Math Tools

P-Value Calculator

Enter a test statistic (z-score) and tail type to compute the p-value from the standard normal distribution.

Test Statistic (z-score)

Tail Type

P-Value

0.0500

Significant at 0.05Yes

CDF Value0.9750

Tail TypeTwo-tailed

What is the P-value Calculator?

A p-value is the probability of observing a test statistic as extreme as (or more extreme than) your observed result, assuming the null hypothesis is true. P-values range from 0 to 1. A small p-value (typically below 0.05) suggests strong evidence against the null hypothesis. It's the foundation of frequentist statistical inference.

How it works

Enter your test statistic (z-score) and choose the tail type (left, right, or two-tailed). The calculator uses the cumulative distribution function (CDF) of the standard normal distribution to compute the p-value. Left-tailed tests compute P(Z less than z). Right-tailed tests compute P(Z greater than z). Two-tailed tests compute twice the smaller tail probability.

Left: p = Φ(z) | Right: p = 1 − Φ(z) | Two-tailed: p = 2 × min(Φ(z), 1 − Φ(z))

Φ(z) is the cumulative distribution function of the standard normal distribution. For left-tailed tests, integrate the normal curve from −∞ to z. For right-tailed, integrate from z to ∞. For two-tailed, double the smaller of the two tail areas to account for extreme values in both directions.

Examples

Input	Result	Notes
z = 1.96 (two-tailed)	p = 0.0500	Classic 5% significance threshold: a z-score of ±1.96 corresponds to exactly p = 0.05 in a two-tailed test.
z = 2.576 (right-tailed)	p = 0.0050	A z-score of 2.576 in a right-tailed test gives p ≈ 0.005, strong evidence against the null hypothesis.
z = −1.645 (left-tailed)	p = 0.0500	A z-score of −1.645 in a left-tailed test corresponds to p ≈ 0.05, the conventional significance level.

How to use the P-value Calculator

Perform your statistical test (t-test, z-test, etc.) to obtain a test statistic

Standardize or convert your test statistic to a z-score if needed

Identify whether your hypothesis is left-tailed, right-tailed, or two-tailed

Enter the z-score into the calculator and select the tail type

Read the p-value from the result box

Compare p-value to your significance level (usually 0.05) to make a conclusion

Benefits

Instant p-value computation — no looking up normal tables or using statistical software

Covers all tail types — left, right, and two-tailed tests in one tool

High precision — reports p-values to 4 decimal places for detailed analysis

Shows CDF value — understand the full picture of your z-score's position in the distribution

Automatic significance check — instantly see if your result meets the 0.05 threshold

Free and client-side — no server dependency, your data stays on your device

Tips & common mistakes

Common mistakes

Confusing z-scores with p-values — a z-score is a standardized test statistic; a p-value is a probability derived from it

Using the wrong tail type — left and right tails give different p-values for the same z; two-tailed doubles the tail area

Misinterpreting small p-values — p < 0.05 suggests evidence against the null, but does not prove it false or prove the alternative true

Forgetting to standardize — if your test statistic is not already a z-score, standardize it first

Tips

Remember the key thresholds: z ≈ 1.645 for p = 0.05 (one-tailed), z ≈ 1.96 for p = 0.05 (two-tailed)

For quality control and scientific research, use p = 0.05; for exploratory work, p = 0.10 is sometimes acceptable

Always report your p-value and the test used (not just 'significant' or 'not significant')

Combine p-values with effect size and confidence intervals for a complete statistical picture

Frequently asked questions

What does p < 0.05 really mean?

It means the probability of observing your test statistic (or more extreme) by chance alone, under the null hypothesis, is less than 5%. This is conventionally considered 'statistically significant,' but it is not proof that your hypothesis is true.

Why are there three tail types?

Left-tailed tests check if a parameter is less than a value. Right-tailed tests check if it is greater. Two-tailed tests check if it is different (either direction). Choose based on your research hypothesis.

Can p-values be negative or greater than 1?

No. P-values are always between 0 and 1 because they represent probabilities. If you get a result outside this range, check your input or calculation.

What's the difference between z-scores and t-scores?

Z-scores assume you know the population standard deviation (or have large n). T-scores are used when estimating from a sample. This calculator uses the normal distribution; for small samples, use a t-table or t-score calculator.

Can I use this for categorical data?

No. This calculator is for z-scores from continuous data. For categorical tests (chi-square, Fisher exact), you need different tools.

What if my z-score is very large (e.g., z = 10)?

Very large z-scores yield p-values extremely close to 0 (for right-tailed) or 1 (for left-tailed). The result is still valid and represents very strong evidence against the null hypothesis.