Probability and Statistics Index > T-Distribution

**Contents:**

## What is the T Distribution?

The **T distribution **(also called **Student’s T Distribution**) is a family of distributions that look almost identical to the normal distribution curve, only a bit shorter and fatter. The t distribution is used instead of the normal distribution when you have small samples (for more on this, see: t-score vs. z-score). The larger the sample size, the more the t distribution looks like the normal distribution. In fact, for sample sizes larger than 20 (e.g. more degrees of freedom), the distribution is almost exactly like the normal distribution.

## How to Calculate the Score for a T Distribution

When you look at the t-distribution tables, you’ll see that you need to know the “df.” This means “degrees of freedom” and is just the sample size minus one.

Step 1: Subtract one from your sample size. This will be your degrees of freedom.

Step 2: Look up the df in the left hand side of the t-distribution table. Locate the column under your alpha level (the alpha level is usually given to you in the question).

**For more detailed steps, including a video**, see: t score formula.

## Uses

The T Distribution (and the associated t scores), are used in hypothesis testing when you want to figure out if you should accept or reject the null hypothesis.

The central region on this graph is the acceptance area and the tail is the rejection region, or regions. In this particular graph of a two tailed test, the rejection region is shaded blue. The area in the tail can be described with z-scores or t-scores. For example, the image to the left shows an area in the tails of 5% (2.5% each side). The z-score would be 1.96 (from the z-table), which represents 1.96 standard deviations from the mean. The null hypothesis will be rejected if z is less than -1.96 or greater than 1.96.

In general, this distribution is used when you have a small sample size (under 30) or you don’t know the population standard deviation. For practical purposes (i.e. in the real world), this is nearly always the case. So, unlike in your elementary statistics class, you’ll likely be using it in real life situations more than the normal distribution. If the size of your sample is large enough, the two distributions are practically the same.

## T distribution on a TI 83

Watch the video or read on below:

Take a look at a traditional textbook T table, and you’ll actually find *many* T tables, which can be a little overwhelming. Instead of poring over tables, you can use a **TI 83 graphing calculator** to assist you in finding T values.

You might be asked to find the area under a T curve, or (like Z scores), you might be given a certain area and asked to find the T score.

## Finding a T-Critical-Value on the TI 83

Click here for our article on finding T Critical Values on the TI 83.

## T distribution on a TI 83: Steps

**Sample problem**: Find the area under a T curve with degrees of freedom 10 for P( 1 ≤ X ≤ 2 ). Use the t distribution on a TI 83.

**Step 1:** Press 2nd VARS 5 to select **tcdf(**.

**Step 2:** Enter the lower and upper bounds, and the degrees of freedom. The lower bound is the lowest number and the upper bound is the highest number: 1,2,10

Your screen should now read **tcdf(1,2,10)**

**Step 3:** Press ENTER. The answer is **.133752549**, or about **13.38%**.

That’s how to find a T distribution on a TI 83!

Check out my YouTube channel for more statistics help and tips.

## T-Distribution on the TI-89

For most T-distribution questions, you’ll usually be given all of the information you need to plug into the **calculator and retrieve the T score**. You might be asked to find the area under a T curve, or (like Z scores), you might be given a certain area and asked to find the T score.

## T Distribution on TI 89 Steps

**Note**: You need to have the STAT/LIST editor installed for these procedures. You can download a copy for free from the TI-website.

**Sample problem**: Find the area under a T curve with degrees of freedom 10 for P( 1 ≤ X ≤ 2 ).

**Step 1:** Press APPS.

**Step 2:** Press ENTER twice to enter the STATS/LIST Editor.

**Step 3:** Press F5 for **F5Distr**.

**Step 4:** Choose 6 for **6:t Cdf**.

**Step 5:** Enter 1 in the box for **Lower Value**.

**Step 6:** Enter 2 in the box for **Upper Value**.

**Step 7:** Enter 10 in the box for **Deg of Freedom, df**.

**Step 8:** Press ENTER. This returns the result **.133753**.

Sample problem: find the T score with a value of 0.25 to the left and df of 10.

**Step 1:** Press APPS.

**Step 2:** Press ENTER twice to enter the STAT/LIST Editor.

**Step 3:** Press F5 for **F5Distr**.

**Step 4:** Press 2 for **Inverse**.

**Step 5:** Press the **right arrow** button.

**Step 6:** Press 2 for **Inverse t** and then press ENTER.

**Step 7:** Enter 0.25 in the **Area** box.

**Step 8:** Enter 10 in the **Deg of Freedom, df** box.

**Step 9:** Press ENTER. The calculator returns the result of **-.699812**.

**Tip**: For P( X ≥ a ), enter 10^99 in the box for Upper Value and for P(X ≤ b), enter 10^99 in the box for Lower Value.

That’s how to find a T Distribution on TI 89!

## More T-Dist Articles

- How to Construct a Confidence Interval From Data Using the t-Dist.
- T test: What it is and how to calculate it.
- Independent Samples T Test.
- The Satterthwaite Approximation.
- T-Score Formula.

Check out our YouTube channel for hundreds of statistics videos.

Hi,

Is the normal distribution you mentioned here is the Standard Normal distribution (mean=0, standard deviation =1)? From the second picture, we can see that with small sample t-distribution curve is lower around the center, it is not taller as in your text.

Thank you for your explanation.

Viet, thanks for your comment! The t-dist is indeed shorter, not taller. There was a typo — sorry for the confusion.

Thank you.

this is good information.