Z Score Calculator
Calculate standard normal z-scores, p-values, and percentile ranks from raw scores, means, and standard deviations.
Privacy First: 100% Client-Side Processing
Processed in your browser. Files are never uploaded.
How to Use the Z Score Calculator – Standard Normal Distribution
- 1Enter the raw data observation value (x).
- 2Enter the population or sample mean (μ).
- 3Enter the standard deviation (σ).
- 4Read the calculated z-score, percentile rank, and cumulative probability P(Z < z).
Formula & How It Works
Standard Normal Deviation Formula
A z-score measures the exact number of standard deviations an individual raw score x falls above or below the distribution mean μ. The cumulative probability is evaluated using the error function (erf) approximation of the standard normal integral.
Worked Examples
Standardized Exam Score Benchmark
Input: Raw Score: 1300, Mean: 1050, Standard Deviation: 200
Explanation: (1300 - 1050) ÷ 200 = 250 ÷ 200 = +1.25. Score is 1.25 standard deviations above average.
Score Falling Below Average
Input: Raw Score: 85, Mean: 100, Standard Deviation: 15 (IQ Scale)
Explanation: (85 - 100) ÷ 15 = -15 ÷ 15 = -1.00. Score is 1 standard deviation below the mean.
Exact Average Observation
Input: Raw Score: 50, Mean: 50, Standard Deviation: 10
Explanation: (50 - 50) ÷ 10 = 0. An observation equal to the mean has a z-score of zero.
Extreme Outlier Detection
Input: Raw Score: 95, Mean: 50, Standard Deviation: 15
Explanation: (95 - 50) ÷ 15 = 45 ÷ 15 = +3.00. Represents a 3-sigma rare outlier.
Frequently Asked Questions
What is a z-score in statistics?▼
A z-score (standard score) indicates how many standard deviations an observation is above (positive) or below (negative) the population mean.
What does a negative z-score mean?▼
A negative z-score means the raw value is smaller than the distribution mean.
What is the empirical rule for normal distributions?▼
The 68-95-99.7 rule states that approximately 68% of observations fall within ±1 standard deviation, 95% within ±2, and 99.7% within ±3.
Can standard deviation be zero or negative?▼
No, standard deviation measures dispersion and must always be a strictly positive number (σ > 0).
What is the p-value P(Z < z)?▼
P(Z < z) is the cumulative probability that a randomly selected observation from a standard normal distribution falls below your z-score.
Does this calculator run client-side without data tracking?▼
Yes, all statistical arithmetic runs locally in your web browser with 100% privacy.
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