Student's t-Distribution Calculator

Compute Student's t PDF, CDF, tail probabilities, critical values, and quantiles from degrees of freedom. This page calculates the distribution itself—not a t-test on sample data.

Calculation mode

Degrees of freedom (df)

df (ν ≥ 1)

1.00 1,000.00

t value

-20.00 20.00

Interval

Lower t (a)

Upper t (b)

Cumulative probability

p (0–1 exclusive)

0.00 1.00

Tail

Significance level α

Custom α

0.00 0.50

Confidence level

Custom confidence %

50.00 99.90

All calculations run in your browser. Inputs are not uploaded. This calculator evaluates the Student's t-distribution. For sample-based tests, use the T-Test Calculator.

Primary result

±t_crit = ±2.228139

Critical t for α = 0.05 (two-tailed), df = 10.

Mode

critical

Detail

—

Distribution properties

—

Calculation steps

df = 10
α = 0.05
tail = two
t_crit = 2.228139
Two-tailed: ±t_crit = ±2.228139
Each tail has probability α/2 = 0.025

Critical t table (two-tailed)

t distribution

How to Use the Student's t-Distribution Calculator

1. Choose the calculation mode: Critical t for a rejection-region cutoff. PDF for the density at a point. CDF for P(T ≤ t) and the upper tail. Tail probability for a one- or two-tailed area. Between for P(a ≤ T ≤ b). Quantile for t = F⁻¹(p). Confidence → t to convert a confidence level into a critical value. t table to generate a reference table.
2. Enter the degrees of freedom: Degrees of freedom determine the shape of the Student's t-distribution. Enter a positive df value supported by the selected calculation. Smaller df produce heavier tails, while larger df make the distribution increasingly similar to the standard normal.
3. Enter the value, probability, or confidence level: In Critical t mode, enter α and choose the tail (left, right, or two-tailed). In PDF, CDF, or Tail modes, enter t. In Between mode, enter the lower and upper t. In Quantile mode, enter p between 0 and 1. In Confidence mode, enter C as a percentage. Table mode uses a fixed df range and α columns defined by the calculator, so no per-run input is needed.
4. Read the primary result: The hero figure shows the primary output for the chosen mode — the critical t (with sign for one-tailed, ± for two-tailed), the PDF value, the cumulative or tail probability, the between probability, the quantile, or the confidence-level critical value. The stats panel repeats the mode and df, and the detail line summarises the numerical result.
5. Review the distribution properties and chart: The properties block reports mean, variance, mode, median, and symmetry at the current df, so you can see how the shape changes. The density chart plots the PDF and marks |t| and/or the critical |t| where applicable. Change df and the chart redraws.
6. Use the t table if you need reference values: The t table mode generates critical values for common df and two-tailed α levels using the same inverse-CDF engine as the calculator. Use it as a quick reference, or switch back to Critical t mode when you need a value for a df or α not listed in the table.

What Is the Student's t-Distribution?

The Student's t-distribution is a continuous, symmetric probability distribution centered at zero. It is used in statistical inference whenever the population standard deviation is unknown and estimated from the sample. Its shape depends on one parameter — the degrees of freedom (df, often written ν) — and it has heavier tails than the standard normal distribution. Heavier tails mean extreme values are more likely under the t-distribution than under the normal, which is why the two distributions give different critical values at the same confidence level.

What This Calculator Does

The calculator answers eight related questions about the t-distribution:

PDF — the density f(t) at a specific t value.
CDF — the lower-tail probability P(T ≤ t) and upper-tail P(T ≥ t).
Tail probability — one of the three tail areas: P(T ≤ t), P(T ≥ t), or two-tailed P(|T| ≥ |t|).
Between — the probability P(a ≤ T ≤ b) between two t values.
Quantile — the t value with a specified cumulative probability, t = F⁻¹(p).
Critical t — the quantile determined by df, α, and tail (left, right, or two-tailed).
Confidence → t — the critical value that matches a two-sided or one-sided confidence level.
t table — a generated critical-value table across common df and α levels.

Every mode uses the same numerical engine, so a value computed in one mode matches the same value when it appears in another. All calculations run in the browser.

t-Distribution Formula

The probability density function (PDF) is:

f(t) = Γ((ν + 1) / 2) / [√(νπ) · Γ(ν / 2)] · (1 + t² / ν)^(-(ν + 1) / 2)

t — the value on the horizontal axis.
ν (df) — degrees of freedom, a positive number.
Γ — the gamma function, which extends the factorial to real numbers.

