How to use the t-table
- Work out your degrees of freedom. For one sample or paired data, that's n − 1.
- Decide whether your test is one-tailed or two-tailed, or whether you need a confidence interval (use the confidence-level row).
- Find your column using the matching header row, for example two-tailed α = 0.05 or 95% confidence.
- Go down to your df row and read the critical value t*. For df = 15 at 95%, t* = 2.131.
- Or enter your exact df in the lookup above. It calculates t* for any df and highlights the closest table cell.
What the critical value means
The critical value t* marks the cutoff in a t distribution beyond which the tail area equals your chosen α. In a hypothesis test you reject the null hypothesis when your test statistic is beyond it; in a confidence interval, it sets the margin of error:
Two-tailed test: reject H0 if |t| > tα/2, df
Confidence interval: x ± t* × sn
- t*
- The critical value read from the table
- α
- The significance level: the total tail area you are willing to treat as unusual (two-tailed tests split it between both tails)
- df
- Degrees of freedom, which set how heavy the tails of the t distribution are
- s, n
- The sample standard deviation and sample size
Worked example
A sample of 16 light bulbs has a mean life of 1,200 hours and a standard deviation of 80 hours. Find the 95% confidence interval for the mean life.
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Degrees of freedom
df = 16 − 1 = 15
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Row 15, 95% confidence column
t* = 2.131
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Margin of error
ME = 2.131 × 8016 = 2.131 × 20 = 42.62
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Interval
1,200 ± 42.62 = 1,157.4 to 1,242.6 hours
What it means: we can be 95% confident the true mean life of these bulbs is between about 1,157 and 1,243 hours. With the z value 1.960 instead of 2.131, the interval would be too narrow, because a sample of 16 doesn't pin down the standard deviation precisely.
Tips for using t critical values
- Smaller samples mean larger t*. With df = 5 the 95% value is 2.571; with df = 100 it is 1.984. The extra width accounts for uncertainty in s.
- Match the header to your test. Using a one-tailed column for a two-tailed test is the most common mistake; it makes results look more significant than they are.
- Compare with your statistic, or get the p-value. The table only tells you whether p is below α. For the exact p-value, use the p-value calculator.
- Known σ? If the population standard deviation is known, use the z-table instead.