Can you do a t-test with n less than 30?

Yes, and here is the rule that replaces "n at least 30"

You can run a t-test with $n = 12$. You can run one with $n = 8$. The t procedure was built for small samples: William Gosset published it in 1908 while working at a brewery, where the batches he could test were tiny and the normal-based methods of the day were too optimistic.

The "n at least 30" line you were taught is not a requirement. It is one of three ways to satisfy a single condition, and the AP course calls that condition normal or large sample. The full sentence is:

The population is approximately normal, or $n \ge 30$, or a graph of the sample data shows no strong skewness and no outliers.

Read the "or" as an actual or. Satisfy any one branch and the condition is met. When $n < 30$, you take the third branch: graph the sample, look at it, and say what you see. That is the sentence the rubric is waiting for, and it is the sentence most students never write because they assumed the procedure was off limits.

The three conditions for a one-sample t procedure

Every t procedure in Unit 4 asks for the same three checks before you compute anything. Small $n$ only affects the third.

The third condition exists for one reason: the t procedure needs the sampling distribution of $\bar{x}$ to be roughly normal. Sample size is not the only thing that delivers that. A normal population delivers it at any $n$, even $n = 3$. A sample that graphs as roughly symmetric with no outliers is evidence that the population is not badly shaped, which is enough in practice.

For paired data, run all three checks on the differences, not on the two original lists. See paired vs two-sample t-test if you are not sure which you have.

What to write on an FRQ when n is under 30

Condition points are given for naming the condition and citing evidence, in context. With a small sample, the graph is your evidence. Write something close to this:

"Normal or large sample: $n = 12 < 30$, so I cannot use the sample size. The problem does not state that the population is normal. A dotplot of the 12 completion times is roughly symmetric with no outliers, so it is reasonable to treat the sampling distribution of $\bar{x}$ as approximately normal."

Three things earn the point there.

1. You named the condition instead of just writing "normality."
2. You ruled out the other two branches out loud, which shows you know there are three.
3. You cited the graph and described its shape, in context, rather than saying "the graph looks fine."

If the problem hands you a dotplot, boxplot, or stemplot, use it: that graph is there because the sample is small and the reader wants you to comment on it. If the problem gives you raw data and no graph, the 1.5 times IQR rule is a fast, defensible outlier check you can show in one line. More on rubric wording in the AP Statistics FRQ guide.

Where "n at least 30" came from, and why it is a slogan

The number 30 is a rule of thumb attached to the central limit theorem. The CLT says the sampling distribution of $\bar{x}$ approaches normal as $n$ grows, whatever the population shape. It never names a sample size, because the sample size you need depends entirely on how skewed the population is. Textbooks picked 30 as a round number that works for most mildly skewed populations, and it hardened into a rule nobody questions.

Two consequences follow, and they point in opposite directions.

Below 30, the procedure is often still fine. t procedures are robust, meaning the true error rate stays close to the stated one even when the population is not exactly normal. For a symmetric population, a t-test with $n = 10$ behaves well.

Above 30, the procedure can still be wrong. A strongly skewed population with one extreme outlier can defeat $n = 40$. The condition is about shape, and $n \ge 30$ is only a shortcut for assuming the shape has been washed out.

So $n = 30$ is neither a floor nor a guarantee. It is a branch of an or-statement. The other two branches work just as hard.

When a small sample really does block the test

Sometimes the answer is genuinely no, and knowing when is worth as many points as knowing when yes.

The condition fails when $n < 30$, the population is not stated to be normal, and the sample shows strong skewness or an outlier. All three at once. That combination means the third branch has nothing to stand on, and you should say the t procedure is not appropriate rather than compute a number you cannot defend.

A harder case: $n < 30$, no normality stated, and the problem gives you a summary such as $n = 11$, $\bar{x} = 14.2$, and $s = 3.1$ but no data and no graph. You have no way to check the shape, so you say that instead of pretending you did. Write that. "The sample size is 11 and the population shape is not given, so the normal or large sample condition cannot be checked." Naming the missing information is the correct answer; inventing a shape is not.

One more trap. A single outlier does not merely dent the condition, it distorts $\bar{x}$ and inflates $s$ at the same time, and with a small $n$ one point carries enormous weight. That is why the rule bans outliers outright instead of asking you to judge how bad they are.

What a small sample costs you

A small sample is allowed, but it is not free. Nothing about the mechanics changes: the statistic is still

with $df = n - 1$. What changes is how much that statistic buys you.

Fewer degrees of freedom means heavier tails in the t distribution and a larger critical value. At 95% confidence, $t^* = 2.201$ with $df = 11$ but only $t^* = 2.045$ with $df = 29$, and $z^* = 1.96$ in the limit. Combined with a larger $s / \sqrt{n}$, a small sample gives a noticeably wider confidence interval and a smaller chance of detecting a real effect. That reduced chance is the definition of low power.

The practical read: a small sample makes it harder to reject the null hypothesis, not easier. So failing to reject with $n = 9$ tells you very little, because the test had little power to detect a difference in the first place. Say that in your conclusion rather than claiming the null is true. Critical values by df are in the t-table, and you can check any small-sample computation against the one-sample t-test calculator.

Mistakes that lose condition points

• Refusing to run the test because the sample is small. The most expensive mistake on this page. When $n < 30$, check the graph instead.
• Switching to a z-test once the sample passes 30. The population standard deviation $\sigma$ is still unknown, so it is still a t-test. See t-test vs z-test.
• Writing "n is large enough" with no number. State $n$, state which branch you are using, and why.
• Rejecting the sample because the dotplot is not bell-shaped. The sample only has to show no strong skew and no outliers. It does not have to look bell-shaped, and with $n = 8$ nothing looks bell-shaped.
• Using n for the degrees of freedom. For a one-sample t procedure $df = n - 1$. With a small sample that off-by-one shifts the p-value, and near $\alpha$ it can flip the decision.
• Checking conditions on the raw groups in a paired design. Check the differences.
• Concluding "there is no difference" after failing to reject with a tiny sample. Say the data do not provide convincing evidence of a difference, and mention that the small sample limits the power to detect one.