The PDF gives the density at a point. For a continuous distribution, that is not the probability of observing exactly t — it is the height of the curve at t. The cumulative distribution function (CDF) is the integral of the PDF from −∞ to t, and it gives P(T ≤ t). The upper-tail probability is P(T ≥ t) = 1 − F(t). For extreme upper tails, the calculator uses a numerically stable survival-function calculation to avoid cancellation. By symmetry, P(T ≤ −t) = P(T ≥ t).

Degrees of Freedom

Degrees of freedom control the shape of the distribution. Small df give a flatter peak and much heavier tails; large df pull the curve toward the standard normal shape. Notable cases:

df = 1 — the t-distribution reduces to the Cauchy distribution. The mean and variance are undefined.
df = 2 — the mean is zero, but the variance is not finite.
df > 2 — mean 0, variance ν / (ν − 2).
df ≈ 30 — already close to the normal in the central region, but still noticeably different in the far tails.
df → ∞ — the t-distribution converges to the standard normal. In practice, t and z values are very close once df exceeds about 100.

Critical Values and Tails

A critical t value is a quantile of the t-distribution that depends on df and α — the significance level. The tail choice determines where the probability sits:

Left-tailed — critical value at F⁻¹(α). Rejection region in the lower tail.
Right-tailed — critical value at F⁻¹(1 − α). Rejection region in the upper tail.
Two-tailed — critical value at F⁻¹(1 − α / 2), symmetric ±t_crit. The α probability is split evenly between the two tails, α / 2 in each.

For a fixed df and α, the two-tailed critical value is always larger than the one-tailed value, because the total rejection probability is divided between both ends of the distribution.

t Table vs Calculated Values

Traditional t tables list critical values at a small set of α levels (usually 0.10, 0.05, 0.025, 0.01, 0.005, 0.001) and a small set of df values. The table mode here generates the same kind of table using the inverse CDF, so it agrees with the underlying math to full precision. The calculated value from the Critical t mode is more flexible — you can use any df, any α, and any tail combination.

Quantiles and Percentiles

A quantile is the inverse of the CDF. Given a cumulative probability p, the quantile is the t value such that P(T ≤ t) = p. The quantile depends on both p and the degrees of freedom, so the same probability can map to different t values at different df. Because the distribution is symmetric about zero, F⁻¹(1 − p) = −F⁻¹(p), so only one side needs to be calculated.

t vs Normal

The t-distribution has heavier tails than the normal because it accounts for the extra uncertainty from estimating the standard deviation from a finite sample. For small samples the difference is large enough to matter — the t critical value at df = 5 and α = 0.05 is 2.571, while the corresponding normal quantile is 1.960. For large samples the two converge: at df = 1000, the t critical value is 1.962, barely distinguishable from the normal.

t-Distribution vs t-Test

These are related but not the same thing. The t-distribution is a probability distribution — the mathematical object this calculator evaluates. A t-test is a statistical procedure that uses the distribution to compute test statistics, p-values, and confidence intervals from sample data. Use this calculator when you need a t quantile, critical value, PDF, CDF, or tail probability. Use a dedicated t-test calculator when you have sample data and want to test a hypothesis about a mean or mean difference.

Worked Examples

Critical t, df = 10, α = 0.05, two-tailed. The calculator returns ±2.2281. Each tail holds α / 2 = 0.025.

Upper-tail probability, t = 2.2281, df = 10. P(T ≥ 2.2281) ≈ 0.025, and the two-tailed probability P(|T| ≥ 2.2281) ≈ 0.05.

CDF at t = 1.8125, df = 10. P(T ≤ 1.8125) ≈ 0.95, so the upper tail is ≈ 0.05.

97.5th percentile, df = 10. F⁻¹(0.975) = 2.2281, matching the two-tailed critical value at α = 0.05.

Probability between t = −1 and t = 1, df = 10. P(−1 ≤ T ≤ 1) ≈ 0.6591, so about 66% of the distribution lies within one unit of zero at this df.

Common Mistakes

Using z instead of t when df is small. The two differ noticeably below df ≈ 30, especially in the tails.
Mixing one-tailed and two-tailed α. A one-tailed α = 0.05 and a two-tailed α = 0.05 use different critical values.
Treating the PDF as a probability. The PDF gives density, not P(T = t) — for continuous distributions, that probability is zero.
Forgetting that df depends on the test. For a one-sample t-test, df = n − 1; for Welch's test, df is fractional; for a paired test, df = number of pairs − 1.
Assuming df = 1 behaves like the normal. At df = 1 the distribution is Cauchy and has undefined moments.

