T Table vs Z Table: Which One to Use and How to Read It

When to use each

Use the z table when the population standard deviation σ is known, or when a problem gives you a normal distribution to standardize. Use the t table when you estimate σ with the sample standard deviation s. OpenStax states that the practice now is to use the Student's t distribution whenever s is used as an estimate for σ.

In real data σ is rarely known, so t is the usual choice for inference about a mean from a sample. The t distribution has more probability in its tails than the standard normal distribution, so its critical values are larger.

Degrees of freedom

A t table has one row per degrees of freedom (df). For a single sample mean, df = n − 1. A sample of 11 values has df = 10. Small df means heavier tails and larger critical values.

One-tailed and two-tailed columns

Many t tables label columns by the area in one tail and by the two-tailed area. A 95% two-tailed interval leaves 0.025 in each tail, so you use the column for one-tail 0.025, which is the same as two-tailed 0.05.

The NIST table labels columns by cumulative probability instead. For a two-sided test with α = 0.05, it uses the 0.975 column (1 − α/2).

Worked lookup

Find the critical value for a 95% two-tailed interval with df = 10. Go to row 10 and the 0.025 one-tail column: t = 2.228.

Example: n = 11 and s = 6. The standard error is 6 / √11 ≈ 1.809, and the margin of error is 2.228 × 1.809 ≈ 4.03. Using z = 1.960 by mistake would give about 3.55 and an interval that is too narrow.

t approaches z as df grows

As df increases, the t distribution looks more like the standard normal. For a 95% two-tailed interval, the critical values are 12.706 at df = 1, 2.228 at df = 10, 2.042 at df = 30, and 1.960 in the infinity row, which matches the z value.

That is why some courses allow z as an approximation for large samples. A t table is still the safer choice when σ is unknown.