Mathematics.

inferential statistics

The t-Distribution

Statistics25 minDifficulty4 out of 10

You should know: normal distribution, confidence intervals

Overview

The t-distribution (Student's t-distribution) is a continuous probability distribution that arises when estimating the mean of a normally distributed population using a small sample and an unknown population standard deviation, which must be estimated from the sample itself. It looks similar to the standard normal distribution — symmetric and bell-shaped — but has heavier tails, reflecting the extra uncertainty introduced by estimating the standard deviation from limited data. The distribution's shape is governed by a single parameter, the degrees of freedom (df = n − 1 for a one-sample test), and as df increases, the t-distribution converges to the standard normal distribution. It underlies the t-test and t-based confidence intervals, which are the standard tools for inference on means when the population standard deviation is unknown and the sample is small.

Intuition

When you know the true population standard deviation σ, the z-statistic (X̄ − μ)/(σ/√n) is exactly standard normal. But in practice σ is rarely known — you must estimate it with the sample standard deviation s, which itself is a noisy estimate, especially for small n. This extra layer of estimation uncertainty widens the tails of the sampling distribution of the resulting statistic, giving the t-distribution its heavier tails compared to the normal. As sample size grows, s becomes a more reliable estimate of σ, and the t-distribution's tails shrink back toward the normal's.

Formal Definition

Definition

If X̄ is the sample mean of n observations from a normal population with unknown standard deviation, estimated by sample standard deviation s:

T=Xˉ−μs/n∼tdf,df=n−1T = \frac{\bar{X} - \mu}{s/\sqrt{n}} \sim t_{df}, \qquad df = n - 1
t-statistic for one-sample test
f(t)=Γ ⁣(df+12)dfπ Γ ⁣(df2)(1+t2df)−df+12f(t) = \frac{\Gamma\!\left(\frac{df+1}{2}\right)}{\sqrt{df\pi}\,\Gamma\!\left(\frac{df}{2}\right)} \left(1 + \frac{t^2}{df}\right)^{-\frac{df+1}{2}}
t-distribution PDF
tdf→df→∞N(0,1)t_{df} \xrightarrow{df \to \infty} N(0,1)
Convergence to standard normal

Worked Examples

  1. Compute the standard error using s and n.

    SE=sn=610≈63.162≈1.897SE = \frac{s}{\sqrt{n}} = \frac{6}{\sqrt{10}} \approx \frac{6}{3.162} \approx 1.897

  2. Compute the t-statistic.

    t=52−501.897≈21.897≈1.054t = \frac{52 - 50}{1.897} \approx \frac{2}{1.897} \approx 1.054

Answer: t ≈ 1.054, with df = n − 1 = 9.

Practice Problems

Difficulty 3/10

A sample of n = 25 has degrees of freedom df = ? for a one-sample t-test.

Difficulty 4/10

A sample of n = 9 has mean 105, sample standard deviation s = 9. Compute the t-statistic for testing H₀: μ = 100.

Difficulty 5/10

Explain why a researcher with only n = 8 data points and unknown population standard deviation should use a t-test rather than a z-test.

Quiz

The t-distribution differs from the standard normal distribution primarily in that it:
As the degrees of freedom increase, the t-distribution:
For a one-sample t-test with sample size n, the degrees of freedom is:

Summary

  • The t-distribution arises when estimating a normal population's mean with an unknown standard deviation, estimated from the sample.
  • It is symmetric and bell-shaped like the normal distribution but has heavier tails, controlled by degrees of freedom (df = n - 1).
  • As df increases, the t-distribution converges to the standard normal distribution, since s becomes a more reliable estimate of σ.

References

  1. WebsiteWikipedia — Student's t-distribution