Z-score Calculator

Statistical Standardization & Normal Distribution Theory

A Z-score (standard score) measures how many standard deviations an individual raw score (x) lies above or below the mean (μ) in a Gaussian normal distribution N(μ, σ^2).

Primary Statistical Equations

• Standard Z-Score:

Z = (x - μ) / (σ)

• Cumulative Normal Distribution Function Φ(Z):

Φ(Z) = (1) / (√(2π)) ∫_-∞^Z e^-t^2/2 dt = (1) / (2) [ 1 + erf( (Z) / (√(2)) ) ]

• Probability Distributions:

• Left-Tail: P(X < x) = Φ(Z)

• Right-Tail: P(X > x) = 1 - Φ(Z)

• Two-Tail: P(|Z| > |z|) = 2 × [1 - Φ(|Z|)]

• Empirical 68-95-99.7 Rule:

• 68.27\% of observations fall within ± 1σ

• 95.45\% fall within ± 2σ

• 99.73\% fall within ± 3σ

How to Use This Calculator

Enter your Raw Value (x) (e.g. 85.0), Mean (μ / x̄) (e.g. 70.0), and Standard Deviation (σ / s) (e.g. 10.0). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Z-score Calculator

Input variables:

rawScoreX — Raw Value (x)

populationMeanMu — Mean (μ / x̄)

standardDeviationSigma — Standard Deviation (σ / s)

Computed outputs:

calculatedZScore — Standardized Z-Score

leftTailProbability — Left-Tail Probability P(X < x)

rightTailProbability — Right-Tail Probability P(X > x)

twoTailProbability — Two-Tailed Probability P(|Z| > |z|)

percentileRankPct — Percentile Rank

Mathematical relationships extracted from calculation logic:

z = safeDiv(x - mu, sigma)

t = 1.0 / (1.0 + 0.5 * Math.abs(zVal))

phiZ = 0.5 * (1 + erf(safeDiv(z, Math.sqrt(2))))

percentile = leftTail * 100

Standard: ASA and NIST statistical best practices

Worked Example

A student scoring 85 on an exam with a class average of 70 and standard deviation of 10 earns a Z-score of +1.50, outperforming 93.32% of students.

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.