The formula
How to calculate p value
A p-value is the probability of seeing a result at least as extreme as the one observed, assuming the null hypothesis is true. It is a measure of how surprising the data would be if nothing were going on.
The exact calculation requires the normal cumulative distribution function, which has no elementary closed form. The logistic approximation used here is accurate to within roughly 0.01 across the usual range of z-scores.
What to enter:
- z-score
The result updates on every keystroke. The URL updates too, which makes the filled-in version easy to bookmark or send to someone else.
Where more decimal places matter than the fields above display, the underlying calculation is not rounded until the final figure is shown — the precision used internally is higher than what is printed, which matters for anyone chaining this result into a further calculation of their own.
Why p value matters
A p value calculation gets used both to check work already done by hand and to explore how a formula behaves without redoing the algebra every time an input changes — this page exists for both, since the underlying arithmetic is the same either way.
It is useful for checking a manual calculation before submitting or acting on it, and equally useful for building intuition about a formula by adjusting one input at a time and watching how the result moves in response — a much faster way to understand a relationship than working through several versions of the algebra by hand.
A formula like this one is rarely the last step in a piece of work — the figure it produces usually feeds into a further calculation, a comparison against a published value, or a write-up that needs to state both the result and how confident it is. Getting this step right the first time, rather than propagating a small arithmetic slip through several more steps, is the main practical reason to check a manual calculation against a tool like this one before building on top of it.
In practice, a formula like this one is most often reached for at the exact moment a manual calculation needs checking against a deadline — a lab report due, a problem set to submit — which is precisely the situation where a small arithmetic slip is easiest to miss and most costly to leave uncorrected. Running the same inputs through an independent calculator catches that class of error reliably.
Worked example
Here is the calculation with the starting values:
- z-score: 2.1
That gives:
- Two-tailed p-value: 0.03573
- One-tailed p-value: 0.01786
- Confidence level: 96.42713 %
The figures above are the calculator's own default values, shown purely so the working is visible rather than hidden — the same steps apply exactly to your own numbers, entered in the fields at the top of this page.
Reading the result
Below 0.05 is conventionally called significant, below 0.01 highly significant. Those thresholds are conventions inherited from Fisher, not laws — a p-value of 0.049 and one of 0.051 say almost exactly the same thing.
Where this goes wrong. A p-value is not the probability that the hypothesis is true, and it is not a measure of effect size. A tiny, useless difference will produce a very small p-value given a large enough sample.
A result that is wrong by an exact factor of ten, a hundred or a similar round number is almost always a units error rather than a mistake in the formula itself — checking each input against the unit stated beside it is the fastest way to track it down.
That data this extreme would occur less than 5% of the time if the null hypothesis were true. It does not mean there is a 95% chance the alternative is correct.
Two-tailed unless you had a directional hypothesis before seeing the data. Switching to one-tailed after the fact halves the p-value and is a well-known way to manufacture significance.
The headline figure is two-tailed p-value. With 2.1 z-score, that comes to 0.03573. Change any field and the figure moves with it.
Generally, no more than the least precise input justifies — a result reported to six decimal places from inputs measured to two significant figures is implying a precision the calculation does not actually have. The calculator shows full precision so you can round appropriately for your own use.
Yes — the equation shown in the formula section above is the standard form used in textbooks and reference material for this calculation, not a simplified or approximate version.
Yes, in the sense that it applies the correct standard formula and returns an accurate result for the inputs given — but check your own course or publication's requirements for how results should be rounded, presented and referenced, since those conventions vary and are not something a calculator can know on your behalf.