Z-Score Calculator
Calculate the z-score, p-value, and percentile rank for any data point.
Results
Z-score
0.5000
(x − μ) / σ
Percentile
69.15%
of values below x
P (left tail)
0.6915
P(X ≤ x)
P (right tail)
0.3085
P(X > x)
Two-tailed p-value: 0.6171
Interpretation: Z = 0.50 is not statistically significant at α = 0.05.
How z-scores and p-values work
A z-score (standard score) measures how many standard deviations a value is from the mean: z = (x − μ) / σ. A z-score of 0 means the value equals the mean; ±1 is one standard deviation away. The p-value (left tail) gives the probability that a randomly sampled value would be less than or equal to x — the percentile rank. Two-tailed p-value covers both extremes.
For standard deviation, try the Standard Deviation Calculator. For general statistics, see Average Calculator.
Built and maintained by Meet Shah · Last updated
What this tool is used for
- Standardising a value against a mean and standard deviation.
- Getting a percentile for a measurement against a reference population.
- Comparing values from two differently scaled distributions.
- Checking whether an observation is unusual.
- Standardising a value so it can be compared with one from another scale.
Frequently Asked Questions
- What does the z-score itself measure?
- Distance from the mean expressed in standard deviations: z = (x − μ) / σ. It is unit-free, which is the point — a z of 2 means the same thing for blood pressure in mmHg as for an exam mark, so two measurements on incompatible scales become directly comparable.
- Why must the standard deviation be greater than zero?
- Because σ is the divisor. A σ of 0 means every observation is identical, so there is no spread to measure a distance against and the formula divides by zero. The tool blocks it with an explicit message rather than displaying Infinity, which would look like a very significant result.
- How is the percentile calculated from the z-score?
- It is the left-tail probability Φ(z) × 100, computed with the Abramowitz & Stegun approximation of the normal CDF (error under 7.5×10⁻⁸). The IQ preset shows the shape: 130 against a mean of 100 and σ of 15 is z = 2.0, which is the 97.7th percentile — of people scoring 130 or below, not exactly 130.
- When is the two-tailed p-value the one I want?
- When your question is 'is this unusual?' rather than 'is this unusually high?'. Two-tailed doubles the smaller tail, so z = 2 gives 0.0455 rather than the one-tailed 0.0228. Choosing the one-tailed value after seeing which direction the data went is the classic way to manufacture significance that is not there.
- Does this assume my data is normally distributed?
- The z-score itself does not — it is just a rescaling and is defined for any distribution. The percentile and the p-values do, because they read probabilities off the normal curve. For strongly skewed data such as income, the z is still meaningful as a distance while the percentile attached to it is not.
Common errors and gotchas
- Assuming normality, without which the percentile is not meaningful.
- Confusing the sample and population standard deviation.
- Reading a z-score as a probability rather than a distance in standard deviations.
- Comparing z-scores across distributions with very different shapes.
- Using a small sample's statistics as if they were the population's.