Student's t-Distribution (t-table)

The table below gives the critical values of Student's $t$-distribution. Column headers show two quantities: the central area A (the probability enclosed between −$t$ and +$t$) and the corresponding two-tailed P value (= 1 − A). The row gives the degrees of freedom (DF). The last row (DF = ∞) equals the standard normal critical values.

How to use the table

The $t$-distribution is used whenever a test statistic or confidence interval is based on a sample mean and an estimated standard deviation, and the sample is small. As the degrees of freedom increase the $t$-distribution converges to the standard normal: at DF = ∞ the critical values equal those in the $z$-table.

To use the table, identify the degrees of freedom (usually $n$ − 1 for a one-sample problem or $n$ − 2 for regression) and the significance level or confidence level you are targeting, then read off the critical value at their intersection.

Example 1 — Confidence interval for a mean

Systolic blood pressure reduction is measured in 10 patients after treatment with a new antihypertensive drug: mean = 8 mmHg, SD = 5 mmHg. Construct a 95% confidence interval for the mean reduction.

Degrees of freedom: $\text{DF} = n - 1 = 9$. From the table at DF = 9 and A = 0.95: $t = 2.262$. The standard error is $\text{SE} = 5/\sqrt{10} = 1.581$, so the margin of error is $2.262 \times 1.581 = 3.58$ mmHg. The 95% CI is:

Example 2 — One-sample $t$-test

Mean cholesterol in 16 diabetic patients is 212 mg/dL (SD = 18 mg/dL). Is this significantly different from the population reference of 200 mg/dL?

The $t$-statistic is:

Degrees of freedom: $\text{DF} = 15$. From the table at DF = 15 and P = 0.05, the critical value is $t = 2.131$. Because $2.67 > 2.131$, the result is statistically significant at the 5% level: the patients' mean cholesterol differs from 200 mg/dL (P < 0.05).