Why is 1.96 the number, not 2?

Because the two-tailed p at z = 1.96 is 0.0499958, just under the 5% line, while z = 2 gives 0.0455. The exact cut is 1.959964. Tables rounded it to 1.96 in the 1920s and it stuck; 2 is the mental shortcut, 1.96 is the value.

What does the p-value actually mean?

The probability of seeing data at least this extreme if the null hypothesis were true. It is not the probability the null is true, not the probability your result is a fluke, and not a measure of effect size. The American Statistical Association issued a formal statement in 2016 specifically because those three misreadings are so common.

How far into the tail is this accurate?

To the digits shown, out past z = 6, where the two-tailed p is about 1.97 in a billion. The two-tailed branch is computed as 1 − erf(|z|/√2) instead of 2(1 − Φ(|z|)) precisely so the tail does not lose its significant digits to cancellation. Beyond z = 8 report the result as a bound, not a figure.

What is "five sigma"?

z = 5, a two-tailed p of about 5.7 in ten million, the evidence threshold particle physics adopted for claiming a discovery. It is roughly a million times stricter than the 0.05 used across most of the social and life sciences.

Z-Score to P-Value Calculator

With z-score 1.96, z-score to p-value comes to 0.0499958 — p-value. It is reached in 5 steps, the last of which is 1 * (1 - erf(abs(1.96) / sqrt(2))) + 0 * (1 - 0.9750021) + 0 * 0.9750021, and each one is printed on the page with its numbers filled in. The formula is the one published by NIST/SEMATECH e-Handbook of Statistical Methods, not an approximation fitted to it.

Exact p-value from a z-score, one-tailed or two-tailed, with the percentile the score sits at.

Formula and sources checked 17 August 2026 · How we check

Z-score 1.96, Test Two-tailed — is it different?

0.0499958

P-value for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.

It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.

Z-scoreHow many standard deviations the observation sits from the mean.Test

P-value

0.0499958

Φ(z), area to the left: 0.5 * (1 + erf(1.96 / sqrt(2)))0.975
Percentile: 0.9750021 * 10097.5 %
P-value: 1 * (1 - erf(abs(1.96) / sqrt(2))) + 0 * (1 - 0.9750021) + 0 * 0.97500210.05
Confidence level this clears: (1 - 0.0499958) * 10095 %
Odds against, 1 in: 1 / 0.049995820.002

Ask about this in the chat Z-Score to P-Value Table

Worked example

A z of 1.96 gives a two-tailed p of 0.0499958, which is why 1.96 and not 2 is the number behind a 95% confidence interval. The exact 5% cut is z = 1.959964.

How to work it out yourself

1.Enter the z-score your test produced: the observation minus the mean, divided by the standard error.
2.Choose the test. Two-tailed asks whether the result differs from the null in either direction; one-tailed commits to a direction before seeing the data, and must be chosen for that reason rather than because it gives a smaller number.
3.The p-value is the area in the tail: the probability of a result at least this extreme if the null hypothesis were true.

Two-tailed p-value by z-score

Two-tailed p-value by z-score
Z-score	P-value	Percentile
1	0.317311	84.13 %
1.5	0.133614	93.32 %
1.645	0.0999698	95 %
1.96	0.0499958	97.5 %
2	0.0455003	97.72 %
2.33	0.0198062	99.01 %
2.576	0.00999506	99.5 %
3	0.0026998	99.87 %
3.291	0.000998319	99.95 %
4	6.33425e-5	100 %
5	5.73303e-7	100 %

The rows people actually look up: 1.645 is the 10% two-tailed cut, 1.96 the 5%, 2.576 the 1%, and 5 the "five sigma" of particle physics.

The formula

Φ(z), area to the left0.5 * (1 + erf(1.96 / sqrt(2)))
Percentile0.9750021 * 100
P-value1 * (1 - erf(abs(1.96) / sqrt(2))) + 0 * (1 - 0.9750021) + 0 * 0.9750021
Confidence level this clears(1 - 0.0499958) * 100
Odds against, 1 in1 / 0.0499958

Source: NIST/SEMATECH e-Handbook of Statistical Methods — cumulative distribution function of the standard normal, American Statistical Association — statement on p-values

Questions people actually ask

Why is 1.96 the number, not 2?: Because the two-tailed p at z = 1.96 is 0.0499958, just under the 5% line, while z = 2 gives 0.0455. The exact cut is 1.959964. Tables rounded it to 1.96 in the 1920s and it stuck; 2 is the mental shortcut, 1.96 is the value.
What does the p-value actually mean?: The probability of seeing data at least this extreme if the null hypothesis were true. It is not the probability the null is true, not the probability your result is a fluke, and not a measure of effect size. The American Statistical Association issued a formal statement in 2016 specifically because those three misreadings are so common.
One-tailed or two-tailed?: Two-tailed unless you committed to a direction before collecting data and a result in the other direction would be treated as no effect at all. A one-tailed test halves the p-value, which is why switching to one after seeing the numbers is a well-known way to manufacture significance.
How far into the tail is this accurate?: To the digits shown, out past z = 6, where the two-tailed p is about 1.97 in a billion. The two-tailed branch is computed as 1 − erf(|z|/√2) instead of 2(1 − Φ(|z|)) precisely so the tail does not lose its significant digits to cancellation. Beyond z = 8 report the result as a bound, not a figure.
What is "five sigma"?: z = 5, a two-tailed p of about 5.7 in ten million, the evidence threshold particle physics adopted for claiming a discovery. It is roughly a million times stricter than the 0.05 used across most of the social and life sciences.

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