T Distribution Calculator: Online Statistical Table
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 FrequentlyAsked
Questions or review the Sample
Problems.
To learn more about Student's t distribution, go to Stat Trek's
tutorial on the t distribution.
Instructions: To find the answer to a frequentlyasked
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
score 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.
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 firstgraders. 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.

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 randomlyselected 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 twostep 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) ] = ( 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 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 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.

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 randomlyselected 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.

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.