Z-Score Calculator
Z-Score: -
Understanding Z-Scores
What is a Z-Score?
A z-score (also called a standard score) measures how many standard deviations away from the mean a data point is:
z = (x - μ) / σ
where:
- x = raw score
- μ = population mean
- σ = population standard deviation
Properties of Z-Scores
Mean
Z-scores have a mean of 0
Standard Deviation
Z-scores have a standard deviation of 1
Distribution
Follows the standard normal distribution
Scale
Linear transformation of raw scores
Empirical Rule (68-95-99.7 Rule)
| Z-Score Range | Percentage | Description |
|---|---|---|
| ±1σ | 68% | Within one standard deviation |
| ±2σ | 95% | Within two standard deviations |
| ±3σ | 99.7% | Within three standard deviations |
Interpreting Z-Scores
- Positive z-score: Value is above the mean
- Negative z-score: Value is below the mean
- Zero z-score: Value equals the mean
- Magnitude indicates distance from mean in standard deviations
Advanced Concepts
Percentile Conversion
Z-scores can be converted to percentiles using standard normal table
Probability Calculation
Area under normal curve between z-scores
Sample vs Population
Use of n-1 for sample standard deviation
Common Applications
- Educational Testing:
- Standardized test scores
- Grade normalization
- Performance comparison
- Quality Control:
- Process capability
- Defect analysis
- Control limits
- Research:
- Outlier detection
- Data standardization
- Comparative analysis
Real-World Applications
Education
Standardizing test scores and comparing student performance
Manufacturing
Quality control and process monitoring
Research
Data analysis and outlier detection
Finance
Risk assessment and portfolio analysis