Standardize a value to a z-score and read off the probability and percentile, find the chance of landing between two values, or reverse a percentile back to a raw score — with a shaded curve.
Z-score
2
Percentile
97.72%
Calculator
Z-score
2
x = 130 is 2 standard deviations from the mean.
P(X < x) — left
0.9772
P(X > x) — right
0.0228
Two-tailed P(|Z| > |z|)
0.0455
How z-scores and the normal curve work
A z-score measures how many standard deviations a value lies from the mean: z = (x − μ) / σ. The standard normal curve then turns that z into a probability. The area to the left of z, Φ(z), is the chance of being below x; the area to the right is 1 − Φ(z); the two-tailed area is the chance of being at least |z| away from the mean in either direction.
Reversing the process, a percentile is converted back to a z-score with the inverse normal function Φ⁻¹, and then to a raw value with x = μ + z·σ. The shaded region under the curve shows exactly which probability you are reading.
What is a good or bad z-score?
A z-score has no good or bad on its own — it just says how unusual a value is. Roughly 68% of values fall within ±1, 95% within ±2 and 99.7% within ±3 standard deviations of the mean.
When do I use the two-tailed probability?
Use it when you care about being far from the mean in either direction — for example a two-sided hypothesis test. For z = 1.96 the two-tailed probability is about 0.05, the basis of a 95% confidence level.
How accurate are the probabilities?
The cumulative probability uses a standard error-function approximation accurate to about seven decimal places, and the inverse uses a rational approximation accurate to roughly nine — far beyond what printed z-tables provide.
Results are estimates. Verify with a professional for important decisions.
About this calculator
This calculator converts a raw value into a z-score and shows how that value compares to the rest of a normally distributed population. Use it to find the probability of a result falling below, above, or between two values, to read off a percentile rank, or to reverse-solve a score from a target percentile.
How to read your results
The headline figure is the z-score, which tells you how many standard deviations the value sits above or below the mean. The stat strip also shows the corresponding percentile. Below the inputs a shaded normal curve highlights the area of interest — left-shaded for a single value, between-shaded when you enter two values. The result card lists the left-tail, right-tail, and two-tailed probabilities so you can pick the one that matches your question.
Worked example
An IQ score of 130 on a test with mean 100 and standard deviation 15.
The z-score is 2.00. The left-tail probability is 0.9772, meaning 97.72 % of the population scores below 130. The right-tail probability is 0.0228 and the two-tailed probability is 0.0455.
Frequently asked questions
What does a z-score tell me?
A z-score measures distance from the mean in units of standard deviation. A z of 0 means the value equals the mean; a z of 1 means it is one standard deviation above; a z of −1 means one standard deviation below. This makes it possible to compare values from different distributions on the same scale.
When should I use left-tail, right-tail, or two-tailed probability?
Use left-tail when asking "what fraction of the population scores less than X?" Use right-tail for "what fraction scores more than X?" Use two-tailed when testing whether a value is unusual in either direction — for example, in a hypothesis test with no predetermined direction.
Does this assume a normal distribution?
Yes. All probabilities and percentiles here are calculated under the assumption of a perfect standard normal curve. For real data that is heavily skewed or has long tails, the results are approximate and a distribution-specific tool may be more appropriate.
How it's calculated
The z-score formula is z = (x − μ) / σ, where x is the observed value, μ the population mean, and σ the standard deviation. Cumulative probabilities come from the standard normal CDF Φ(z), computed with the Abramowitz & Stegun erf approximation (7.1.26, maximum error ~1.5 × 10⁻⁷). The inverse CDF Φ⁻¹(p) uses Acklam's rational approximation to map a percentile back to a z-score. The raw value is then recovered as x = μ + z·σ.
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