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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