P-Value Calculator
P-value from a z, t, chi-square or F statistic and its degrees of freedom, left-, right- or two-tailed, with an optional comparison against alpha. How it works
Use the P-Value Calculator
About the P-Value Calculator
A z-score of 1.96 has a two-tailed p of 0.049996, which rounds to 0.05000, and a right-tailed p of 0.02500. This p-value calculator turns a z, t, chi-square or F statistic into the tail probability that goes with it, using the degrees of freedom you enter, and can compare the result with a significance level α.
P-values for common z-scores
| z | Two-tailed p | Right-tailed p |
|---|---|---|
| 1 | 0.3173 | 0.1587 |
| 1.282 | 0.1998 | 0.09992 |
| 1.645 | 0.09997 | 0.04998 |
| 1.96 | 0.05000 | 0.02500 |
| 2 | 0.04550 | 0.02275 |
| 2.326 | 0.02002 | 0.01001 |
| 2.576 | 0.009995 | 0.004998 |
| 3 | 0.002700 | 0.001350 |
| 3.291 | 0.0009983 | 0.0004992 |
| 4 | 6.334 × 10^-5 | 3.167 × 10^-5 |
| 5 | 5.733 × 10^-7 | 2.867 × 10^-7 |
The left-tailed p for a negative z equals the right-tailed p for the same positive z.
t critical values for a two-tailed test at 0.05, and the p each one gives
| Degrees of freedom | t | Two-tailed p |
|---|---|---|
| 1 | 12.706 | 0.05000 |
| 2 | 4.303 | 0.04999 |
| 5 | 2.571 | 0.04997 |
| 10 | 2.228 | 0.05001 |
| 20 | 2.086 | 0.05000 |
| 30 | 2.042 | 0.05003 |
| 60 | 2.000 | 0.05003 |
| 120 | 1.980 | 0.04999 |
Critical values are rounded to three decimals, as in printed tables, so p comes out within 0.0001 of 0.05 and is sometimes just above it.
Chi-square critical values for a right-tailed test at 0.05
| Degrees of freedom | Chi-square | Right-tailed p |
|---|---|---|
| 1 | 3.841 | 0.05001 |
| 2 | 5.991 | 0.05001 |
| 3 | 7.815 | 0.04999 |
| 4 | 9.488 | 0.04999 |
| 5 | 11.070 | 0.05001 |
| 10 | 18.307 | 0.05000 |
How to use this tool
- Choose the type of test statistic: z, t, chi-square or F.
- Type the statistic. Add the degrees of freedom for t and chi-square, or the numerator and denominator degrees of freedom for F.
- Choose the tail: two-tailed, right-tailed or left-tailed for z and t, and right- or left-tailed for chi-square and F.
- To compare with a significance level, enter α, such as 0.05. The tool then says whether p is below it.
Frequently asked questions
- How do I find the p-value from a z-score?
- Take the area under the standard normal curve beyond your z. For z = 2.5 the right-tailed p is 0.006210 and the two-tailed p is 0.01242. A negative z in a left-tailed test gives the mirror-image area: z = -1.96, left-tailed, is 0.02500.
- How do I find the p-value from a t statistic?
- Choose t, then enter the statistic and its degrees of freedom. t = 2.5 with 10 degrees of freedom gives a two-tailed p of 0.03145 and a right-tailed p of 0.01572. Fewer degrees of freedom mean a larger p for the same statistic: a value of 2 gives 0.07339 with 10 degrees of freedom, 0.05927 with 20 and 0.04550 on the normal curve.
- Should I use a one-tailed or a two-tailed p-value?
- That follows from the alternative hypothesis, which is set before looking at the data. If a difference in either direction would count, use two-tailed; if only one direction would, use that tail. For z and t the two-tailed value is twice the smaller one-tailed value: z = 2 is 0.02275 right-tailed and 0.04550 two-tailed.
- What does it mean if p is less than 0.05?
- A statistic at least this extreme would turn up less than 5% of the time if the null hypothesis were true. The American Statistical Association’s 2016 statement adds that a p-value does not measure the probability that the hypothesis is true, or the size or importance of an effect, and that conclusions should not rest only on whether p passes a threshold. When you enter α, the tool reports that p is below α or not below it, and nothing else.
- Which degrees of freedom do I enter?
- That depends on the test. A one-sample t test on n values uses n − 1: with 15 values, t = 2.5 and 14 degrees of freedom give a two-tailed p of 0.02547. A regression slope uses n − 2. A chi-square goodness-of-fit test with k categories uses k − 1, so a die tested for fairness has 5, and a chi-square of 12 then gives 0.03479. A test of independence in a table with r rows and c columns uses (r − 1) × (c − 1), so a 2 × 3 table has 2.
- Why is there no two-tailed option for chi-square and F?
- The tool offers the right and left tails only, because the usual chi-square and F tests use the right tail: a chi-square of 10 with 4 degrees of freedom gives p = 0.04043, and F = 4.96 with 1 and 10 degrees of freedom gives 0.05009. A two-sided test of one variance, or of the ratio of two, does exist: it puts α/2 in each tail, which is the same as doubling the smaller one-tailed p (capped at 1). F = 2.5 with 20 and 20 degrees of freedom is 0.02329 right-tailed, so the two-sided p is 0.04658.
Related Tools
View all toolsLinear Regression Calculator
Least-squares line y = mx + b for your (x, y) data, with Pearson r, R², the number of points used and predictions of y from x or x from y.
Average Calculator
Calculate mean, median, mode, range and sum for a list of numbers separated by commas, spaces or new lines, up to 10,000 values.
Permutation and Combination Calculator
Counts permutations nPr and combinations nCr for n and r up to 1000, with or without repetition. Every digit is exact and the formula is worked through.
Sample Size Calculator
Survey sample size from the confidence level, margin of error and expected proportion, or the margin of error for a sample you have: 95% and ±5% need 385.
Standard Deviation
Standard deviation calculator for a list of numbers: population σ and sample s side by side, both variances, plus mean, median, minimum, maximum, range and sum.
Scientific Calculator
Keypad or keyboard calculator with brackets, sin, cos, tan and their inverses in degrees or radians, log, ln, √, x^y, 10ˣ, memory and Ans, in the usual order.