T-Distribution Calculator (tcdf & invT)

This free t-distribution calculator finds probabilities and critical values from Student’s t-distribution: tcdf for left, right, between or two-tailed areas, invT for critical t* values (left, right or center), tpdf for the height of the curve, and a t-score from x̄, μ, s and n with one- and two-tailed p-values. It draws a shaded t-curve next to the matching normal curve, shows step-by-step working with your numbers, a critical t* table, and the exact TI-84 keystrokes. Results update as you type.

Contents

How to Use the T-Distribution Calculator

1. Enter the degrees of freedom df. For a one-sample mean this is n − 1. A two-sample (Welch) df can be a decimal; any value greater than 0 is allowed.
2. Pick a tab. tcdf finds an area (probability). invT finds the t* that cuts off a given area — use CENTER and 0.95 (or 95%) for a 95% confidence interval. tpdf gives the height of the curve. t-score turns x̄, μ₀, s and n into a t statistic and p-values.
3. Read the purple boxes and the shaded curve. The dashed line is the standard normal, so you can see how much heavier the t tails are at your df.
4. Open the extras: step-by-step working with your numbers, the TI-84 keys (including the invT left-tail conversion), and a critical t* table for common confidence levels.

Results update as you type. Copy share link saves your inputs in the URL. Need the full interval x̄ ± t* · s/√n? Use the confidence interval calculator. For a reject / fail-to-reject decision, use the hypothesis test calculator.

What Is the t-Distribution?

Student’s t-distribution is the sampling distribution of the t-statistic

t = (x̄ − μ) / (s / √n)  with  df = n − 1

when the data are roughly normal (or n is reasonably large) and the population standard deviation σ is unknown and replaced by the sample standard deviation s. The curve is bell-shaped and symmetric about 0, like the standard normal, but it has heavier tails: more probability sits far from 0, because estimating σ from the sample adds extra uncertainty. That is why a 95% t critical value is larger than the familiar z* = 1.96, especially at small df.

The single extra parameter is the degrees of freedom. As df grows, the t-curve tightens toward the standard normal; by a few hundred df the two are almost indistinguishable. At df = 1 (the Cauchy distribution) the tails are so heavy that the mean is not even defined.

tcdf, invT and tpdf: Which One Do You Need?

The TI-84 DISTR menu has three t commands. Use this table to pick the right one:

Command	What it answers	TI-84 syntax
tcdf	Probability / area under the t-curve between two bounds	tcdf(lower, upper, df)
invT	The t value with a given area to its left	invT(area, df)
tpdf	The height of the curve at a point (not a probability)	tpdf(x, df)

The most common mix-up is asking invT for a 95% confidence interval and typing invT(0.95, df). That is the left-tail 0.95 cutoff (one-sided). A two-sided 95% interval needs a left-tail area of 0.975: invT(0.975, df). This calculator’s CENTER option does that conversion for you. The normal-distribution analogue is the normal distribution calculator (normalcdf and invNorm); use that when σ is known, and this page when you only have s.

T-Distribution Formula

The probability density (what tpdf returns) is

f(t) = Γ((df+1)/2) / (√(df π) · Γ(df/2)) · (1 + t²/df)^−(df+1)/2

The cumulative probability P(T ≤ t) is the integral of that density from −∞ to t, which is written in terms of the regularized incomplete beta function. You do not need to evaluate this by hand; tcdf and invT do it. Useful facts:

• Mean = 0 when df > 1 (undefined at df = 1).
• Variance = df / (df − 2) when df > 2 (undefined at df ≤ 2). At df = 24 that is 24/22 ≈ 1.091, so the SD is about 1.044, a little more than the normal’s 1.
• The curve is symmetric: P(T ≤ −t) = P(T ≥ t), and P(−t ≤ T ≤ t) = 1 − 2 P(T ≥ t).

Worked Example: Critical t* for a 95% CI (n = 25)

The calculator opens with this standard problem: you have a sample of n = 25 observations and want the critical t* for a 95% confidence interval for the mean. What is t*, and what is P(−2.0639 ≤ T ≤ 2.0639)?

