Z Score and Normal Distribution Calculator

From a value to its z, its percentile, and the picture, and back again

A z score answers “how many standard deviations from the mean?” and the normal curve turns that into the answers people actually want: the percentile (what share falls below), the tail probabilities, and the share between two values. This calculator works in both directions: value, mean and standard deviation in, z and every probability out, with the region drawn shaded on the curve; or a probability in and the z (and the value it implies) back. The tails are handled explicitly rather than by table interpolation, and the between-two-values tool covers the question course questions ask most.

Value to z and probabilities
Between two values
Probability back to z and value (inverse)

Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice. Probabilities assume a normal distribution; the cumulative function is evaluated to about six decimal places.

What this calculator does

Standardising (subtract the mean, divide by the standard deviation) converts any normal value into the standard normal’s language, where one table, one curve and one set of landmarks serve every problem. This page does the conversion and then answers in probability: below the value (the percentile), above it, and between it and the mean’s mirror. The between tool takes the two-sided form directly, and the inverse tool inverts the cumulative function so a percentile can be turned back into the score that earns it, which is how cut-offs and grading boundaries are actually set. The drawn curve is not decoration: the shaded region is your probability, to scale, and reading area off a bell is the skill the whole topic is secretly teaching.

How to use it

  • Mean and SD describe the population you are comparing against. Test scores against their cohort, heights against their population: the z is only ever as meaningful as the reference group whose mean and spread you enter.
  • Left, right and between are different questions. Percentile means below; exam top-decile questions mean above; the between tool takes two values as they come and orders them itself.
  • Use the inverse for thresholds. “What score is the 90th percentile?” is probability 0.90 into the inverse, and its value output is the threshold at the entered mean and SD.
  • Check normality before believing tails. Far-tail probabilities (beyond ±3) are exquisitely sensitive to the normal assumption; real data with heavier tails will betray the third decimal long before the first.

The method

  • z = (x − mean) / SD. The cumulative probability Φ(z) is evaluated through the error function with a high-accuracy rational approximation (absolute error below 1.5 × 10⁻⁷), not by rounding z to a two-decimal table.
  • Tails: left = Φ(z); right = 1 − Φ(z); between two values = Φ(z₂) − Φ(z₁); within ±z of the mean = 2Φ(|z|) − 1.
  • Inverse: z from a left-tail probability by Acklam’s rational approximation (error in z under about 5 × 10⁻⁴ across the range, much finer near the centre), then value = mean + z × SD.
  • The graph plots the standard normal density from −4 to +4 and shades the exact region whose area the headline probability reports.

Worked examples

85 on a test with mean 70, SD 10. z = (85 − 70)/10 = 1.50. Φ(1.50) = 0.933193: the score sits at the 93.32nd percentile, with 6.68 percent above it and 86.64 percent between it and the symmetric 55 below. Every one of those numbers is a row of the result, because different questions ask for each.

The middle of the distribution. Between 60 and 80 on the same test (z = −1 and +1): Φ(1) − Φ(−1) = 0.841345 − 0.158655 = 0.682689, the famous 68 percent of the 68-95-99.7 rule, computed rather than recited. And its inverse at 0.975 returns z = 1.96, the threshold that fences off the top 2.5 percent.

Reading your results

Read the percentile as a rank in a crowd of a hundred: the 93rd percentile outperformed about 93 of them. Read z itself as distance with a sign: +1.50 is one and a half spreads above the centre, and the symmetry of the curve means every negative z mirrors a positive one, which the between-within row exploits. Read far-tail results with the humility the limitations demand: a computed 0.000032 assumes the world is exactly normal out where the data are sparsest, and the honest use of such figures is orders of magnitude, not decimals.

Limitations

  • Normality is an assumption, not a fact of nature. Incomes, wait times and city sizes are not normal; applying z machinery to them gives precise wrong answers, and a histogram of your data is the entry fee this page cannot collect for you.
  • Sample SD versus population SD. The formula divides by the SD you enter; with a small sample, that SD is itself uncertain, and t distributions exist precisely for that case (a companion topic, not this page’s).
  • Percentile definitions vary at the edges. This page’s percentile is the strict area below the value under a continuous curve; reported percentiles for discrete test scores follow test-makers’ conventions, which can differ by a point.
  • Approximation error is stated, not hidden. The cumulative evaluation errs below 1.5 × 10⁻⁷ and the inverse below roughly 5 × 10⁻⁴ in z; both are finer than the tables they replace and are printed in the method so the precision claim can be checked.

Frequently asked questions

What does a z score of 1.5 mean?

The value lies one and a half standard deviations above the mean of its reference distribution: on the worked example’s test, 85 against a mean of 70 and SD of 10. Converted to standing, that is about the 93rd percentile, meaning roughly 93 percent of the reference group falls below. The z is the distance, the percentile is the rank, and questions that seem to ask for one usually grade the other, which is why the results print both.

How do I find the probability between two values?

Standardise both, and subtract the cumulative at the lower from the cumulative at the upper: Φ(z₂) − Φ(z₁). The between tool does exactly this with your mean and SD attached; the 60-to-80 example lands on 0.6827 because those values are exactly one SD either side. A common error is adding the two tail figures or reading the right tail at the upper value, and the drawing exists to make the correct region visible: between means the slice in the middle, shaded.

What z value cuts off the top 10 percent?

Top 10 percent means a left-tail probability of 0.90, whose inverse gives z ≈ 1.2816. In value terms at mean 70, SD 10, the cut-off is 70 + 1.2816 × 10 ≈ 82.82. Symmetrically the bottom 10 percent sits at z ≈ −1.2816, and the pair fence the middle 80 percent: the inverse tool exists because threshold questions are how percentiles get used in the wild, from grade boundaries to growth charts.

Why does the 68-95-99.7 rule work?

Because the normal curve’s area accumulates at those rates: within one SD of the mean holds Φ(1) − Φ(−1) = 68.27 percent of the area, within two holds 95.45, within three 99.73. The rule is the curve’s area read at integer distances, not a separate fact, and this page’s between tool reproduces each figure from the cumulative function itself, which is the difference between memorising a chant and owning the instrument.

Related calculators

Method: z by standardisation, cumulative probabilities by a high-accuracy error-function approximation, the inverse by Acklam’s rational approximation, tails and between probabilities by explicit differences, with the reported region shaded on the standard curve. Last reviewed: 15 October 2026.