T-Distribution | What It Is and How To Use It (With Examples)
The t-distribution, also known as Student’s t-distribution, is a way of describing data that follow a bell curve when plotted on a graph, with the greatest number of observations close to the mean and fewer observations in the tails.
In statistics, the t-distribution is most often used to:
- Find the critical values for a confidence interval when the data is approximately normally distributed.
- Find the corresponding p-value from a statistical test that uses the t-distribution (t-tests, regression analysis).
What is a t-distribution?
The t-distribution is a type of normal distribution that is used for smaller sample sizes. Normally-distributed data form a bell shape when plotted on a graph, with more observations near the mean and fewer observations in the tails.
The t-distribution is used when data are approximately normally distributed, which means the data follow a bell shape but the population variance is unknown. The variance in a t-distribution is estimated based on the degrees of freedom of the data set (total number of observations minus 1).
It is a more conservative form of the standard normal distribution, also known as the z-distribution. This means that it gives a lower probability to the center and a higher probability to the tails than the standard normal distribution.
T-distribution and the standard normal distribution
As the degrees of freedom (total number of observations minus 1) increases, the t-distribution will get closer and closer to matching the standard normal distribution, a.k.a. the z-distribution, until they are almost identical.
Above 30 degrees of freedom, the t-distribution roughly matches the z-distribution. Therefore, the z-distribution can be used in place of the t-distribution with large sample sizes.
The z-distribution is preferable over the t-distribution when it comes to making statistical estimates because it has a known variance. It can make more precise estimates than the t-distribution, whose variance is approximated using the degrees of freedom of the data.
T-distribution and t-scores
In statistics, t-scores are primarily used to find two things:
- The upper and lower bounds of a confidence interval when the data are approximately normally distributed.
- The p-value of the test statistic for t-tests and regression tests.
T-scores and confidence intervals
Confidence intervals use t-scores to calculate the upper and lower bounds of the prediction interval. The t-score used to generate the upper and lower bounds is also known as the critical value of t, or t*.
T-scores and p-values
Statistical tests generate a test statistic showing how far from the null hypothesis of the statistical test your data is. They then calculate a p-value that describes the likelihood of your data occurring if the null hypothesis were true.
The test statistic for t-tests and regression tests is the t-score. While most statistical programs will automatically calculate the corresponding p-value for the t-score, you can also look up the values in a t-table, using your degrees of freedom and t-score to find the p-value.
The t-score which generates a p-value below your threshold for statistical significance is known as the critical value of t, or t*.
Frequently asked questions
- What is a t-distribution?
The t-distribution is a way of describing a set of observations where most observations fall close to the mean, and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.
The t-distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation.
- What is the difference between the t-distribution and the standard normal distribution?
In this way, the t-distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance, you will need to include a wider range of the data.
- What is a t-score?
A t-score (a.k.a. a t-value) is equivalent to the number of standard deviations away from the mean of the t-distribution.
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