1. df = n − 1 = 24.
2. A 95% interval leaves 5% outside, split as 2.5% in each tail, so the left-tail area for invT is 0.975.
3. t* = invT(0.975, 24) = 2.0639. The two critical values are −2.0639 and 2.0639. The interval is x̄ ± 2.0639 × s/√25.
4. Check the area: tcdf(−2.0639, 2.0639, 24) = 0.9500 (95.00%), as required.
5. The usual trap: invT(0.95, 24) = 1.7109, which is the cutoff for a one-sided 95% area (or a 90% two-sided interval). That is what the TI-84 returns if you forget to convert 0.95 into 0.975.
6. The matching z* for a 95% interval is 1.960. The t value is larger because s was estimated from 25 observations. At this df the difference is about 0.10.
7. The peak of the curve is tpdf(0, 24) = 0.3948, a little below the normal peak 0.3989.

	t (df = 24)	z (normal)
95% two-sided critical value	2.0639	1.9600
90% two-sided / 95% one-sided	1.7109	1.6449
99% two-sided critical value	2.7969	2.5758

invT Tails: Left, Right and Center

On a TI-84, invT(area, df) always uses the area to the left of the point — there is no RIGHT or CENTER argument, unlike invNorm on a TI-84 Plus CE. This calculator adds those options and shows the left-tail area you should type on the calculator:

• LEFT — the TI-84 default. invT(0.05, 24) = −1.7109, the lower cutoff for a left-tailed test at α = 0.05.
• RIGHT — area in the upper tail. A right-tailed test at α = 0.05 is invT(0.95, 24) = 1.7109. The calculator converts this for you.
• CENTER — the middle area, which is what a confidence interval or a two-sided test needs. CENTER 0.95 with df = 24 returns ±2.0639, and the TI-84 command is invT(0.975, 24).

Quick conversions:

C% confidence interval → t* = invT((1 + C)/2, df)
Right-tailed test at α → t* = invT(1 − α, df)
Two-tailed test at α → t* = invT(1 − α/2, df)

invT itself was added in TI-84 Plus OS 2.30. It is not on the TI-83 or on a TI-84 that has never been updated; in that case use the critical t* table on this page (or a printed t-table).

How to Find a t-Score

A t-score (t-statistic) standardises a sample mean when σ is unknown:

t = (x̄ − μ₀) / (s / √n)  df = n − 1

The t-score tab opens with a second worked example: a class mean of x̄ = 72 from n = 25 students, s = 9, tested against μ₀ = 70.

1. SE = s/√n = 9/√25 = 1.8.
2. t = (72 − 70) / 1.8 = 1.1111, with df = 24.
3. Right-tail p = tcdf(1.1111, 1E99, 24) = 0.1388.
4. Left-tail p = tcdf(−1E99, 1.1111, 24) = 0.8612.
5. Two-tailed p = 2 × 0.1388 = 0.2775 (27.75%).

That two-tailed p-value is well above 0.05, and |t| = 1.1111 is far below the 95% critical value 2.0639, so the mean is not significantly different from 70 at the usual levels. If σ had been known and equal to 9, the z-score would also be 1.1111 but the two-tailed normal p-value would be 0.2665 — slightly smaller, because the normal does not charge you for estimating σ. For the official test with a null and alternative and a reject / fail-to-reject wording, switch to the hypothesis test calculator.

t vs z: When the t-Curve Approaches Normal

Use t when the population SD is unknown and you plug in s. Use z (the z-score calculator or normal calculator) when σ is given, or for a proportion (the standard error then comes from a known p or from p̂(1 − p̂), not from an estimated σ). The old “n ≥ 30, just use z” rule is a shortcut, not a theorem: at n = 30 (df = 29) the 95% t* is still 2.045, not 1.960. The table below shows how quickly t* falls toward 1.960 as df grows.

df	5	10	24	29	60	120	∞ (z)
95% t*	2.5706	2.2281	2.0639	2.0452	2.0003	1.9799	1.9600

The dashed normal curve on the plot makes the same point visually: at df = 24 the t-curve is only a little lower in the middle and a little higher in the tails; at df = 1 the difference is obvious.

