Tdistribution: what it is and how to use it
The tdistribution, also known as Student’s tdistribution, 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.
It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.
In statistics, the tdistribution is most often used to:
 Find the critical values for a confidence interval when the data is approximately normally distributed.
 Find the corresponding pvalue from a statistical test that uses the tdistribution (ttests, regression analysis).
What is a tdistribution?
The tdistribution is a type of normal distribution that is used for smaller sample sizes. Normallydistributed data form a bell shape when plotted on a graph, with more observations near the mean and fewer observations in the tails.
The tdistribution 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 tdistribution 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 zdistribution. This means that it gives a lower probability to the center and a higher probability to the tails than the standard normal distribution.
Tdistribution and the standard normal distribution
As the degrees of freedom (total number of observations minus 1) increases, the tdistribution will get closer and closer to matching the standard normal distribution, a.k.a. the zdistribution, until they are almost identical.
Above 30 degrees of freedom, the tdistribution roughly matches the zdistribution. Therefore, the zdistribution can be used in place of the tdistribution with large sample sizes.
The zdistribution is preferable over the tdistribution when it comes to making statistical estimates because it has a known variance. It can make more precise estimates than the tdistribution, whose variance is approximated using the degrees of freedom of the data.
Tdistribution and tscores
A tscore is the number of standard deviations from the mean in a tdistribution. You can typically look up a tscore in a ttable, or by using an online tscore calculator.
In statistics, tscores are primarily used to find two things:
 The upper and lower bounds of a confidence interval when the data are approximately normally distributed.
 The pvalue of the test statistic for ttests and regression tests.
Tscores and confidence intervals
Confidence intervals use tscores to calculate the upper and lower bounds of the prediction interval. The tscore used to generate the upper and lower bounds is also known as the critical value of t, or t*.
Tscores and pvalues
Statistical tests generate a test statistic showing how far from the null hypothesis of the statistical test your data is. They then calculate a pvalue that describes the likelihood of your data occurring if the null hypothesis were true.
The test statistic for ttests and regression tests is the tscore. While most statistical programs will automatically calculate the corresponding pvalue for the tscore, you can also look up the values in a ttable, using your degrees of freedom and tscore to find the pvalue.
The tscore which generates a pvalue below your threshold for statistical significance is known as the critical value of t, or t*.
Frequently asked questions about the tdistribution
 What is a tdistribution?

The tdistribution 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 tdistribution 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 tdistribution and the standard normal distribution?

The tdistribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the zdistribution).
In this way, the tdistribution 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 tscore?

A tscore (a.k.a. a tvalue) is equivalent to the number of standard deviations away from the mean of the tdistribution.
The tscore is the test statistic used in ttests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a tdistribution.
 What is a test statistic?

A test statistic is a number calculated by a statistical test. It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.
The test statistic tells you how different two or more groups are from the overall population mean, or how different a linear slope is from the slope predicted by a null hypothesis. Different test statistics are used in different statistical tests.
 What is a critical value?

A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval, or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data (i.e. 90%, 95%, 99%).
If you are constructing a 95% confidence interval and are using a threshold of statistical significance of p = 0.05, then your critical value will be identical in both cases.