Z-Score Calculator
Calculate z-score, percentile, and probability for normal distributions
What is a Z-Score?
A z-score (also called a standard score) measures how many standard deviations a value is from the mean of a distribution. It allows you to compare values from different normal distributions and determine their relative positions within the distribution.
Z-Score Formula
The formula for calculating z-score is:
Where:
- z = Z-score
- x = Value
- μ = Mean
- σ = Standard deviation
Interpreting Z-Scores
- z = 0: Value equals the mean
- z > 0: Value is above the mean
- z < 0: Value is below the mean
- |z| = 1: Value is 1 standard deviation from the mean
- |z| = 2: Value is 2 standard deviations from the mean
- |z| = 3: Value is 3 standard deviations from the mean
How to Use the Calculator
- Enter the value you want to convert to z-score
- Enter the mean of the distribution
- Enter the standard deviation of the distribution
- Click "Calculate" to get the z-score
- View the step-by-step solution and percentile
Example Calculations
Example 1:
Value = 85
Mean = 75
Standard Deviation = 10
z = (85 - 75) / 10 = 1
This value is 1 standard deviation above the mean
Example 2:
Value = 60
Mean = 75
Standard Deviation = 10
z = (60 - 75) / 10 = -1.5
This value is 1.5 standard deviations below the mean
Common Z-Score Values and Percentiles
| Z-Score | Percentile | Probability | Interpretation |
|---|
| -3.0 | 0.13% | 0.0013 | Very low |
| -2.0 | 2.28% | 0.0228 | Low |
| -1.0 | 15.87% | 0.1587 | Below average |
| 0.0 | 50.00% | 0.5000 | Average |
| 1.0 | 84.13% | 0.8413 | Above average |
| 2.0 | 97.72% | 0.9772 | High |
| 3.0 | 99.87% | 0.9987 | Very high |
Applications of Z-Scores
- Standardization: Compare values from different distributions
- Outlier Detection: Identify unusual values (typically |z| > 2 or 3)
- Percentile Calculation: Find what percentage of values are below a given point
- Quality Control: Monitor process performance
- Academic Testing: Compare test scores across different tests