Z-Score Calculator - Normal Distribution Probability & Percentile

Calculate standard Z-scores, normal distribution curve probabilities (P-values), left-tail, right-tail, two-tailed values, and population percentile rank.

⚙️ Configuration & Inputs

Raw Score (x) The individual observation or test score

Population Mean (μ) Average of the population (e.g. 100 for IQ)

Standard Deviation (σ) Spread of the distribution (e.g. 15 for IQ)

📊 Results & Analysis

Standard Z-Score +1.0000 1.00 standard deviations above mean

Left-Tail P(X < x) 0.8413 84.13th Percentile

Right-Tail P(X > x) 0.1587 Top 15.87% of population

Two-Tailed P(|Z| > |z|) 0.3173 Probability of extreme deviation

Between Mean & Score 0.3413 34.13% between μ and x

68-95-99.7 Empirical Rule Within 1σ 68.27% of values fall within ±1σ

Understanding Z-Scores & Standard Normal Distribution

A Z-score (standard score) measures exactly how many standard deviations an individual observation ($x$) lies above or below the population mean ($\mu$).

1. The Fundamental Z-Score Formula

Z = (x - μ) / σ

Positive Z-score: The observation is above the average.
Z = 0: The observation equals the population mean.
Negative Z-score: The observation is below the average.

2. The 68-95-99.7 Empirical Rule

In any bell-shaped normal distribution curve:

68.27% of data points fall within 1 standard deviation ($|Z| \le 1$).
95.45% of data points fall within 2 standard deviations ($|Z| \le 2$).
99.73% of data points fall within 3 standard deviations ($|Z| \le 3$).
Observations with $|Z| > 3$ are typically categorized as statistically significant outliers.

Frequently Asked Questions

A Z-score of +2.0 means your score is 2 full standard deviations above the average. In a normal distribution, this places you at the 97.72nd percentile, meaning your score exceeds 97.72% of the entire population.

The cumulative distribution function (CDF) is computed by integrating the standard normal probability density function from negative infinity to Z using the error function: Φ(Z) = 0.5 × (1 + erf(Z / √2)).

Yes. Any score lower than the population mean produces a negative Z-score. For example, if average test score is 80 and you score 70 with σ = 10, your Z-score is (70 - 80) / 10 = -1.0.