P Value Calculator
The p value is the probability of seeing a result this extreme if the null hypothesis were true. Pick your test, enter the statistic, and the calculator returns the exact p value with the working shown.
Enter your test statistic and this p value calculator returns the exact probability, with every step shown. It works for a z score, a t value, a chi-square statistic, an F value from ANOVA, or a correlation coefficient. A p value tells you one thing: if the null hypothesis were true, how likely is a result this extreme or more so.
A small p value (below your significance level, usually 0.05) means the result is unlikely to be chance, so you reject the null. A large p value means you fail to reject it. Below you will find worked examples using the exact numbers people search for, quick reference tables for z, t and chi-square, and the rules for one-tailed versus two-tailed tests that decide the answer.
What is a p value in simple terms?
A p value is the probability of getting a result at least as extreme as the one you observed, assuming the null hypothesis is true. It is a number between 0 and 1. The smaller it is, the stronger your evidence against the null.
Think of the null hypothesis as the boring explanation: nothing is going on, any difference is just random noise. The p value scores how well that boring explanation survives your data. If your data would be very surprising under “nothing is going on,” the p value drops, and at some point you stop believing the boring story.
People read too much into the number, so here is what a p value is not. It is not the probability that the null hypothesis is true. It is not the probability your result happened purely by chance. And it says nothing about how big or important an effect is. A tiny, useless difference can still land below 0.05 with a large enough sample.
What does the p value mean when you get a result?
Once you have a p value, the reading is straightforward. A low value is evidence against the null. A high value means your data fits the null comfortably. Here is how the common ranges are usually described.
| P value | What it means | Decision at α = 0.05 |
|---|---|---|
| p ≤ 0.01 | Very strong evidence against the null | Reject |
| 0.01 < p ≤ 0.05 | Significant, the usual cutoff | Reject |
| 0.05 < p ≤ 0.10 | Marginal, weak evidence | Fail to reject |
| p > 0.10 | No real evidence against the null | Fail to reject |
The 0.05 line is a convention, not a law of nature. A p value of 0.049 and one of 0.051 are almost identical in strength, so treat values near the cutoff with judgment rather than a hard yes or no.
How do you find a p value from a test statistic?
Every p value comes from the same three decisions. Get these right and the calculator does the arithmetic.
1. Pick the distribution. A z score uses the standard normal curve. A t value uses the t distribution, which needs degrees of freedom. Chi-square and F have their own right-skewed curves. A correlation coefficient gets converted into a t value first.
2. Choose the tail. Right-tailed, left-tailed, or two-tailed. This single choice can double or halve your p value, so it matters more than people expect.
3. Read the area under the curve. The p value is the area in the tail beyond your statistic. That is the calculation you would normally look up in a z table or t table. This tool does it to full precision instead of rounding to two decimals.
How do you find the p value from a z score?
For a right-tailed test the p value is the area to the right of your z score. For a left-tailed test it is the area to the left. For a two-tailed test, take the area beyond the absolute value of z and double it.
A z score of 1.96 is the classic case. The area to its right is 0.025. Double it for a two-tailed test and you get exactly 0.05, the standard 5% cutoff. That is why 1.96 shows up everywhere in statistics.
Worked example: z = 2.43685, one-sided
Area to the right of z = 2.43685 under the normal curve.
One-tailed p = 0.0074
Well below 0.05, so a one-sided test rejects the null. Two-tailed it would be 0.0148, still significant.
Worked example: z = 2.5, two-tailed
p = 2 × P(Z > 2.5) = 2 × 0.0062
Two-tailed p = 0.0124
Worked example: z = 1.5, right-tailed
Area to the right of z = 1.5.
One-tailed p = 0.0668
Above 0.05, so this one does not reach significance.
Worked example: z = −2.05, left-tailed
Area to the left of z = −2.05.
