t-Table (Student's t)

What Is a t-Table?

A t-table (also called Student's t-distribution table) displays critical values for the t-distribution at various degrees of freedom and significance levels. It's used when the population standard deviation is unknown and must be estimated from sample data.

The t-distribution was developed by William Sealy Gosset in 1908, who published under the pseudonym "Student" while working at the Guinness Brewery.

When to Use the t-Distribution

Use t critical values instead of z values when:

• Sample size is small (typically n < 30)
• Population standard deviation is unknown (using sample s instead of population σ)
• Data is approximately normally distributed

As the degrees of freedom increase, the t-distribution approaches the standard normal (z) distribution.

How to Read the t-Table

The table is organized with:

• Rows: Degrees of freedom (df)
• Columns: Significance level (α) for one-tailed tests

To find the critical t-value for a two-tailed test at α = 0.05 with 20 degrees of freedom:

1. Find the row for df = 20
2. For two-tailed tests, use α/2 = 0.025 column
3. The value is t = 2.086

Degrees of Freedom

For different tests, degrees of freedom are calculated as:

• One-sample t-test: df = n - 1
• Independent two-sample t-test: df = n₁ + n₂ - 2
• Paired t-test: df = n - 1 (number of pairs minus 1)
• Welch's t-test: Uses the Welch-Satterthwaite approximation

Common Applications

One-Sample t-Test

Testing if a sample mean differs from a hypothesized value:

$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

Two-Sample t-Test

Comparing means of two independent groups:

$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

Confidence Intervals

For a 95% CI with df = 15: t* = 2.131

$CI = \bar{x} \pm t^* \times \frac{s}{\sqrt{n}}$

t vs. z: A Comparison

The t-distribution has heavier tails than the normal distribution, which accounts for the extra uncertainty from estimating the standard deviation.

Historical Note

Gosset developed the t-distribution while working on quality control at Guinness Brewery. The company prohibited employees from publishing, so he used the pen name "Student." Ronald Fisher later expanded the work and introduced the terminology we use today.

Related Tools

• One-Sample t-Test Calculator — Perform the test
• Two-Sample t-Test Calculator — Compare two means
• Critical Value t Calculator — Get exact critical values
• T-Distribution Calculator — Find probabilities
• Confidence Interval Calculator — Build intervals

Summary

The t-table provides critical values for hypothesis testing and confidence interval construction when working with sample data. Unlike the z-table, the t-table accounts for the additional uncertainty of estimating population variance from a sample. As sample size increases, t critical values converge toward z values, but for small samples, using the correct t critical value is essential for accurate inference.