Calculate z-scores from raw values, find percentiles from z-scores, and convert between standard scores and raw data points. Includes two-tailed probability and normal distribution visualisation.
| Z-Score | Percentile (Left) | Upper Tail | Two-Tailed (Between Β±Z) |
|---|---|---|---|
| 0.00 | 50.00% | 50.00% | 0.00% |
| 0.50 | 69.15% | 30.85% | 38.29% |
| 1.00 | 84.13% | 15.87% | 68.27% |
| 1.28 | 89.97% | 10.03% | 79.95% |
| 1.645 | 95.00% | 5.00% | 90.00% |
| 1.96 | 97.50% | 2.50% | 95.00% |
| 2.00 | 97.72% | 2.28% | 95.45% |
| 2.326 | 99.00% | 1.00% | 98.00% |
| 2.576 | 99.50% | 0.50% | 99.00% |
| 3.00 | 99.87% | 0.13% | 99.73% |
| 3.29 | 99.95% | 0.05% | 99.90% |
| 4.00 | 99.997% | 0.003% | 99.994% |
A z-score (also called a standard score) is a statistical measurement that tells you how many standard deviations a data point is from the mean of a distribution. The formula is simple: z = (X − μ) / σ, where X is the raw value, μ (mu) is the population mean, and σ (sigma) is the standard deviation.
A z-score of 0 means the value is exactly average. A z-score of +1 means the value is one standard deviation above average, while a z-score of −1 means one standard deviation below. Most real-world data falls between z = −3 and z = +3 in a normal distribution.
The standard normal distribution (also called the Z-distribution) is a special normal distribution with a mean of 0 and a standard deviation of 1. When you calculate a z-score, you are essentially converting any normal distribution into this standard form. This lets you use a single set of tables (or this calculator) to find probabilities for any normal distribution.
The bell curve is symmetric around zero. The total area under the curve equals 1 (or 100%). The area to the left of a given z-score is called the cumulative probability or percentile. Our calculator uses the complementary error function (erfc) approximation to compute these probabilities to four decimal places.
One of the most useful properties of the normal distribution is the empirical rule:
This rule applies to any normally distributed variable β heights, test scores, measurement errors, financial returns, and more.
IQ scores follow a normal distribution with mean μ = 100 and standard deviation σ = 15 (the Wechsler scale, used in both the USA and UK). If someone scores 130:
The SAT (Scholastic Assessment Test) is the primary college admissions test in the United States. The College Board redesigned it in 2016; total scores range from 400 to 1600. The approximate mean score is around 1010 with a standard deviation of roughly 210. If a student scores 1220:
The ACT scores range from 1 to 36 with an approximate mean of 20 and SD of about 5. A score of 30 gives z = (30 − 20) / 5 = 2.00, placing a student in the top 2.3%.
In England, Wales, and Northern Ireland, A-level grade boundaries are set by examination boards such as AQA, Edexcel (Pearson), and OCR. While raw z-scores are not published by default, they are used internally in standardisation. A typical A-level unit might have a mean mark of 45 and a standard deviation of 12. A student scoring 63 marks would have:
GCSE grades (9–1 scale) are also standardised using similar statistical methods. The grade 5 boundary (the "strong pass") is set at roughly the C/D boundary of the old lettered system.
In finance, the Sharpe ratio is effectively a z-score of portfolio returns. It measures how many standard deviations of excess return an investment achieves per unit of risk. The formula is: Sharpe = (R − Rf) / σ, which is identical in form to the z-score formula where Rf is the risk-free rate.
A Sharpe ratio above 1.0 is generally considered good, above 2.0 is very good, and above 3.0 is exceptional. An S&P 500 index fund has historically returned a Sharpe ratio of around 0.5 to 0.6 over long periods.
The Altman Z-Score is a different application in credit analysis, using five financial ratios to predict the probability of a company going bankrupt within two years. A score above 2.99 suggests financial safety; below 1.81 indicates distress.
