Guess the Critical t

Quick answer: the critical t (t*) is the t-table value in the row for your degrees of freedom and the column for your α. Two-tailed α = 0.05 with df = 10 gives t* = 2.228; one-tailed α = 0.05 with df = 10 gives t* = 1.812. For a confidence interval for one mean, df = n − 1.

This game makes that lookup automatic: 10 short rounds, a running score and streak, and the right t* with the row and column it came from after every answer.

Play: Guess the Critical t

Ten quick rounds mixing two-tailed tests, one-tailed tests, and confidence intervals (where you work out df = n − 1). Tap the t* you think is right, or type it (3 decimals, or 2 decimals rounded). Every answer is a value from the t-table on our homepage.

Worked example

Question: you want a 95% confidence interval for a mean from n = 16 measurements. Which t* do you use?

Step 1. Degrees of freedom: df = n − 1 = 15.

Step 2. A 95% interval leaves 5% in the two tails, so use the two-tailed table, column α = 0.05 (95%).

Answer: row df = 15 gives t* = 2.131. A common slip is reading the one-tailed α = 0.05 column, which gives 1.753 (that is a 90% interval).

FAQ

How do I find the critical t value?

Pick the table that matches your test (one-tailed or two-tailed), go to the column for your α, and read across from your degrees of freedom. A two-tailed test with α = 0.05 and df = 10 gives t* = 2.228; a one-tailed test with α = 0.05 and df = 10 gives t* = 1.812.

Why is the two-tailed critical t bigger than the one-tailed one?

A two-tailed test splits α between both tails, so each tail only holds α/2 and the cutoff moves further out. Two-tailed α = 0.05 uses the same t* as one-tailed α = 0.025.

What degrees of freedom do I use for a confidence interval for one mean?

df = n − 1. A 95% interval from n = 16 observations uses df = 15 and t* = 2.131.

What if my df is not in the table?

Use the next smaller df in the table (it gives a slightly larger, more conservative t*), or get the exact value with the T-table calculator.

How does critical t relate to the z critical value?

As df grows, t* shrinks toward z*. The ∞ row of the t-table is the normal z value, for example 1.960 for two-tailed α = 0.05. For z-scores and areas see the z-table and its Guess the Z game.

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