One-tailed p = 0.0202
Quick reference: z score to p value
| Z score | One-tailed p | Two-tailed p |
|---|---|---|
| 1.00 | 0.1587 | 0.3173 |
| 1.28 | 0.1003 | 0.2005 |
| 1.645 | 0.0500 | 0.1000 |
| 1.96 | 0.0250 | 0.0500 |
| 2.33 | 0.0099 | 0.0198 |
| 2.576 | 0.0050 | 0.0100 |
| 3.00 | 0.0013 | 0.0027 |
How do you find the p value from a t score?
A t score needs one extra input: degrees of freedom, usually n minus 1. With that, the p value is the tail area under the t curve, applied the same way as z (right, left, or doubled for two-tailed).
The t distribution has fatter tails than the normal curve, so the same statistic gives a larger p value at low degrees of freedom. As the sample grows, the t curve tightens and starts to look like the normal one.
Worked example: t = 2.262, df = 9, two-tailed
Area beyond t = 2.262 with 9 degrees of freedom, doubled.
Two-tailed p = 0.0500
Sits right on the 0.05 line. This is the t equivalent of z = 1.96.
Worked example: t = 3.2, df = 12, two-tailed
Area beyond t = 3.2 with 12 df, doubled.
Two-tailed p = 0.0076
Worked example: t = 1.7, df = 24, right-tailed
Area to the right of t = 1.7 with 24 df.
One-tailed p = 0.0510
Just misses 0.05. A good reminder that being close is not the same as significant.
Watch what df does to the same t value. With t = 2.262, moving from df = 9 to a much larger sample shrinks the p value, because the tails get thinner. This is exactly why you cannot read a t statistic off a z table when your sample is small, you would report a p value that looks more significant than it really is.
Quick reference: t value for two-tailed p = 0.05
| Degrees of freedom | t value (two-tailed 0.05) |
|---|---|
| 10 | 2.228 |
| 20 | 2.086 |
| 30 | 2.042 |
| 100 | 1.984 |
| Infinite (z) | 1.960 |
How do you calculate the p value from chi-square?
The chi-square test is always right-tailed. Feed in your chi-square statistic and the degrees of freedom, and the p value is the area to the right. Because chi-square is built from squared differences, only large values count as evidence, so there is no left tail to worry about.
For a test of independence, degrees of freedom equal (rows minus 1) times (columns minus 1). For a goodness of fit test, it is the number of categories minus 1. Get df wrong and your p value is wrong, this is the most common chi-square mistake. Our chi-square calculator works out the statistic itself if you only have observed and expected counts.
Worked example: chi-square = 9.48773, df = 4
Area to the right of 9.48773 on the chi-square curve with 4 df.
p = 0.0500
Exactly at the threshold. 9.488 with 4 df is the chi-square critical value for 0.05.
Worked example: chi-square = 15.0, df = 5
Area to the right of 15.0 with 5 df.
p = 0.0104
Below 0.05, so you reject the null of no association.
Quick reference: chi-square critical values (p = 0.05)
| Degrees of freedom | Chi-square at p = 0.05 |
|---|---|
| 1 | 3.841 |
| 2 | 5.991 |
| 3 | 7.815 |
| 4 | 9.488 |
| 5 | 11.070 |
A chi-square statistic can never be negative, because it adds up squared differences. If you got a negative number, recheck your observed and expected values.
How do you find the p value from an F value?
The F test in ANOVA is right-tailed. It needs two degrees of freedom: df1 for the numerator (groups minus 1) and df2 for the denominator (total observations minus number of groups). The p value is the area to the right of your F value.
F compares variance between groups against variance within groups. A big F means the group means are spread out more than random noise would explain, which is why only large F values push the p value down. For example, F = 4.256 with df1 = 3 and df2 = 12 gives p = 0.0290, significant at 0.05.
How do you get a p value from a correlation coefficient?
To test whether a correlation r is significant, the calculator converts it to a t statistic using t = r times the square root of (df divided by 1 minus r squared), where df = n minus 2. Then it reads the p value off the t distribution. For example, r = 0.7 with n = 10 gives t = 2.77 and a two-tailed p of 0.0242, so the correlation is significant at 0.05.
Can you find a p value from the mean and standard deviation?
Not directly. A p value always comes from a test statistic, so you first turn your mean and standard deviation into a z or t score, then read the p value from that.
