What Is a t Distribution? — AP Statistics
1. Definition of the t Distribution ★★☆☆☆ ⏱ 3 min
The t distribution (also called Student's t-distribution) is a bell-shaped, symmetric probability distribution developed by William Gosset (writing under the pseudonym "Student") for use when working with small samples and unknown population standard deviation.
This is the foundational concept for all t-confidence intervals and t-tests for means, which make up 12-15% of the total AP Statistics exam score, and appear in both multiple-choice and free-response sections.
2. Degrees of Freedom ★★☆☆☆ ⏱ 3 min
Degrees of freedom (abbreviated df) is the parameter that controls the shape of the t distribution. For all one-sample inference procedures for a population mean, degrees of freedom are calculated as:
df = n - 1
We subtract 1 from $n$ because we use the sample mean $\bar{x}$ to estimate the unknown population mean $\mu$ when calculating the sample standard deviation $s$. The sum of deviations from $\bar{x}$ is always 0, so only $n-1$ deviations are free to vary, giving us $n-1$ independent pieces of information. As df increases, the t distribution becomes narrower and closer to the standard normal distribution.
Exam tip: Always write your degrees of freedom explicitly on FRQs; AP graders require it to award full credit.
3. Shape Comparison: t vs. Standard Normal (z) Distribution ★★★☆☆ ⏱ 4 min
Both the t distribution and standard normal (z) distribution are unimodal, symmetric, and centered at 0. They differ in spread: the t distribution has heavier tails than the z distribution for any finite degrees of freedom. This extra spread accounts for the additional uncertainty from estimating $\sigma$ with the sample standard deviation $s$, which varies from sample to sample.
As degrees of freedom increase, the t distribution converges to the z distribution. When $df \to \infty$, $s$ is almost identical to $\sigma$, so the extra uncertainty disappears. Even for $df = 30$, the t distribution is very close to z, but it is never exactly the same for finite sample sizes.
Exam tip: If you forget whether t has fatter tails than z, remember: smaller sample = more uncertainty = more spread = fatter tails. This works for any MCQ comparison question.
4. Conditions for Using the t Distribution ★★★☆☆ ⏱ 4 min
We use the t distribution for inference about a population mean (or difference in two population means) any time the population standard deviation $\sigma$ is unknown, which is almost always true in AP exam problems. Two core conditions must be satisfied for t procedures to be valid:
- **Random**: The data comes from a random sample from the population of interest, or a randomized comparative experiment.
- **Normal/Large Sample**: The sampling distribution of $\bar{x}$ is approximately normal. This is true if either the sample size is large ($n \geq 30$, by the Central Limit Theorem), or the population distribution is approximately normal. For small samples ($n < 30$), we check the sample for strong skewness or extreme outliers; if none exist, we can assume normality.
Exam tip: On AP FRQs, you must justify using the t distribution by mentioning two things: $\sigma$ is unknown, AND the random and normality conditions are met. Don't stop at just "we use t because $\sigma$ is unknown."
Common Pitfalls
Why: Students confuse the rare case of known $\sigma$ with the common real-world/AP exam case of unknown $\sigma$
Why: Students forget that one degree of freedom is lost when estimating the population mean with the sample mean to calculate $s$
Why: Students mix up t distribution properties with the right-skewed chi-square distribution
Why: Students confuse inference for quantitative means (t) with inference for categorical proportions (z)
Why: Students confuse the Central Limit Theorem normality condition with the requirement to use t