Z Score to P Value Calculator: How to Find It

This calculator converts a z score into a p value (one-tailed and two-tailed) using the standard normal distribution. You enter the z score, choose the tail type, and get the exact probability that a standard normal variable falls beyond that value. The result tells you how statistically extreme the z score is under the null hypothesis.

Formula

The p value comes from the cumulative distribution function (CDF) of the standard normal distribution, denoted Φ(z). For a given z score:

• One-tailed (right): p = 1 − Φ(z)
• One-tailed (left): p = Φ(z)
• Two-tailed: p = 2 × (1 − Φ(|z|))

Here Φ(z) is the probability that a standard normal random variable is less than or equal to z. The standard normal distribution has mean 0 and standard deviation 1. The calculator uses the same Φ function that appears in statistics tables.

Worked Example

Using the defaults: z = 1.96, two-tailed. First find Φ(1.96). From standard normal tables, Φ(1.96) = 0.9750. Then the two-tailed p = 2 × (1 − 0.9750) = 0.0500. The calculator shows p = 0.0500, which matches the classic 5% significance threshold. For a one-tailed test, p = 1 − 0.9750 = 0.0250.

When to Use a Z Score to P Value Calculator

You use this when you have a z score from a hypothesis test and need the corresponding p value. Common situations include:

• Testing a sample mean against a known population mean (z test)
• Checking if a proportion differs from a hypothesized value
• Converting a test statistic to a probability for reporting in research

The z score itself is usually computed as (observed value − expected value) / standard error. This calculator assumes the z score follows a standard normal distribution, which holds when the sample size is large (typically n > 30) or the population standard deviation is known.

How to Choose the Right Tail Type

The tail type depends on your research hypothesis. Use a one-tailed test when you predict a specific direction (e.g., “the new drug increases response rate”). Use a two-tailed test when you only care about any difference, regardless of direction. Two-tailed p values are always larger than one-tailed for the same absolute z, making them more conservative. Many scientific fields default to two-tailed tests to avoid inflating significance.

Common Mistakes and Misconceptions

A frequent error is mixing up one-tailed and two-tailed p values. If your hypothesis is directional but you use a two-tailed test, you reduce power. Conversely, using a one-tailed test without a strong prior direction can be seen as p-hacking. Another mistake is forgetting that the p value is not the probability that the null hypothesis is true. It is the probability of observing a z score at least as extreme as yours, assuming the null is true. Also, be careful with negative z scores: the two-tailed formula uses the absolute value, so z = −1.96 gives the same p as z = 1.96.

Limitations and Alternatives

This calculator works only for the standard normal distribution. If your test statistic follows a t distribution (common when the sample size is small and the population standard deviation is unknown), you need a t to p calculator instead. Similarly, chi-square and F tests require their own conversions. The normal approximation also breaks down for very small sample sizes or extreme proportions (e.g., when np or n(1−p) is less than 5). In those cases, exact tests like Fisher’s exact test are more appropriate.

Frequently Asked Questions

What does a p value of 0.05 mean?

It means that, if the null hypothesis is true, there is a 5% chance of obtaining a z score at least as extreme as the one observed. It does not mean there is a 95% chance the alternative is true. The threshold for significance is arbitrary; 0.05 is a convention, not a law.

Can a p value be negative?

No. Probabilities range from 0 to 1. If your calculation yields a negative p, you likely used the wrong formula (e.g., forgot to subtract from 1). A p value of 0 is technically possible but usually indicates rounding or an extremely large z score.

How do I find the z score from a p value?

You use the inverse standard normal function, often denoted Φâ�»Â¹(p). For a one-tailed p, z = Φâ�»Â¹(1 − p). For a two-tailed p, first divide p by 2, then find the z that leaves that probability in the tail. Many calculators have this built in as “inverse normal” or “z critical value.”

What is the difference between p value and significance level?

The significance level (α) is the threshold you set before the test, usually 0.05. The p value is the probability computed from your data. If p ≤ α, you reject the null hypothesis. The p value is a continuous measure of evidence, while α is a fixed cutoff.

Why do two-tailed p values double the one-tailed value?

Because the normal distribution is symmetric. The probability of being in either tail is the sum of the probabilities of the left tail and the right tail. Since both tails have the same area for a given absolute z, you multiply the one-tailed probability by 2. This is only exact for symmetric distributions like the normal.

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in z | Z score | - | 1.96 | 0
sel tail | Tail type | One-tailed=1, Two-tailed=2 | 2 | 
out p | p value | - | 2 * (1 - normcdf(abs(z))) | 4
out p_one | One-tailed p | - | 1 - normcdf(abs(z)) | 4