What does a t statistic say about a hypothesis? To find its p-value, use the statistic, its degrees of freedom, and the direction of the alternative hypothesis to locate the relevant tail area. A printed table often gives a range for that area rather than an exact decimal.
For example, with t = 2.00 and 10 degrees of freedom, a two-sided test gives a p-value between 0.05 and 0.10 using the table below. The same table can be used to find a probability from a t statistic in one tail, but the answer is different because the question is different.
How to Find a P Value From a T Table: Identify the Tail
A t statistic measures a difference relative to its estimated standard error. To calculate the p-value from a t statistic, select the appropriate t distribution using the degrees of freedom; these depend on the test. For a one-sample t test with 11 independent observations, for example, the usual degrees of freedom are 11 minus 1, or 10. Other tests can use different formulas, so take the degrees of freedom from the test being performed.
Choose the alternative hypothesis before inspecting the result. An alternative that the mean is greater than a reference value calls for a right-tailed test. An alternative that it is less calls for a left-tailed test. An alternative that it differs in either direction calls for a two-sided test. With a two-sided test, both unusually high and unusually low statistics count as extreme.
When deriving a p-value from a t table, its heading matters. Some tables label the area to the right of a positive critical value; others label the combined area in both tails or the cumulative area to the left. The Penn State t table uses upper-tail areas. The excerpt here uses those one-tail headings and adds the equivalent two-tail areas for a symmetric t distribution.
Work through t = 2.00 with 10 degrees of freedom
Imagine a one-sample test of a population mean against a stated reference value. The alternative says the mean differs in either direction. Suppose the test gives t = 2.00 with 10 degrees of freedom. The data and calculation are hypothetical; the table lookup below is a worked example.
| Upper-tail area | Combined two-tail area | Critical t, df = 10 |
|---|---|---|
| 0.050 | 0.100 | 1.812 |
| 0.025 | 0.050 | 2.228 |
To see how to find a p value from t table entries, locate the row for 10 degrees of freedom and compare the positive statistic with the critical values. Here, 1.812 < 2.00 < 2.228. These neighboring cutoffs let the reader estimate the p-value associated with the observed t. A larger positive t leaves less area to its right, so the right-tail area for 2.00 is between 0.025 and 0.050. For this two-sided test, double that area: 0.05 < p < 0.10.
The two-tail column in the excerpt already gives the doubled areas. Reading that column leads directly to the same bounds; do not double it again. The table does not show the precise p-value for a t score of 2.00, so report the interval rather than calling either endpoint the exact p-value.
One tail, two tails, and a negative statistic
If the alternative had been “greater than” and was chosen in advance, the relevant right-tail p-value for the same t would be between 0.025 and 0.050. If the alternative had been “less than,” the relevant area would be to the left of positive 2.00, between 0.950 and 0.975. Choosing a direction after seeing the sign changes the test being reported.
To obtain a p-value for a negative t statistic, use the symmetry of the distribution. For t = −2.00 and 10 degrees of freedom, the left-tail area equals the right-tail area beyond +2.00. It is between 0.025 and 0.050. A two-sided test again combines both tails and gives a p-value between 0.05 and 0.10. The sign still matters for explaining the observed direction, even though the two-sided calculation uses the magnitude.
To practice how to find a p value from t, first write the alternative in words and mark the relevant tail or tails. A table of positive values can then be read using symmetry. This step is particularly helpful when an observed negative result points away from a preselected right-tailed alternative: the right-tail area is then large, not the small area beyond the corresponding positive magnitude.
Interpret the result in context
A p-value describes the probability, assuming the null hypothesis and the test model, of a statistic at least as extreme as the one observed in the direction specified by the alternative. NIST’s explanation makes this conditional interpretation explicit. It is not the probability that the null hypothesis is true, nor the size or practical importance of a difference.
If the significance level was set at 0.05 in advance, the example’s two-sided p-value is above that threshold. The test therefore does not reject the null at the 5% level. That result does not prove the null is true: it says the observed evidence did not meet the chosen threshold under the stated test. Report the estimate and a confidence interval when available, so readers can also assess the size and uncertainty of the difference.
A calculator or statistical program can give a more precise tail probability. A table remains useful for seeing how to find a p value from t and for checking whether a reported result is plausible. Enter the correct degrees of freedom and tail choice in either case; a precise decimal from the wrong test is still the wrong answer.
A reader checking how to get a p value from t should also verify that the t procedure is appropriate for the study design and assumptions. The t-table lookup only converts the statistic into a p-value under the model; it cannot repair an incorrectly calculated statistic or an unsuitable test.
Quick checklist
- Record the t statistic, degrees of freedom, and alternative hypothesis.
- Identify whether the table labels one-tail, two-tail, or cumulative areas.
- Locate the correct degrees-of-freedom row and bracket the statistic.
- Read or combine tails according to the alternative, using symmetry for negative values.
- Report the resulting value or interval with the null hypothesis and test context.
