t Distribution Calculator: Get One-Tailed and Two-Tailed Probabilities Fast (With Examples)

Q: What does this result mean?

It’s the area under the t curve . For probabilities, it tells you “how much of the distribution is left/right of your t.” For two-tailed p-values, it tells you “how likely a t value at least this extreme is in either direction.”

Q: What is df in a t distribution?

df means degrees of freedom , and in many intro stats settings it’s n − 1 for one-sample t procedures. df affects how heavy the tails are.

Q: What’s the difference between one-tailed and two-tailed probabilities?

One-tailed looks in one direction only (left or right). Two-tailed looks for “extreme on either side,” so it uses both tails beyond ±∣t∣\pm|t|.

Q: Which test should I use: z-test or t-test?

Use a t-test/t distribution when the population standard deviation σ is unknown (you use sample s) and/or when the sample is small (depending on your course rules). Use z when σ is known or your class tells you to use z for that situation.

Q: Why is my answer different from my teacher’s?

Common reasons: You used df = n instead of df = n − 1 . You used one-tailed but the problem is two-tailed (or vice versa). Your teacher rounded differently (tables often round t values and areas). You used P(T≤t)P(T \le t) when they wanted P(T≥t)P(T \ge t) (or the opposite).

Q: How does this connect to confidence intervals?

Confidence intervals use critical t : xˉ±t∗(sn)\bar{x} \pm t^*\left(\frac{s}{\sqrt{n}}\right)So if you can find t∗t^* (given df and α), you can build the interval endpoints.

Q: Does the t distribution become normal?

Yes— as df increases , the t distribution approaches the standard normal distribution (the curves become very similar).

Stuck on df, tails, and t-table lookups? This t distribution calculator helps you find left-tail, right-tail, and two-tailed probabilities from a t value and degrees of freedom (df). It also gives critical t from df and alpha (α), which you’ll use for hypothesis tests and confidence intervals.

You’ll see both the math notation and plain-language labels (so you don’t have to guess what $P(T \le t)$ means).

t Distribution Calculator

Use this tool to get t probabilities or critical t values (based on df and tails).

What this means

The t distribution is used when your sample is small and the population standard deviation is unknown. Larger df makes the t curve closer to the normal curve.

Next, explore the Distributions hub (internal link) to compare t vs z and see when each distribution is used.

What this calculator does

This calculator has two modes:

Probability from t (given t and df)
It returns probabilities such as:

Left-tailed: area to the left of your t value, $P(T \le t)$ P(T≤t)
Right-tailed: area to the right of your t value, $P(T \ge t)$ P(T≥t)
Two-tailed p-value: both tails beyond $\pm|t|$ ±∣t∣, $P(|T|\ge|t|)$ P(∣T∣≥∣t∣)

Critical t (given df and alpha)
It returns the cutoff t value(s) for:

One-tailed tests (right or left)
Two-tailed tests (± critical t)

Quick df refresher (because df causes 80% of the confusion)

For most intro stats t-problems, df is tied to sample size:

One-sample t procedures: df = n − 1
(Example: if n = 12, df = 11)

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When to use it (and when not to)

Use this calculator when:

Your problem involves the t distribution (common in small samples).
You’re doing inference about a mean and the population standard deviation is unknown.
You need one-tailed or two-tailed probabilities, p-values, or critical t.

Don’t use it when:

You’re explicitly told to use a z distribution (or you truly know the population σ and your course treats that as z).
Your question is about chi-square, F, or normal probabilities instead.
Your professor requires a t-table-only answer format (you can still use this to check, but your final submission might need table steps).

How it works (simple explanation)

The t distribution is a family of curves controlled by degrees of freedom (df):

With small df, the curve has fatter tails (more extreme values are more likely).
As df increases, the t curve gets closer to the standard normal (z) curve.

When you input a t value, the calculator is finding the area under the t curve:

Left-tailed = area left of t
Right-tailed = area right of t
Two-tailed = both tails beyond $\pm|t|$

When you input df and α, the calculator finds the cutoff t value where the tail area equals α (or α/2 for two-tailed).

Confidence intervals preview (where critical t shows up)

A common t-based confidence interval for a mean looks like: $\bar{x} \pm t^* \left(\frac{s}{\sqrt{n}}\right)$

$t^*$ = critical t from df and confidence level
$\frac{s}{\sqrt{n}}$ = standard error (SE)

So if you can find critical t, you’re already halfway to building a confidence interval.

Step-by-step example

Example A: Probability from t (one-tailed and two-tailed)

Suppose you have:

df = 11
t = 1.25

Step 1: Identify what you need.

