How to calculate a z-score and p-value
- Enter a value, the mean and the standard deviation of the distribution, or type a z-score directly.
- Read the z-score, the left-tail and right-tail probabilities, the two-tailed p-value and the percentile. The curve shades the area to the left of your value.
- To go the other way, switch to p → critical z, enter a significance level such as 0.05 and choose the tail to get critical values for hypothesis tests and confidence intervals.
What a z-score means
A z-score tells you how many standard deviations a value lies from the mean: z = (x − μ) ÷ σ. With IQ scores (mean 100, standard deviation 15), a score of 130 has z = 2, which is higher than about 97.7% of the population. For a normally distributed variable, the area under the standard normal curve to the left of z is the probability of getting a value at most that large. That area is also the one-tailed p-value in a z-test; the two-tailed p-value doubles the smaller tail.
Familiar critical values come from the inverse calculation: ±1.96 for 95% confidence (α = 0.05, two-tailed), ±2.576 for 99%, and 1.645 for a one-tailed test at 5%. Probabilities are computed with a high-precision approximation of the normal distribution (accurate to about 7 decimal places) and the inverse uses Acklam's algorithm refined by a Newton step. Use the t-distribution instead for small samples with an estimated standard deviation. Everything runs in your browser.
What z-score is the 95th percentile?
About 1.645. The 97.5th percentile is 1.96.
Can a z-score be negative?
Yes. Negative z-scores are below the mean; z = −1 is the 15.9th percentile.
Is a p-value below 0.05 significant?
By the common convention, yes, but significance thresholds should be chosen before the test and interpreted with effect sizes.