Six Sigma is a quality management methodology used extensively in manufacturing and services. The name comes from the goal of achieving process quality within six standard deviations (σ) of the mean β corresponding to fewer than 3.4 defects per million opportunities (DPMO). This is a z-score of approximately 4.5 (with a 1.5-sigma shift allowed).
| Sigma Level | Z-Score | Defects per Million | Yield |
|---|---|---|---|
| 1σ | 1.00 | 690,000 | 31.0% |
| 2σ | 2.00 | 308,538 | 69.1% |
| 3σ | 3.00 | 66,807 | 93.3% |
| 4σ | 4.00 | 6,210 | 99.4% |
| 5σ | 5.00 | 233 | 99.977% |
| 6σ | 6.00 | 3.4 | 99.99966% |
The z-test uses z-scores to test hypotheses about population means. When the population standard deviation is known and the sample size is large (n β₯ 30), the z-test is appropriate. The test statistic is:
z = (&xΜ; − μ0) / (σ / √n)
where &xΜ; is the sample mean, μ0 is the hypothesised population mean, and n is the sample size.
A one-tailed test tests whether a value is significantly greater than (or less than) a reference. The critical z-value for α = 0.05 is Β±1.645.
A two-tailed test tests whether a value is significantly different from a reference in either direction. The critical z-value for α = 0.05 is Β±1.96, meaning 2.5% in each tail.
Most statistical software and published research uses two-tailed p-values as the default, which is why you'll often see p < 0.05 referenced with z = 1.96.
The percentile corresponding to a z-score is the percentage of data points that fall below that value in the distribution. For example:
In clinical medicine, growth charts use z-scores to assess whether a child's height or weight is within healthy ranges. The WHO growth standards use z-scores −2 to +2 as the normal range for most measurements.
| Scale | Mean | SD | Typical Use |
|---|---|---|---|
| Z-score | 0 | 1 | General statistics, z-test |
| T-score | 50 | 10 | Personality tests, bone density (DEXA) |
| IQ (Wechsler) | 100 | 15 | Intelligence tests (US & UK) |
| SAT (post-2016) | 1010 | ~210 | US college admissions |
| Stanine | 5 | ~2 | Older educational testing |
A z-score measures how many standard deviations a data point is from the mean of its distribution. A positive z-score means the value is above average; negative means below average. It standardises scores from different distributions so they can be compared on the same scale.
A z-score of 1.96 corresponds to the 97.5th percentile in a standard normal distribution. It is the critical value for a two-tailed hypothesis test at the 5% significance level (Ξ± = 0.05), meaning values beyond Β±1.96 lie in the rejection region for a 95% confidence test.
Use the standard normal cumulative distribution function (CDF). For example, z = 1.00 gives a CDF of 0.8413, meaning the 84.13th percentile. Our calculator computes this automatically using the error function approximation: P = 0.5 Γ (1 + erf(z / β2)).
Yes. A negative z-score simply means the value is below the mean. For instance, a z-score of β1.5 means the value is 1.5 standard deviations below average. Negative z-scores are completely normal and correspond to percentiles below 50%.
IQ tests use ΞΌ = 100 and Ο = 15. A z-score of 0 equals IQ 100 (average). A z-score of +2 equals IQ 130 (top 2.3%), which is often cited as the MENSA qualifying threshold. A z-score of β2 equals IQ 70, which is a clinical threshold for intellectual disability screening.
A z-score uses the known population standard deviation (Ο) and is appropriate for large samples (n β₯ 30). A t-score uses the estimated sample standard deviation (s) and is used for small samples. As sample size increases, the t-distribution approaches the normal (z) distribution.
Two-tailed probability is the area in both tails of the normal distribution beyond Β±z. It answers: "What proportion of data falls more extreme than this z-score in either direction?" For z = 1.96, the two-tailed probability is 5% (2.5% in each tail), which is the most common significance level in research.
Use a z-test when: (1) the population standard deviation is known, (2) the sample size is large (n β₯ 30), or (3) you are testing proportions. Use a t-test when the population SD is unknown and must be estimated from the sample, especially for small samples where the t-distribution's heavier tails are important.
Results are for educational and informational purposes only. Z-scores assume a normal distribution; results may not be meaningful for non-normal data. Consult a qualified statistician or researcher before making decisions based on these calculations.