T Distribution Calculator
The t-distribution calculator makes it easy to compute the cumulative probability associated with a t score or with a sample mean. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems.
To learn more about Student's t-distribution, read Stat Trek's lesson on the t-distribution.
Frequently-Asked Questions
Instructions: To find the answer to a frequently-asked question, simply click on the question.
What is a t-score?
A t-score 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.
Which statistic should I use - the t-score or the mean score"?
The t-distribution calculator accepts two statistics as input: a t-score or a sample mean. Choose the option that works for you. Here are some things to consider.
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If you choose to work with a t-score, the calculator reports probability for a t-score from a t-distribution with
degrees of freedom that you specify. To use this option, you need to know the t-score or you need to compute the
t-score from sample data, using this equation:
t = ( x̅ - μ ) / [ s / sqrt(n) ]
where x̅ is the sample mean, μ is the population mean, s is the standard deviation of the sample, and n is sample size. - If you choose to work with the sample mean, the calculator computes the t-score for you behind the scenes. But you need to provide additional inputs. Specifically, you need to input values for the sample mean, population mean, and sample standard deviation. The calculator estimates sample size, based on the degrees of freedom. It assumes that sample size is equal to degrees of freedom plus one. If sample size in your application is not equal to degrees of freedom plus one, you should compute the t-score manually as shown above and select the "t-score" option in the Statistics dropdown box.
For an example that uses the t-score, see SampleProblem 1. For an example that uses the sample mean, see Sample Problem 2.
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.
From the Probability dropdown box in the calculator, you can choose any of eight cumulative probabilities:
- P( T ≤ t ) This is the probability that a random t-score from the t distribution will be less than or equal to the value t.
- P( T ≥ t ) This is the probability that a random t-score from the t distribution will be greater than or equal to the value t.
- P( t1 ≤ T ≤ t2 ). This is the probability that a random t-score from the t distribution fall between t1 and t2.
- P( T ≤ t1) + P(T ≥t2 ). This is the probability that a random t-score from the t distribution is less than t1 or greater than t2.
- P( X ≤ x̅ ). This is the probability that a random mean score from a normal distribution will be less than or equal to the value x̅.
- P( X ≥ x̅ ). This is the probability that a random mean score from a normal distribution will be greater than or equal to the value x̅.
- P( x̅1 ≤ X ≤ x̅2 ). This is the probability that a random mean score from a normal distribution will fall between x̅1 and x̅2.
- P( X ≤ x̅1) + P(X ≥ x̅2 ). This is the probability that a random mean score from a normal distribution will be less than x̅1 or greater than x̅2.
What are degrees of freedom?
In statistics, degrees of freedom (df) refer to the number of independent values that are free to vary given certain constraints. For example,
- Given a sample of n observations, suppose the mean is constrained to equal 5. Here, the degrees of freedom are one less than sample size (df = n - 1). This is because only n - 1 observations can independently vary, since the value of the last observation is determined by the constraint that the mean equal 5.
Degrees of freedom are an important input to the t-distribution calculator, because they affect the shape of the t-distribution. The magnitude of the effect is most pronounced when sample size is small.
How do I calculate degrees of freedom?
The exact way to calculate degrees of freedom depends on the specific analysis you are conducting. For example, a common application is a one-sample t-test, which compares a sample mean to a known population mean. For that application, the degrees of freedom formula is:
df = n - 1
where df is degrees of freedom and n is sample size.
In other situations, the degrees of freedom formula would be different. Elsewhere on this site, we explain how to compute degrees of freedom for other situations (e.g. in hypothesis testing, regression analysis, and the calculation of confidence intervals).
Note: As sample size increases, the t distribution more closely resembles a normal distribution. The larger the sample size, the closer the resemblance. As a practical matter, the shape of the t distribution is not greatly affected by degrees of freedom when sample size is large. Other things being equal, probabilities for a t-score with 100 degrees of freedom are about the same as probabilites for a t-score with 1000 degrees of freedom.
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.
With the t-distribution calculator, standard deviation refers to the standard deviation of a random sample from which a sample mean is computed.
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 are the related probabilities?
The related probabilities are probabilities of events that can be derived from knowledge of other events. Here's the logic for computing related probabilities:
- The t-distribution is symmetric. Therefore, P(T ≤ -t) = P(T ≥ t).
- From the rule of subtraction, we know P(T ≤ t) = 1 - P(T ≥ t).
Given these relationships, if we know the cumulative probability for one event, we can compute cumulative probabilities for other related events. For example, suppose we know P(T ≤ t). Then, we can compute the following:
- P(T ≥ t) = 1 - P(T ≤ t)
- P(T ≤ -t) = P(T ≥ t)
- P(-t ≤ T ≤ t) = P(T ≤ t) - P(T ≤ -t)
When the t-distribution calculator computes the probability for one event, it also displays probabilities for other related events.
Sample Problems
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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
buys 14 randomly-selected chains. What is the probability that the average breaking strength
for the customer's' chains will be 19,800 pounds or less?
Solution:
One strategy would be a two-step approach:
- Compute a t statistic, assuming that the breaking strength for the customer's chains 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 mean breaking strength in the customer's chains, μ is the population mean, s is the standard deviation, 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 Statistic dropdown box; and "P(T ≤ t)", from the Probability dropdown 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 about 0.338.
Therefore, there is a 33.8% chance that the average breaking strength for the customer's chains 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 "mean score" mode. That strategy may be a little bit easier. It is illustrated in the next example.
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Let's look one more time 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. A customer buys 14 randomly-selected chains. What is the probability that the average breaking strength for the customer's chains 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 Statistic dropdown box in the t-distribution Calculator , and "P(X ≤ x̅)" from the Probability dropdown box. Then, we plug our known inputs (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 about 0.338. Thus, there is a 33.8% probability that the average breaking strength for the customer's chains will be 19,800 pounds or less.
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 manual computation of a t statistic.
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The principal at a local high school claims that the average IQ of his faculty is higher
than the actual IQ of 90% of teachers in the district.
The school board administered an IQ test to 15 randomly selected teachers at a his high school. They
found that the average IQ score was 115 with a standard deviation of 11. Assume that the principal's
claim about his faculty is correct. What would be the average IQ of a teacher in the district?
Note: In terms of cumulative probability, the notion that 90% of teachers in the district have an IQ of 115 or less implies that P(IQ≤115) is 0.90.
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 14. (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 15 - 1 or 14.)
First, we select "Sample mean" from the dropdown box in the T Distribution Calculator, and "P(X ≤ x̅)" from the Probability dropdown box. 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 about 111.2.
Thus, if the principal's assessment of the IQ of his faculty is correct, we conclude that the average IQ of a teacher in the district is 111.2.