Z-Score Calculator

A z-score (standardized score) measures how many standard deviations an observation is from the mean. Formula: z = (x − μ) / σ. A positive z-score means the value is above average; negative means below.

z = (x − μ) / σ, where x is the value, μ is the population mean and σ is the standard deviation. If σ = 0, the z-score is undefined (the calculator returns 0).

Standardization lets you compare values from different distributions or with different units: IQ scores, SAT results, centile charts, machine-learning feature normalization, and financial return analysis all rely on z-scores.

The normal (Gaussian) distribution is a symmetric bell curve described by mean μ and standard deviation σ. A z-score converts any value to the standard normal distribution N(0,1), enabling use of the CDF Φ to find probabilities and percentiles.

The percentile shows what fraction of the population scores below a given value. For z = 0 the percentile is 50%, for z = 1 about 84.1%, for z = 2 about 97.7%, for z = 3 about 99.9%. The conversion uses the CDF Φ(z) = P(Z ≤ z).

A common rule: values with |z| > 3 (three standard deviations from the mean) are potential outliers. In a normal distribution only about 0.27% of observations fall outside ±3σ. Some fields use stricter thresholds such as |z| > 2.5 or |z| > 2.

The empirical rule: about 68% of observations fall within ±1σ (z ∈ [−1, 1]), about 95% within ±2σ and about 99.7% within ±3σ. This helps quickly assess how unusual an observation is relative to the whole population.

The formula is the same — z = (x − x̄) / s — using the sample mean x̄ and sample standard deviation s (computed with Bessel correction, dividing by n−1). Enter your pre-calculated values into the calculator.

Z-score assumes data are approximately normally distributed. For heavily skewed or multimodal distributions the result can be misleading. In small samples (n < 30) the standard deviation is uncertain, reducing reliability.

Z-score has mean 0 and SD 1. T-score rescales it to mean 50, SD 10: T = 50 + 10z. IQ scores use mean 100, SD 15: IQ = 100 + 15z. All three describe the same concept with different presentation scales.