How to Use tcdf and invT on a TI-84

tcdf (area / probability)

1. Press 2nd VARS (DISTR).
2. Choose 6:tcdf(.
3. Enter lower, upper and df. For a left tail use lower -1E99; for a right tail use upper 1E99. Type −1E99 as (−) 1 2nd , 9 9.
4. Highlight Paste, press ENTER, then ENTER again.

For the default area: tcdf(-2.0639,2.0639,24) returns 0.9500. On a TI-84 Plus CE you can also shade the curve with DISTR ▸ DRAW ▸ Shade_t( and the same three arguments.

invT (critical value)

1. Press 2nd VARS (DISTR) and choose 4:invT(.
2. Enter the left-tail area and df. For a 95% confidence interval that area is 0.975, not 0.95.
3. Paste and press ENTER twice. invT(0.975,24) returns 2.0639.

invT is not on the TI-83, and not on a TI-84 running a system older than OS 2.30. Update the OS from Texas Instruments, or use the critical t* table above.

tpdf (height)

DISTR 5:tpdf( takes x and df and returns the height of the curve. That is useful for a sketch, not for a probability — areas always come from tcdf. tpdf(0,24) returns 0.3948.

How This Differs from the Normal, Z-Score, CI and Hypothesis Test Pages

• The normal distribution calculator is the right tool when the variable is normal with a known mean and σ, or when you already have a z-score. It uses normalcdf, invNorm and normalpdf. This page is the t analogue: same kinds of area and inverse questions, but with df instead of μ and σ.
• The z-score calculator converts a raw score to z, or z to a percentile, under a normal model. A t-score uses s and n instead of σ, and its p-value comes from the t-curve, not the z-curve.
• The confidence interval calculator builds the actual interval x̄ ± t* · s/√n (and intervals for proportions and two samples). This page is where you look up t* itself, check an area under the t-curve, or see why invT(0.95, df) is the wrong keystroke for a 95% interval.
• The hypothesis test calculator runs a complete t-test (or z-test, proportion test, two-sample or paired test) with H₀, H_a and a reject / fail-to-reject decision. This page stops at the t-score and the p-value, which is what you want when the question is “find t*” or “find P(T ≥ 1.71).”

Browse every tool on the math and statistics calculators hub, or try the same numbers on the TI-84 Plus CE calculator online.

Common Mistakes

• Typing invT(0.95, df) for a 95% confidence interval. That returns the one-sided cutoff (1.7109 at df = 24). Use invT(0.975, df), or CENTER 0.95 in this calculator, to get 2.0639.
• Using z / invNorm when σ is unknown. If you only have s, the critical value and the p-value should come from t, not z. The difference is noticeable below about 30 df.
• Forgetting df = n − 1 (or using n, or n − 2). A 95% interval with n = 25 must use df = 24, not 25.
• Giving tcdf only one bound. tcdf always wants a lower and an upper. Left tail: tcdf(-1E99, t, df). Right tail: tcdf(t, 1E99, df).
• Treating tpdf as a probability. tpdf is a height. The area (the probability) is tcdf.
• Using t for a proportion. Tests and intervals for a proportion use z, not t.
• Invoking “n ≥ 30 so use z.” You can still use t at any n; it is the more accurate choice whenever σ is estimated from the sample.

Why This T-Distribution Calculator Is Different

• tcdf, invT, tpdf and a t-score tab in one place, with left, right, between and two-tailed areas — the four questions students actually get.
• invT LEFT, RIGHT and CENTER, with the left-tail area written out so the TI-84 keystrokes match. CENTER is the option the calculator itself does not have.
• A shaded t-curve next to the normal curve, so the heavier tails are visible at your df, plus a critical t* table for common confidence levels.
• Step-by-step working with your numbers and the exact tcdf, invT, tpdf and Shade_t keys, including the OS 2.30 / TI-83 note for invT.
• Accurate tails and inverse values, checked against SciPy’s Student t on thousands of random cases (df from 1 up, including non-integer Welch df).
• Dark mode, share links and a phone-friendly layout.

This is the page to use when the question is about the t-curve, t*, tcdf or invT. Use the confidence-interval and hypothesis-test calculators when you want the interval or the decision built on top of that t*.

T-Distribution Calculator FAQ

TI 84 Calc PRO is not affiliated with or endorsed by Texas Instruments. TI-84 Plus CE is a trademark of Texas Instruments.