# T Distribution Calculator

The t distribution calculator makes it easy to compute cumulative probabilities, based on t statistics; or to compute t statistics, based on cumulative probabilities. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems.

• In the dropdown box, describe the random variable.
• Enter a value for degrees of freedom.
• Enter a value for all but one of the remaining text boxes.
• Click the Calculate button to compute a value for the blank text box.
Random variable
Degrees of freedom
t score
Probability: P(T < t)

Instructions: To find the answer to a frequently-asked question, simply click on the question. If you don't see the answer you need, read Stat Trek's tutorial on Student's t distribution or visit the Statistics Glossary.

### Which random variable should I use - the t statistic or the sample mean"?

The t distribution calculator accepts two kinds of random variables as input: a t score or a sample mean. Choose the option that is easiest. Here are some things to consider.

• If you choose to work with t statistics, you may need to transform your raw data into a t statistic. You can accomplish this transformation by using the following equation:

t = [ x - μ ] / [ s / sqrt( n ) ]

where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, n is the sample size, and t is the t statistic.
• If you choose to work with the sample mean, you can avoid the "transformation" step. This is often easiest.

For an example that uses t statistics, see Sample Problem 1. For an example that uses the sample mean, see Sample Problem 2

### What are degrees of freedom?

Degrees of freedom can be described as the number of scores that are free to vary. For example, suppose you tossed three dice. The total score adds up to 12. If you rolled a 3 on the first die and a 5 on the second, then you know that the third die must be a 4 (otherwise, the total would not add up to 12). In this example, 2 die are free to vary while the third is not. Therefore, there are 2 degrees of freedom.

In many situations, the degrees of freedom are equal to the number of observations minus one. Thus, if the sample size were 20, there would be 20 observations; and the degrees of freedom would be 20 minus 1 or 19.

### What is a standard deviation?

The standard deviation is a numerical value used to indicate how widely individuals in a group vary. It is a measure of the average distance of individual observations from the group mean.

### What is a t statistic?

A t statistic is a statistic whose values are given by

t = [ x - μ> ] / [ s / sqrt( n ) ]

where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, n is the sample size, and t is the t statistic.

### What is a population mean?

A mean score is an average score. It is the sum of individual scores divided by the number of individuals. A population mean is the mean score of a population.

### What is a sample mean?

A mean score is an average score. It is the sum of individual scores divided by the number of individuals. A sample mean is the mean score of a sample.

### What is a probability?

A probability is a number expressing the chances that a specific event will occur. This number can take on any value from 0 to 1. A probability of 0 means that there is zero chance that the event will occur; a probability of 1 means that the event is certain to occur. Numbers between 0 and 1 quantify the uncertainty associated with the event. For example, the probability of a coin flip resulting in Heads (rather than Tails) would be 0.50. Fifty percent of the time, the coin flip would result in Heads; and fifty percent of the time, it would result in Tails.

### What is a cumulative probability?

A cumulative probability is a sum of probabilities. In connection with the t distribution calculator, a cumulative probability refers to the probability that a t statistic or a sample mean will be less than or equal to a specified value.

Suppose, for example, that we sample 100 first-graders. If we ask about the probability that the average first grader weighs exactly 70 pounds, we are asking about a simple probability - not a cumulative probability.

But if we ask about the probability that average weight is less than or equal to 70 pounds, we are really asking about a sum of probabilities (i.e., the probability that the average weight is exactly 70 pounds plus the probability that it is 69 pounds plus the probability that it is 68 pounds, etc.). Thus, we are asking about a cumulative probability.

Note: The t distribution calculator only reports cumulative probabilities (e.g., the probability that a t statistic is less than or equal to a specified value.)

## Sample Problems

1. The Acme Chain Company claims that their chains have an average breaking strength of 20,000 pounds, with a standard deviation of 1750 pounds. Suppose a customer tests 14 randomly-selected chains. What is the probability that the average breaking strength in the test will be no more than 19,800 pounds?

