Math & Engineering
Z-Score Calculator
Calculate how many standard deviations a data point is from the mean
Reviewed by the calculator.uk.com Team · Last reviewed 5 May 2026 · Our methodology
Enter values to calculate the z-score
Related to Z-Score Calculator
The Z-Score Calculator determines how many standard deviations away from the mean a data point is. It converts raw scores into a standardized form that allows for comparison across different datasets. The formula used is:
Z-Score Formula
Z = (X - μ) / σ
Where:
- X is the raw score (value)
- μ (mu) is the population mean
- σ (sigma) is the population standard deviation
The calculator also converts the z-score into a percentile using the cumulative standard normal distribution function. This tells you what percentage of the data falls below your value in a normal distribution.
Z-scores help you understand where a value stands in relation to the mean of a dataset. The interpretation depends on the z-score value:
Common Z-Score Values
- Z = 0: The value equals the mean
- Z = 1: The value is one standard deviation above the mean
- Z = -1: The value is one standard deviation below the mean
- Z = 2: The value is two standard deviations above the mean
- Z = -2: The value is two standard deviations below the mean
The percentile tells you the proportion of values that fall below your data point. For example, a percentile of 84.13% means your value is higher than 84.13% of the data in a normal distribution.
1. What is a z-score used for?
Z-scores are used to standardize data and make comparisons across different datasets. They're commonly used in statistics, academic grading, quality control, and any field where you need to understand how far a value deviates from the mean.
2. What is considered a high z-score?
In a normal distribution, approximately 68% of the data falls within ±1 standard deviation, 95% within ±2, and 99.7% within ±3. Z-scores beyond ±3 are often considered unusual or significant, depending on the context.
3. Can z-scores be negative?
Yes, z-scores can be negative. A negative z-score indicates that the value is below the mean, while a positive z-score indicates it's above the mean. The magnitude tells you how many standard deviations away from the mean the value is.
4. How is the percentile calculated?
The percentile is calculated using the cumulative distribution function of the standard normal distribution. It represents the area under the normal curve up to the z-score value, multiplied by 100 to convert to a percentage.
5. What is the scientific source for this calculator?
This calculator computes the z-score directly from z = (x - mean) / standard deviation, the standard formula for expressing how many standard deviations a value lies from the mean. The percentile conversion uses a polynomial approximation to the error function, the same one published as formula 7.1.26 in Abramowitz and Stegun's Handbook of Mathematical Functions, to evaluate the cumulative standard normal distribution without requiring lookup tables.