Z-Score Calculator
Calculate the standard score (z-score) to determine how many standard deviations a raw observation lies above or below the population or sample mean. Computes corresponding normal curve percentiles.
The specific raw measurement, test score, or data point
Arithmetic mean of the distribution
Spread or dispersion parameter (strictly positive)
Number of standard deviations from the mean (z = (x − μ)/σ)
How to Use This Calculator
- Enter your raw test or measurement score (x).
- Enter the benchmark mean (μ).
- Enter the standard deviation (σ).
- Inspect your z-score and corresponding percentile ranking.
Formula & Calculation Methodology
The z-score standardizes any normal distribution into a standard normal curve N(0, 1). The cumulative probability Φ(z) is evaluated via error function approximation.
Worked Example
Scoring 85 on an exam with mean 70 and standard deviation 10
Observation 1 standard deviation below average (x = 40, mean = 50, sd = 10)
Frequently Asked Questions
What does a negative z-score indicate?↓
A negative z-score means the raw observation is lower than the arithmetic mean (e.g. z = −1.5 is 1.5 standard deviations below average).
What is the empirical rule (68-95-99.7)?↓
In a normal distribution, approximately 68% of data falls within z = ±1, 95% falls within z = ±2, and 99.7% falls within z = ±3.
What occurs if standard deviation is zero?↓
If standard deviation is zero, there is zero dispersion and z-scores are undefined because division by zero is impossible.
What is a limitation of the z-score?↓
Z-scores assume the underlying population follows an approximately symmetric bell-shaped normal distribution. For skewed distributions, percentiles may diverge.
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Source Citation & Disclaimer
Informational tool only; not licensed tax, legal, or financial advice.