Left-tailed probability: $P(T \le 1.25)$
Right-tailed probability: $P(T \ge 1.25)$
Two-tailed p-value: $P(|T| \ge 1.25)$

Step 2: Use the complement rule for the right tail. $P(T \ge t) = 1 – P(T \le t)$

Step 3: Use symmetry for two tails. $P(|T| \ge |t|) = 2 \cdot P(T \ge |t|)$

Step 4: Read the results (rounded to 4 decimals).
Typical values for this setup are:

Left-tailed: $P(T \le 1.25) \approx 0.8814$
Right-tailed: $P(T \ge 1.25) \approx 0.1186$
Two-tailed: $P(|T|\ge|1.25|) \approx 0.2372$

Example B: Critical t for a two-tailed test

Suppose:

df = 11
α = 0.05 (two-tailed)

Step 1: Split alpha into two tails. $\alpha/2 = 0.025$

Step 2: Find the t cutoff where the right tail is 0.025.
That gives $t^*$ , and the critical values are ± $t^*$ .

Common mistakes

Using the wrong df. For one-sample mean problems, df is usually n − 1, not n.
Mixing up one-tailed vs two-tailed. Two-tailed means “extreme in either direction,” so you use both tails.
Forgetting to split α for two-tailed critical values. Two-tailed critical t uses α/2 in each tail.
Using the wrong sign for left-tail vs right-tail. Negative t values live on the left; positive on the right.
Confusing probability with critical value. Probabilities are areas; critical values are x-axis cutoffs.
Rounding too early. Keep t and df as-is; round only the final probability/critical value.

FAQs: t distribution calculator

What does this result mean?

It’s the area under the t curve. For probabilities, it tells you “how much of the distribution is left/right of your t.” For two-tailed p-values, it tells you “how likely a t value at least this extreme is in either direction.”

What is df in a t distribution?

df means degrees of freedom, and in many intro stats settings it’s n − 1 for one-sample t procedures. df affects how heavy the tails are.

What’s the difference between one-tailed and two-tailed probabilities?

One-tailed looks in one direction only (left or right).
Two-tailed looks for “extreme on either side,” so it uses both tails beyond $\pm|t|$ .

Which test should I use: z-test or t-test?

Use a t-test/t distribution when the population standard deviation σ is unknown (you use sample s) and/or when the sample is small (depending on your course rules). Use z when σ is known or your class tells you to use z for that situation.

Why is my answer different from my teacher’s?

Common reasons:
You used df = n instead of df = n − 1.
You used one-tailed but the problem is two-tailed (or vice versa).
Your teacher rounded differently (tables often round t values and areas).
You used $P(T \le t)$ when they wanted $P(T \ge t)$ (or the opposite).

How do I get a p-value from a t value?

Pick the tail type from the problem statement:
Right-tailed: $p = P(T \ge t)$
Left-tailed: $p = P(T \le t)$
Two-tailed: $p = P(|T|\ge|t|)$

How does this connect to confidence intervals?

Confidence intervals use critical t:
$\bar{x} \pm t^*\left(\frac{s}{\sqrt{n}}\right)$ So if you can find $t^*$ (given df and α), you can build the interval endpoints.

Does the t distribution become normal?

Yes—as df increases, the t distribution approaches the standard normal distribution (the curves become very similar).

Related tools

If you’re building skills around inference, these are natural next steps:

Distributions Hub
critical value calculator
t distribution table

References

Diez, D. M., Barr, C. D., & Çetinkaya-Rundel, M. (2023). OpenIntro statistics (4th ed.). OpenIntro.

OpenIntro Statistics: Fourth Edition

Black and white paperback edition, with this new edition released in May 2019
The OpenIntro project was founded in 2009 to improve the quality and availability of education by producing exceptional books and teaching tools that are free to use and easy to modify
Our inaugural effort is OpenIntro Statistics
Probability is optional, inference is key, and we feature real data whenever possible
Files for the entire book are freely available at openintro

NIST/SEMATECH. (2013). e-Handbook of statistical methods (Student’s t distribution; confidence intervals). National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/

Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & statistics for engineers & scientists (9th ed.). Pearson.

Probability and Statistics for Engineers and Scientists (9th Edition)

This classic text provides a rigorous introduction to basic probability theory and statistical inference, with a unique balance of theory and methodology
Interesting, relevant applications use real data from actual studies, showing how the concepts and methods can be used to solve problems in the field
This revision focuses on improved clarity and deeper understanding