Frequently Asked Questions

Q: What is a t-distribution calculator?

A t-distribution calculator computes mathematical quantities from the Student's t-distribution — PDF values, cumulative probabilities, tail probabilities, critical t values, and quantiles. It works from degrees of freedom and does not require sample data. This is a distribution calculator, not a t-test.

Q: What is a t critical value?

A t critical value is a quantile of the t-distribution that corresponds to a specified tail probability. It is determined by degrees of freedom and α (or a confidence level). For a two-tailed test the critical value is symmetric, ±t_crit, and each tail holds α / 2 probability.

Q: How do I calculate a critical t value?

For a right-tailed test, t_crit = F⁻¹(1 − α). For a left-tailed test, t_crit = F⁻¹(α). For a two-tailed test, t_crit = F⁻¹(1 − α / 2). F⁻¹ is the inverse CDF of the t-distribution with the specified df. This calculator does all three from the tail selector.

Q: What is the difference between one-tailed and two-tailed t values?

A one-tailed critical value puts the entire α probability into a single tail. A two-tailed critical value splits α evenly, α / 2 in each tail. For the same df and α, the two-tailed value is larger because the total rejection probability is divided between two sides.

Q: What is the t-distribution PDF?

The PDF is the density of the distribution at a point t. It gives the height of the curve, not the probability of observing exactly t — for a continuous distribution, that probability is zero. The PDF formula uses the gamma function and t² / ν inside a power term.

Q: What is the t-distribution CDF?

The CDF gives P(T ≤ t), the probability that a t-distributed random variable falls at or below t. It is the integral of the PDF from −∞ to t, and it ranges from 0 to 1. The upper tail is 1 − CDF(t).

Q: What is a t-distribution quantile?

A quantile is the inverse CDF: the t value such that P(T ≤ t) = p for a given cumulative probability p. The quantile depends on both p and the degrees of freedom, so the same probability can map to different t values at different df. Because the t-distribution is symmetric about zero, F⁻¹(1 − p) = −F⁻¹(p).

Q: How are confidence levels related to critical t values?

For a two-sided confidence level C, α = 1 − C, and the critical value is the upper quantile F⁻¹(1 − α / 2). For a one-sided confidence level, the critical value is F⁻¹(1 − α). This calculator's Confidence → t mode takes C directly and returns the matching t value.

Q: Why does the t-distribution have heavier tails than the normal?

The t-distribution accounts for the extra uncertainty from estimating the population standard deviation from a finite sample. This makes extreme values more likely than under the standard normal, especially when the sample — and hence df — is small. At df = 5, α = 0.05, the two-tailed critical t is 2.571 versus a normal quantile of 1.960.

Q: When does the t-distribution approach the normal?

As df increases, the t-distribution approaches the standard normal. The convergence is fast in the central region and slower in the far tails. By df = 30 the two are similar in the body; by df = 1000 the critical values differ by less than 0.002. For small df, the t should not be replaced by the normal.

Q: What happens when df = 1?

With one degree of freedom, the t-distribution is equivalent to the Cauchy distribution. The tails are extremely heavy, and neither the mean nor the variance is defined. The usual rules of thumb for the normal approximation do not apply at df = 1.

Q: Is the t-distribution the same as a t-test?

No. The t-distribution is a continuous probability distribution — a mathematical object used in statistical inference. A t-test is a statistical procedure that applies the distribution to sample data to test a hypothesis about a mean or mean difference. Use this calculator for distribution quantities; use a dedicated t-test calculator for sample-based tests.

Q: What is the difference between a t value and a critical t value?

A t value is any point on the horizontal axis of the t-distribution. A critical t value is a specific quantile chosen by df and α (or a confidence level) that defines the boundary of a rejection region. Only the critical value depends on the significance level; any real number is a valid t value.

Q: Can I calculate a t value from a probability?

Yes. The Quantile mode computes t = F⁻¹(p) from any cumulative probability p between 0 and 1 (exclusive). This is also called calculating a t percentile or inverse t. The result depends on df as well as p.

Q: How is the t table generated?

The t table is generated by evaluating the inverse t CDF at each combination of df and two-tailed α listed in the table's rows and columns. The values are computed by the same engine as the single-value Critical t mode, so the table and the calculator always agree.