P-value calculator
A z-score of 1.96 gives a two-tailed p of 0.05, which is where the convention comes from rather than the other way round. The same z one-tailed is 0.025. A t statistic needs its degrees of freedom before it means anything: 1.96 with 10 of them is p = 0.078, not 0.05.
From a z-score, a t statistic or a chi-square, with the degrees of freedom that change the answer. Every value here is checked by test against the critical values printed in statistical tables, not against itself.
Checked against published critical values for t and χ² · How we check
p-value
0.050012
- Area to the left
- 0.974994
- At α = 0.05
- not below
- At α = 0.01
- not below
- As a frequency
- about 1 in 20
That frequency is how often data this extreme would arise if the null hypothesis were true. It is not the probability that the null is true, and it is not the probability that your result is a fluke — those are different quantities and a p-value cannot give either.
What the table rounds away
A t table gives 2.228 as the two-tailed 5% critical value at ten degrees of freedom. Put exactly 2.228 in and the p-value is 0.050012 — just above 0.05, on the wrong side of the line it is meant to mark. The exact critical point is 2.22814; the table rounded it down.
The same holds for chi-square: the tabled 3.841 at one degree of freedom returns 0.050014. Neither is an error. It is what happens when a continuous quantity is published to three decimal places and then used as if it were a boundary.
Questions people actually ask
- Can I use this as a chi-square calculator?
- Yes. Pick chi-square as the distribution, enter your statistic and the degrees of freedom, and it returns the p-value. For a goodness-of-fit test the degrees of freedom are the number of categories minus one; for a contingency table they are (rows − 1) × (columns − 1). A 2×2 table therefore has one degree of freedom, and a chi-square of 3.84 there is exactly p = 0.05.
- Does p = 0.03 mean there is a 3% chance the result is wrong?
- No, and this is the most common misreading there is. A p-value is the probability of data at least this extreme assuming the null hypothesis is true. It says nothing about the probability that the null is true, because that quantity depends on how plausible the hypothesis was before you collected anything — which the p-value has no access to.
- One tail or two?
- Two, unless you decided otherwise before seeing the data. A one-tailed test asks whether the effect goes in one specific direction and has no power at all to detect the opposite. Switching to one tail after looking at which way the result went halves your p-value by construction and is not a test of anything.
- What are degrees of freedom?
- How many values were free to vary once the estimates were fixed. A one-sample t-test on n observations has n − 1, because the sample mean uses one up. Fewer degrees of freedom means fatter tails and a larger p-value for the same statistic: at df = 10 a t of 2.228 gives 0.05, at df = 100 it gives 0.028.
- Why does chi-square have no two-tailed option?
- Because it is a sum of squared deviations, so it is already one-sided: large values mean the data are far from expectation, and that is the only direction the test can find anything in. A very small chi-square means the fit is suspiciously good, which is a different question and not a smaller p-value.
- Is 0.049 meaningfully different from 0.051?
- No. The threshold is a convention, not a discovery, and the difference between those two numbers is nothing next to the uncertainty in almost any real sample. Reading the exact critical value off a table makes the point: 2.228 at ten degrees of freedom is the tabled figure, and it actually gives p = 0.050012, on the wrong side of a line it is supposed to define.