This z-score calculator standardizes a value using its mean and positive standard deviation. You can also reverse a score, calculate normal-model areas, or find z from a left-tail percentile.
Interpret a z-score and the normal model
A z-score is the signed distance between a value and its reference mean, measured in standard deviations. Positive scores are above the mean, negative scores below it, and zero is at the mean. The value, mean and standard deviation must describe the same quantity in the same unit. Standard deviation must be positive. The standardization formula is z = (x − μ) / σ. Here x is the value, μ is the reference mean and σ is its standard deviation. This arithmetic works without assuming a normal distribution. A normal percentile or curve area needs an additional normal-model assumption. The calculator leaves that assumption off for score conversion; optional settings can enable it. The separate normal probability and inverse-percentile tasks explicitly use the standard normal model. The NIST normal-distribution reference defines the density and cumulative areas.
Use a reference mean and positive SD that belong to the population or dataset you mean to compare.
Keep value, mean and SD in the same measurement unit. The resulting score is dimensionless.
A normal-model percentile is not the observed rank of a value in an arbitrary dataset.
Check a standardized score step by step
Enter value 85, mean 75 and standard deviation 5. Subtract the mean from the value, then divide that difference by the standard deviation. The engine returns z = 2. The value is 2 standard deviations above the reference mean.
With no normal model selected, the result does not infer a percentile. If you explicitly choose a normal model, the same score has left area 0.9772498681, right area 0.02275013195 and combined area outside the symmetric bounds 0.0455002639. These are model probabilities. They do not establish that a dataset is normal or that a hypothesis test is valid.
Standard normal z-score landmarks
These are areas under the standard normal curve. Left and right refer to the signed z-score; the final column is the combined area outside the symmetric bounds −|z| and +|z|.
| Z-score | Area to the left | Area to the right | Area outside ±|z| |
|---|---|---|---|
| -3 | 0.1349898032% | 99.8650102% | 0.2699796063% |
| -2.5 | 0.6209665326% | 99.37903347% | 1.241933065% |
| -2 | 2.275013195% | 97.72498681% | 4.55002639% |
| -1.5 | 6.680720127% | 93.31927987% | 13.36144025% |
| -1 | 15.86552539% | 84.13447461% | 31.73105079% |
| -0.5 | 30.85375387% | 69.14624613% | 61.70750775% |
| 0 | 50% | 50% | 100% |
| 0.5 | 69.14624613% | 30.85375387% | 61.70750775% |
| 1 | 84.13447461% | 15.86552539% | 31.73105079% |
| 1.5 | 93.31927987% | 6.680720127% | 13.36144025% |
| 2 | 97.72498681% | 2.275013195% | 4.55002639% |
| 2.5 | 99.37903347% | 0.6209665326% | 1.241933065% |
| 3 | 99.8650102% | 0.1349898032% | 0.2699796063% |
Model percentages, not empirical ranks in an arbitrary dataset. Display values are rounded after calculation.
Choose left, right, between or outside normal areas
A left area is P(Z ≤ b), the cumulative probability Φ(b). A right area is P(Z ≥ b), evaluated directly by the survival function. Between two bounds a and b, the area is Φ(b) − Φ(a). Outside those bounds, it is Φ(a) + P(Z ≥ b). The lower bound must not exceed the upper bound. Equal bounds have zero between-area. The two symmetric tails outside the negative and positive absolute z-score bounds are a different region from a single right tail.
The between-area preset uses lower z = -1 and upper z = 1. Its model probability is 0.6826894921, equivalent to 68.26894921%. The probability result is a fraction of total area; the detail note converts it to a percentage. It is not automatically a test p-value. A hypothesis test also needs a null model, a statistic and a preselected alternative.
For inverse calculation, enter a left-tail percentage number without a sign. The inverse preset uses 97.5%, so left probability is 0.975 and z = 1.959963985. Inverse percentiles exclude the endpoints because finite normal cutoffs do not contain all or none of the curve. The Python NormalDist documentation specifies the same inverse cumulative meaning.
Reverse a score and understand numerical limits
To reconstruct a value, use x = μ + σz with the same reference mean and SD. The reverse example enters z = 2, mean 75 and SD 5, returning value 85. This value retains the original measurement unit.
If you start with observations rather than a reference mean and SD, the Mean Median Mode Calculator describes their center and the Standard Deviation Calculator computes spread with explicit sample and population conventions. The Interquartile Range Calculator measures the middle portion of an ordered dataset. None of those summaries alone proves normality.
This tool does not use clinical reference charts, bone-density or BMI standards, the Altman financial formula, modified robust z-scores, or a standard error for a sample-mean test. It does not assign automatic outlier labels or remove observations. Those jobs need different inputs or formulas.
Local input policy allows decimals, scientific notation and simple fractions, with at most 128 characters per numeric token. Nonzero input magnitudes range from 1e-100 to 1e+100. Values are IEEE754 floating-point numbers. Differences below their representable precision cannot be recovered. The display rounds to 10 significant digits after calculation.
Small right-tail probabilities are computed directly. Narrow intervals use density integration to avoid subtracting nearly equal cumulative areas. Very extreme areas can still round to one or underflow to zero; this is a numerical limit, not certainty or impossibility. Numerical methods use the NIST error-function series, continued fraction, and Simpson quadrature.
When the measurement has a meaningful zero, the Coefficient of Variation Calculator compares SD to a nonzero mean. It distinguishes signed CV from absolute percentage RSD and warns when the measurement scale makes relative interpretation inappropriate.
The Pooled Standard Deviation Calculator combines sample spreads with degrees-of-freedom weights. Keep groups separate; its within-group result differs from the standard deviation of all observations together.