T-Distribution Table

Type of t-table

Degrees of Freedom

What is the T-Distribution?

The t-distribution, also called Student's t-distribution, is a probability distribution used when sample sizes are small or population standard deviation is unknown. It's similar to the normal distribution but has heavier tails, accounting for additional uncertainty. The shape depends on degrees of freedom (df = n - 1), approaching the normal distribution as df increases. It's essential for t-tests and constructing confidence intervals.

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How to Read the T-Table

The table shows critical t-values for different significance levels (α) and degrees of freedom (df). The left column lists degrees of freedom, while column headers show significance levels for one-tailed and two-tailed tests. For a 95% confidence interval with 20 df, use the 0.025 two-tailed column (α/2 = 0.025). The intersection gives the critical value. Values beyond this threshold indicate statistical significance.

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T-Distribution Properties

The t-distribution is symmetric around zero with mean = 0. Variance equals df/(df-2) for df > 2. As degrees of freedom increase, the distribution converges to the standard normal distribution (z-distribution). With df ≥ 30, t-values approximate z-values. The heavier tails account for estimation uncertainty when using sample standard deviation instead of population standard deviation.

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Common Applications

T-distributions are used in one-sample t-tests (comparing sample mean to population mean), two-sample t-tests (comparing two group means), paired t-tests (comparing before/after measurements), confidence intervals for means with unknown population variance, regression analysis for testing coefficient significance, and ANOVA. Essential when working with small samples (n < 30) or unknown population parameters.

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Degrees of Freedom

Degrees of freedom represent the number of independent pieces of information available for estimation. For single sample tests, df = n - 1. For two-sample tests, df calculation depends on whether variances are equal. Lower df means wider distribution and higher critical values, reflecting greater uncertainty. Critical values determine rejection regions for hypothesis tests and confidence interval widths.

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