. The normal CDF calculator and the Gaussian distribution calculator are the right tools for z-scores when σ is genuinely known.

The t procedures assume the data come from a roughly normal population, or that the sample is large enough for the central limit theorem to help. With very skewed data and a small sample, the t test can mislead, and a rank-based test is safer. For counts of successes in a fixed number of trials, the binomial distribution calculator is the exact tool.

What Mistakes Do People Make With t Calculations?

• Using the wrong degrees of freedom. One sample is n − 1, not n. A pooled two-sample test is n₁ + n₂ − 2. A paired test uses the number of pairs minus one.
• Mixing up one-tailed and two-tailed. Doubling the right-tail area gives the two-tailed p-value. Forgetting to double it halves your p-value.
• Entering a percentage as a decimal. A 95% confidence level means a two-tailed α of 0.05, not 0.95. The calculator accepts both, but check which box you are in.
• Dropping the sign. A negative t on a right-tailed test gives a p-value above 0.5, which is the correct signal that the data point the other way.
• Using the standard deviation when you need the standard error. The t statistic divides by s / √n, not by s. The mean, median and mode calculator and the coefficient of variation calculator help you check the inputs.
• Treating a large p-value as proof of no effect. It only means the data do not rule out the null hypothesis.

How Do You Calculate the t Distribution in Excel, R, Python and a TI-84?

Every tool has functions for the CDF and its inverse. Excel is the one with the most variants, so pay attention to which tail each function returns.

The older Excel functions TDIST and TINV still work, but TDIST only accepts positive t values and TINV is two-tailed by default. The newer names make the tail explicit. The calculator prints these lines with your numbers filled in, so you can paste one into a spreadsheet and compare. invT is on newer TI-84 Plus CE software; on older calculators use a solver or the table.

How Does the t Distribution Compare With the Normal Curve?

At the center, the t curve is a little lower than the normal curve, and in the tails it is higher. That moves probability out of the middle and into the extremes. The effect fades as the degrees of freedom increase.

Three facts are worth remembering. First, the variance is ν / (ν − 2), always above 1, so a t variable is more spread out than a z-score. Second, the t distribution has no variance at all for ν ≤ 2, because the tails are so heavy, and no mean for ν = 1, where it becomes the Cauchy distribution. Third, for ν above about 30 the 95% critical values sit only about 4% above the normal one, which is why old textbooks tell you to switch to z at n = 30. Software makes that shortcut unnecessary, so use the t value that matches your df.

If you want a cut-off for a normal distribution instead, the inverse normal calculator gives the z value and the x value for any tail.

Where Else Does the t Distribution Show Up?

• Regression. Every coefficient in a linear regression has a t statistic with n − k − 1 degrees of freedom, and the p-values in the output table come from it.
• Paired comparisons. Before-and-after measurements are reduced to differences, then tested as a one-sample t.
• Quality checks. A lab testing eight samples of a batch against a target uses df = 7 and a critical t of 2.3646 at the 95% level.
• Prediction intervals. A range for a single future value uses t with a factor of √(1 + 1/n) on the standard error.

In each case the pattern is the same. Find the t statistic, find the degrees of freedom, and look up the tail area, or ask for the critical value and build the interval.

Frequently Asked Questions

Sources: Student (W. S. Gosset), "The Probable Error of a Mean", Biometrika 6(1), 1908. NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.4 Student's t Distribution. SciPy documentation, scipy.stats.t. R documentation, TDist.

Related Calculators

Working with a known standard deviation instead? These cover the normal distribution and the wider library, or browse the full list of free statistics calculators:

Inverse Normal Distribution CalculatorFind the cut-off value for any area, tail or interval. Normal CDF CalculatorArea under the normal curve between any two bounds. Normal Curve CalculatorPlot the bell curve and shade any region. All StatsSolve CalculatorsBrowse the full directory of free statistics tools.