Solution:

One strategy would be a two-step approach:

• Compute a t statistic, assuming that the mean of the sample test is 19,800 pounds.
• Determine the cumulative probability for that t statistic.

We will follow that strategy here. First, we compute the t statistic:

t = [ x - μ ] / [ s / sqrt( n ) ]
t = (19,800 - 20,000) / [ 1750 / sqrt(14) ]
t = ( -200 ) / [ (1750) / (3.74166) ]
t = ( -200 ) / (467.707) = -0.4276

where x is the sample mean, μ is the population mean, s is the standard deviation of the sample, n is the sample size, and t is the t statistic.

Now, we can determine the cumulative probability for the t statistic. We know the following:

• The t statistic is equal to -0.4276.
• The number of degrees of freedom is equal to 13. (In situations like this, the number of degrees of freedom is equal to number of observations minus 1. Hence, the number of degrees of freedom is equal to 14 - 1 or 13.)

Now, we are ready to use the T Distribution Calculator. Since we have already computed the t statistic, we select "t score" from the drop-down box. Then, we enter the t statistic (-0.4276) and the degrees of freedom (13) into the calculator, and hit the Calculate button. The calculator reports that the cumulative probability is 0.338. Therefore, there is a 33.8% chance that the average breaking strength in the test will be no more than 19,800 pounds.

Note: The strategy that we used required us to first compute a t statistic, and then use the T Distribution Calculator to find the cumulative probability. An alternative strategy, which does not require us to compute a t statistic, would be to use the calculator in the "Sample mean" mode. That strategy may be a little bit easier. It is illustrated in the next example.
1. Let's look again at the problem that we addressed above in Example 1. This time, we will illustrate a different, easier strategy to solve the problem.

Here, once again, is the problem: The Acme Chain Company claims that their chains have an average breaking strength of 20,000 pounds, with a standard deviation of 1750 pounds. Suppose a customer tests 14 randomly-selected chains. What is the probability that the average breaking strength in the test will be no more than 19,800 pounds?

Solution:

We know the following:

• The population mean is 20,000.
• The standard deviation is 1750.
• The sample mean, for which we want to find a cumulative probability, is 19,800.
• The number of degrees of freedom is 13. (In situations like this, the number of degrees of freedom is equal to number of observations minus 1. Hence, the number of degrees of freedom is equal to 14 - 1 or 13.)

First, we select "Sample mean" from the dropdown box, in the T Distribution Calculator. Then, we plug our known input (degrees of freedom, sample mean, standard deviation, and population mean) into the T Distribution Calculator and hit the Calculate button. The calculator reports that the cumulative probability is 0.338. Thus, there is a 33.8% probability that an Acme chain will snap under 19,800 pounds of stress.

Note: This is the same answer that we found in Example 1. However, the approach that we followed in this example may be a little bit easier than the approach that we used in the previous example, since this approach does not require us to compute a t statistic.
1. The school board administered an IQ test to 25 randomly selected teachers. They found that the average IQ score was 115 with a standard deviation of 11. Assume that the cumulative probability is 0.90. What population mean would have produced this sample result?

Note: In this situation, a cumulative probability of 0.90 suggests that 90% of the random samples drawn from the teacher population will have an average IQ of 115 or less. This problem asks you to find the true population IQ for which this would be true.

Solution:

We know the following:

• The cumulative probability is 0.90.
• The standard deviation is 11.
• The sample mean is 115.
• The number of degrees of freedom is 24. (In situations like this, the number of degrees of freedom is equal to number of observations minus 1. Hence, the number of degrees of freedom is equal to 25 - 1 or 24.)

First, we select "Sample mean" from the dropdown box, in the T Distribution Calculator. Then, we plug the known inputs (cumulative probability, standard deviation, sample mean, and degrees of freedom) into the calculator and hit the Calculate button. The calculator reports that the population mean is 112.1.

Here is what this means. Suppose we randomly sampled every possible combination of 25 teachers. If the true population mean were 112.1, we would expect 90% of our samples to have a sample mean of 115 or less.