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The t-distribution (also called Student’s t-distribution) is a probability distribution used to estimate population parameters when the sample size is small and the population standard deviation is unknown. It is bell-shaped and symmetric like the normal distribution, but it is shorter at the peak and has heavier (fatter) tails. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution and becomes almost indistinguishable from it.
In short, the t-distribution accounts for the extra uncertainty that comes from estimating the standard deviation from a small sample rather than knowing it for the whole population. It is denoted by the letter ‘t’, and it sits at the heart of the t-tests and confidence intervals that researchers rely on every day.
“It is usual, however, to assume a normal distribution… when we are dealing with the means of small samples, this assumption is not justified.” The t-distribution was introduced by William Sealy Gosset, who published it in 1908 under the pen name ‘Student’ while working as a chemist at the Guinness brewery in Dublin.— Student (W. S. Gosset), “The Probable Error of a Mean”, Biometrika, 1908
The t-distribution (df = 3) has a lower peak and heavier tails than the standard normal; as the degrees of freedom increase, it converges to the normal curve.
Frequently Asked Questions
As the degrees of freedom increase, what does the t-distribution become more like?▾
As the number of degrees of freedom increases, the t-distribution becomes more like the standard normal distribution. Its tails grow thinner and its peak higher, and the difference between the two curves shrinks. By about 30 degrees of freedom they are almost identical, and the t-distribution converges exactly on the standard normal as the degrees of freedom approach infinity.
What is Student’s t-distribution?▾
Student’s t-distribution is a probability distribution used to estimate population parameters from small samples when the population standard deviation is unknown. It is bell-shaped and symmetric like the normal distribution but has heavier tails. It was published in 1908 by William Sealy Gosset under the pen name ‘Student’ while he worked at the Guinness brewery in Dublin.
How do you calculate the t-distribution value?▾
The t-statistic is calculated as t = (x̄ − μ) / (s / √n), where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation and n is the sample size. The term s / √n is the standard error of the mean. You then read the probability or critical value for that t-value from a t-table using df = n − 1 degrees of freedom.
What is the difference between the t-distribution and the normal distribution?▾
Both are symmetric, bell-shaped curves centred on zero, but the t-distribution is shorter and wider with heavier tails, giving more probability to extreme values. The t-distribution depends on degrees of freedom, while the normal distribution has a fixed shape. The t is used for small samples with an unknown population standard deviation; the normal is used for large samples with a known standard deviation.
When should you use a t-test instead of a z-test?▾
Use a t-test when the population standard deviation is unknown and the sample is small (typically fewer than 30 observations). Use a z-test when the population standard deviation is known, or when the sample is large enough that the sample standard deviation reliably estimates it. Because the t-distribution approaches the normal as the sample grows, the two give almost identical results for large samples.
What are the main applications of the t-distribution?▾
The t-distribution underpins one-sample, two-sample and paired t-tests, the construction of confidence intervals for a mean from a small sample, and significance testing of regression coefficients. It is widely used wherever conclusions about a population mean must be drawn from a small sample with an unknown standard deviation.
