Z-Score Calculator
Calculate the z-score for a normal distribution probability. The z-score tells you how many standard deviations a data point is from the mean. This calculator can find the probability (area) between two z-scores or between one z-score and infinity.
Probability Result: 0.3413
-3σ to -2σ
-2σ to -1σ
-1σ to 0
0 to +1σ
+1σ to +2σ
+2σ to +3σ
Standard normal distribution curve
| Z-Score(s) | 0 to 1 |
|---|---|
| Probability | 0.3413 (34.13%) |
| Complementary Probability | 0.6587 (65.87%) |
| Percentile | 84.13% |
About Z-Score Calculator
A Z-score (or standard score) represents the number of standard deviations a data point is from the mean of a normal distribution. Z-scores are commonly used in statistics to compare different data points within the same distribution or across different distributions.
How Z-Score is Calculated
The formula for calculating a z-score is:
Z = (X - μ) / σ
Where:
- X is the raw score
- μ is the population mean
- σ is the population standard deviation
Understanding Z-Scores and Probabilities
The standard normal distribution (z-distribution) has:
- Mean = 0
- Standard deviation = 1
- Total area under the curve = 1 (or 100%)
Key probabilities:
- About 68% of values fall within ±1 standard deviation from the mean
- About 95% within ±2 standard deviations
- About 99.7% within ±3 standard deviations
Common Z-Score Values
| Z-Score | Probability | Percentile |
|---|---|---|
| -3.00 | 0.0013 (0.13%) | 0.13% |
| -2.00 | 0.0228 (2.28%) | 2.28% |
| -1.00 | 0.1587 (15.87%) | 15.87% |
| 0.00 | 0.5000 (50.00%) | 50.00% |
| 1.00 | 0.8413 (84.13%) | 84.13% |
| 2.00 | 0.9772 (97.72%) | 97.72% |
| 3.00 | 0.9987 (99.87%) | 99.87% |
Applications of Z-Scores
Z-scores are widely used in:
- Statistics: Hypothesis testing, confidence intervals
- Quality Control: Process capability analysis
- Finance: Risk assessment, portfolio management
- Psychology: Standardized testing, IQ scores
- Medicine: Growth charts, lab test results