tcdf Calculator
Compute tcdf(lower, upper, df) exactly as a TI-83 or TI-84 does: the area under Student's t-distribution between two values. With one bound left open, the result is the p-value for a t statistic.
Need a critical value instead? Use the invT calculator or the printable t-table.
Part of the TI-84 statistics functions guide, which shows when to use tcdf and every other DISTR-menu function.
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T-Table (Student's t Critical Values)
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What tcdf Does
tcdf is the cumulative distribution function of Student's t-distribution. It returns the area under the t curve between a lower and an upper bound, which is the probability that a t-distributed variable lands in that interval. Its main job in a statistics course is turning a t statistic into a p-value.
tcdf(a, b, df) = P(a ≤ T ≤ b)
Right-tailed p-value → tcdf(t, 1E99, df)
Left-tailed p-value → tcdf(-1E99, t, df)
Two-tailed p-value → 2 × tcdf(|t|, 1E99, df)
How to Use tcdf on a TI-84
- Press 2nd then VARS (DISTR).
- Scroll to 6:tcdf( and press ENTER.
- Enter lower, upper, and df. Use −1E99 or 1E99 for an open tail.
- Select Paste and press ENTER twice.
In Excel or Google Sheets, =T.DIST.RT(t, df) gives the right-tailed p-value and =T.DIST.2T(ABS(t), df) the two-tailed one.
Worked Example: p-Value for a One-Sample t-Test
A sample of n = 16 gives a test statistic of t = 2.1 for the alternative μ > μ₀. What is the p-value?
- Degrees of freedom: df = n − 1 = 15.
- Right-tailed test, so the p-value is tcdf(2.1, 1E99, 15) = 0.026528.
- If the alternative were μ ≠ μ₀, the two-tailed p-value would be twice the unrounded tail area, 2 × 0.0265276 = 0.053055.
At α = 0.05 the one-sided test rejects H₀ but the two-sided test does not — the choice of alternative has to be made before looking at the data. To run the whole test from raw data, use the t-test calculator.
tcdf vs. normalcdf
The t-distribution has heavier tails than the normal curve, so the same cutoff leaves more area in the tails — especially with few degrees of freedom. With df = 5, P(T > 2) ≈ 0.051, while the normal curve gives P(Z > 2) ≈ 0.023. As df grows the two converge; above about df = 100 they agree to two decimal places. Use normalcdf only when the population standard deviation is known.
Frequently Asked Questions
How do I find a p-value with tcdf?
Enter the t statistic as one bound and leave the other open. For a right-tailed test use tcdf(t, 1E99, df); for a left-tailed test use tcdf(-1E99, t, df); for a two-tailed test double the tail area beyond |t|. This page shows all three at once.
What degrees of freedom should I use in tcdf?
For a one-sample or paired t-test, df = n − 1. For a pooled two-sample t-test, df = n₁ + n₂ − 2. For Welch's test, use the Welch–Satterthwaite df, which is usually not a whole number — tcdf accepts decimals.
Why is my tcdf answer different from normalcdf?
The t-distribution has fatter tails than the standard normal to account for estimating the standard deviation from the sample. The difference is largest with small df and fades as df grows.
What does tcdf return if both bounds are equal?
Zero. For a continuous distribution the probability of exactly one value is 0, so tcdf(a, a, df) = 0 for any a.
Can tcdf take a negative t value?
Yes. A negative t means the sample mean is below the hypothesized mean. For a left-tailed test, tcdf(-1E99, t, df) with a negative t gives a small p-value; for a two-tailed test, only |t| matters.
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