What is the t-distribution?

The t-distribution describes the standardized distances of sample means to the population mean when the population standard deviation is not known, and the observations come from a normally distributed population.

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Is the t-distribution the same as the Student’s t-distribution?


What’s the key difference between the t- and z-distributions?

The standard normal or z-distribution assumes that you know the population standard deviation. The t-distribution is based on the sample standard deviation.

t-Distribution vs. normal distribution

The t-distribution is similar to a normal distribution. It has a precise mathematical definition. Instead of diving into complex math, let’s look at the useful properties of the t-distribution and why it is important in analyses.

Like the normal distribution, the t-distribution has a smooth shape.Like the normal distribution, the t-distribution is symmetric. If you think about folding it in half at the mean, each side will be the same.Like a standard normal distribution (or z-distribution), the t-distribution has a mean of zero.The normal distribution assumes that the population standard deviation is known. The t-distribution does not make this assumption.The t-distribution is defined by the degrees of freedom. These are related to the sample size.The t-distribution is most useful for small sample sizes, when the population standard deviation is not known, or both.As the sample size increases, the t-distribution becomes more similar to a normal distribution.

Consider the following graph comparing three t-distributions with a standard normal distribution:


Tails for hypotheses tests and the t-distribution

When you perform a t-test, you check if your test statistic is a more extreme value than expected from the t-distribution.

For a two-tailed test, you look at both tails of the distribution. Figure 3 below shows the decision process for a two-tailed test. The curve is a t-distribution with 21 degrees of freedom. The value from the t-distribution with α = 0.05/2 = 0.025 is 2.080. For a two-tailed test, you reject the null hypothesis if the test statistic is larger than the absolute value of the reference value. If the test statistic value is either in the lower tail or in the upper tail, you reject the null hypothesis. If the test statistic is within the two reference lines, then you fail to reject the null hypothesis.


How to use a t-table

Most people use software to perform the calculations needed for t-tests. But many statistics books still show t-tables, so understanding how to use a table might be helpful. The steps below describe how to use a typical t-table.

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Identify if the table is for two-tailed or one-tailed tests. Then, decide if you have a one-tailed or a two-tailed test. The columns for a t-table identify different alpha levels.If you have a table for a one-tailed test, you can still use it for a two-tailed test. If you set α = 0.05 for your two-tailed test and have only a one-tailed table, then use the column for α = 0.025.Identify the degrees of freedom for your data. The rows of a t-table correspond to different degrees of freedom. Most tables go up to 30 degrees of freedom and then stop. The tables assume people will use a z-distribution for larger sample sizes.Find the cell in the table at the intersection of your α level and degrees of freedom. This is the t-distribution value. Compare your statistic to the t-distribution value and make the appropriate conclusion.