For a one-sample test, the z score is (sample mean minus the hypothesized mean) divided by (standard deviation divided by the square root of n). Say your sample mean is 52, the null value is 50, the standard deviation is 5, and n = 30. That gives z = 2.19, and a two-tailed p of 0.0285. So the path is always the same: mean and standard deviation give you the statistic, and the statistic gives you the p value you enter above.
How do you find a p value in Excel?
Excel has built-in functions that return the p value once you have your data or statistic. Pick the one that matches your test.
| Test | Excel function |
|---|---|
| t-test between two samples | T.TEST(array1, array2, tails, type) |
| Chi-square test | CHISQ.TEST(actual_range, expected_range) |
| z-test | Z.TEST(array, x, sigma) |
| F-test | F.TEST(array1, array2) |
| p from a t statistic | T.DIST.2T(t, df) for two-tailed |
| p from a z score | 1 - NORM.S.DIST(z, TRUE) for right tail |
For T.TEST, set tails to 1 or 2 and type to 1 (paired), 2 (equal variance), or 3 (unequal variance). If you already have the statistic and just need the tail area, the DIST functions are faster than looking up a table.
How do you calculate a p value by hand?
By hand, you compute the test statistic with the right formula, then look up the tail area in a distribution table. The z formula for a mean is (x-bar minus mu) divided by (sigma over root n). The t formula is the same shape but uses the sample standard deviation s and needs df = n minus 1. For chi-square you sum (observed minus expected) squared, divided by expected.
Once you have the statistic, a printed table only gives you the nearest cutoff, not the exact value. That is the limit of doing it by hand: you can confirm whether you beat 0.05, but you cannot report a precise p like 0.0285 without software. The calculator above fills that gap.
One-tailed or two-tailed: which should you use?
Use a two-tailed test when you care about a difference in either direction. Use a one-tailed test only when your hypothesis was directional before you saw the data, for example “the new method is faster,” not just “different.”
The stakes are real. The same z or t value gives a p value twice as large in a two-tailed test as in a one-tailed test. A result that reads as significant one-tailed (p = 0.03) can flip to not significant two-tailed (p = 0.06). Decide the tail before running the test, never after, or you are fishing for significance.
Left-tailed versus right-tailed is about direction. A right-tailed test asks whether your statistic is unusually high, a left-tailed test whether it is unusually low. Chi-square and F are always right-tailed because their statistics only get large when something is going on.
When do you reject the null hypothesis?
Reject the null when the p value is less than your significance level (alpha). If p is greater than or equal to alpha, you fail to reject it. “Fail to reject” is not the same as “prove the null true,” it just means you did not find enough evidence against it.
So with alpha at 0.05: a p value of 0.02 means reject, a p value of 0.08 means fail to reject. The calculator above states this decision for you against the alpha you pick, alongside the exact p value.
Where p values actually get used
Outside the classroom, p values decide real calls. In medical trials, a low p value on a treatment effect is part of the evidence a drug works. In A/B testing, teams check whether a new page or price genuinely lifts conversions or just looks better by luck. In academic research, p values back up whether an experimental result holds. The warning in every one of these fields is the same: a significant p value is a starting point, not proof, and it should sit next to an effect size and a confidence interval, never stand alone.
Common mistakes when calculating a p value
- Wrong test for the data. Using a z-test when the sample is small and the population standard deviation is unknown inflates significance. Small sample, unknown spread, use t.
- Wrong tail. Running a one-tailed test to squeeze under 0.05 after seeing the data is one of the most common ways results get overstated.
- Wrong degrees of freedom. In t and chi-square tests, an off-by-one df error changes the p value. Confirm your df with a t table before trusting the result.
- Rounding too early. Rounding your test statistic to two decimals before finding the p value can shift the answer near the cutoff. Keep full precision until the end.
- Reading the p value as an effect size. A significant p value tells you an effect probably exists, not that it is large or useful. Always look at the effect size too.
Try these next
- Critical Value Calculator
- Confidence Interval Calculator
- Hypothesis Testing Calculator
- Degrees of Freedom Calculator