Z-Score Calculator - Free Standard Score & Probability Calculator
Z-Score Calculator
Calculate z-scores (standard scores) and find probabilities in the normal distribution. Convert between raw scores and standardized values.
Calculate Z-Score from Raw Score
Results
Normal Distribution Curve
Interpretation
Common Z-Score Values
| Z-Score | Percentile | P(X < z) | P(X > z) |
|---|---|---|---|
| -3.0 | 0.13% | 0.0013 | 0.9987 |
| -2.0 | 2.28% | 0.0228 | 0.9772 |
| -1.0 | 15.87% | 0.1587 | 0.8413 |
| 0.0 | 50.00% | 0.5000 | 0.5000 |
| 1.0 | 84.13% | 0.8413 | 0.1587 |
| 2.0 | 97.72% | 0.9772 | 0.0228 |
| 3.0 | 99.87% | 0.9987 | 0.0013 |
How to use this calculator
📊 How to Use This Calculator
- Choose your calculation type from the tabs
- For Z-Score: Enter raw score, mean, and standard deviation
- For Raw Score: Enter z-score, mean, and standard deviation
- For Probability: Enter z-score and select probability type
- View results with normal distribution visualization
📐 Understanding Z-Scores
What is a Z-Score?
A z-score tells you how many standard deviations a value is from the mean. It standardizes different datasets to a common scale.
Z = (X - μ) / σ
Interpreting Z-Scores
- • Z = 0: Exactly at the mean
- • Z > 0: Above the mean
- • Z < 0: Below the mean
- • |Z| > 2: Unusual (top/bottom 5%)
- • |Z| > 3: Very rare (top/bottom 0.3%)
🌟 Real-World Examples
Example 1: Test Scores
Test mean = 75, SD = 10. A score of 85 has z = 1.0
This student scored better than 84% of test-takers
Example 2: Height
Adult male height: mean = 70", SD = 3". Height of 76" has z = 2.0
This person is taller than 97.7% of adult males
Example 3: Quality Control
Product weight: mean = 100g, SD = 2g. Weight of 95g has z = -2.5
This product is unusually light and may be defective
💡 Pro Tips
- • Z-scores allow comparison across different scales
- • About 68% of data falls within Z = ±1
- • About 95% of data falls within Z = ±2
- • About 99.7% of data falls within Z = ±3
- • Z-scores assume normal distribution
- • Outliers typically have |Z| > 3
⚠️ Common Mistakes to Avoid
- • Don't use z-scores for non-normal distributions
- • Remember z-scores have no units
- • Use population SD (σ) not sample SD (s) when appropriate
- • Z-score of 0 doesn't mean the value is 0
- • Negative z-scores are perfectly normal (below mean)
About this calculator
Calculate z-scores, find probabilities, and percentile ranks. Convert between raw scores and z-scores with visual normal distribution curve.
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