p value calculator from z

What this calculator does

This tool converts a z-statistic into a p-value using the standard normal distribution. If you already have a z-score from a z test, proportion test, or large-sample approximation, this is the fastest way to get the p-value you need for hypothesis testing.

Choose the tail type that matches your alternative hypothesis:

• Two-tailed: tests whether the parameter is different from the null value.
• Right-tailed: tests whether the parameter is greater than the null value.
• Left-tailed: tests whether the parameter is less than the null value.

How to use the p value calculator from z

1. Enter your z-score (positive or negative).
2. Select one-tailed or two-tailed testing.
3. Set your significance level (commonly 0.05).
4. Click Calculate p-value to see:
   • the computed p-value,
   • tail probability details, and
   • a simple significance interpretation.

Formula used to convert z to p-value

1) Standard normal CDF

Let Φ(z) be the cumulative distribution function (CDF) of the standard normal random variable Z ~ N(0,1):

2) Tail-specific p-value formulas

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

These formulas are exactly what the calculator implements in JavaScript.

Worked examples

Example A: Two-tailed, z = 1.96

For a two-tailed test with z = 1.96, p is about 0.0500. At α = 0.05, this is right on the common significance boundary.

Example B: Right-tailed, z = 2.33

For a right-tailed test, p = P(Z ≥ 2.33) ≈ 0.0099. Since 0.0099 < 0.05, you would reject H₀ at the 5% level.

Example C: Left-tailed, z = -1.28

For a left-tailed test, p = P(Z ≤ -1.28) ≈ 0.1003. At α = 0.05, this is not statistically significant.

Interpreting the p-value correctly

A p-value is the probability of observing data at least as extreme as yours assuming the null hypothesis is true. It is not the probability that the null hypothesis is true.

• If p ≤ α, results are often called statistically significant.
• If p > α, you generally fail to reject the null hypothesis.

Always pair p-values with effect size, confidence intervals, and context. Statistical significance alone does not guarantee practical importance.

Common mistakes to avoid

• Using a one-tailed test when your hypothesis is actually two-sided.
• Choosing the tail direction after looking at the data.
• Interpreting a large p-value as “proof” the null is true.
• Ignoring assumptions behind z-based methods (independence, known/large-sample variance conditions).

FAQ

Can I use this for t-tests?

No. This calculator is for z-scores under the standard normal distribution. For small samples with unknown population variance, use a t distribution based calculator.

Why can p-values become extremely small?

Very large |z| values put your statistic deep in the tails of the distribution, which naturally leads to tiny tail probabilities.

Does this support negative z-scores?

Yes. Negative z-values are fully supported and interpreted correctly for left, right, and two-tailed tests.