Z-Score Calculator
Enter a mean and a standard deviation, paste one raw score or a whole column of them, and get the z-score, the percentile and the tail probabilities off the normal curve. The same page also inverts z back to a raw score, converts a probability into its critical z, measures the area between two bounds, and standardizes a dataset. Everything is computed on your device.
About this tool
Z-Score Calculator converts between raw values, z-scores, normal-curve probabilities, and critical values. In the default mode, enter a mean μ, a standard deviation σ, and one or more raw scores. The output shows each score's z-score, percentile, left-tail probability, right-tail probability, and two-tailed p-value.
A z-score is the number of standard deviations a value is from the mean: z = (x - μ) / σ. Positive z-scores are above the mean, negative z-scores are below it, and z = 0 is exactly at the mean. When Sample size (n) is greater than 1, the denominator becomes the standard error σ / sqrt(n), which is the form used for testing a sample mean against a known population mean.
The other modes cover common lookup-table workflows: convert z-scores back to raw values, turn left-tail probabilities into critical z values, calculate the area between two bounds, or standardize a pasted dataset by deriving its own mean and standard deviation. Numbers may be separated by spaces, commas, semicolons, or newlines.
Worked example
For an IQ-style scale with mean 100 and standard deviation 15, a score of 130 gives:
x = 130
z = 2
percentile = 97.724987%
left tail P(X < x) = 0.977249868052
right tail P(X > x) = 0.0227501319482
two-tailed p = 0.0455002638964
That means 130 is two standard deviations above the mean, about the 97.7th percentile under the normal model. Use Decimal places to control display precision; very tiny tail probabilities keep significant digits rather than rounding all the way to zero.
This is a calculator for the normal-distribution arithmetic only. It does not decide whether your data are actually normal, and it does not replace study-specific statistical judgment.
FAQ
What is the difference between z-score, percentile, and p-value?
The z-score is the standardized distance from the mean. The percentile is the left-tail area under the normal curve, so z = 0 is the 50th percentile and z ≈ 1.96 is about the 97.5th percentile. A p-value is a tail probability used for a hypothesis test; the two-tailed p-value reported here is 2 * min(left tail, right tail).
When should I set sample size n above 1?
Use n > 1 when the value you entered is a sample mean, not a single observation, and you know the population standard deviation. The standard error is σ / sqrt(n), so the same distance from the population mean becomes more unusual as the sample size grows. Leave n = 1 for ordinary single-score z-scores.
What does critical mode expect?
Critical mode expects left-tail probabilities between 0 and 1. For example, 0.975 returns about 1.959964, the familiar two-sided 95% cutoff, and 0.025 returns the matching negative value. It is the inverse of the standard normal CDF, not a percent string; enter 0.975, not 97.5.
How is dataset mode different from a full normalization tool?
Dataset mode derives the mean and standard deviation from the numbers you paste, then reports z-scores plus normal-curve probabilities for those values. It is intentionally narrow: min-max scaling, robust scaling, CSV column selection, and bulk feature preprocessing belong to dedicated normalization tools. Turn on sample when you want the sample standard deviation, dividing by N - 1.
What limits should I know about?
The input accepts up to 10,000 numbers. Standard deviation must be greater than zero, n must be at least 1, and decimal places are limited to 0 through 12. Between mode requires exactly two bounds, and critical mode rejects probabilities at or outside 0 and 1 because the corresponding z values are infinite.
Developer & Automation Access
Run it from the terminal
Same engine as this page, headless — via the gizza CLI:
gizza tool z-score-calculator "130
85
115"New to the CLI? Get gizza →
Open it by URL
Pre-fill and auto-run this tool with query parameters — the names match the API/CLI:
https://gizza.ai/tools/z-score-calculator/?values=130%0A85%0A115&mode=score&mean=100&std_dev=15&n=1&sample=true&decimals=6Machine-readable descriptor: tool.json — title + parameters JSON Schema for agents.