Z-Score Calculator with Probability Visualization (Bell Curve)
Convert raw scores to Z-scores and find the P-value probability. Interactive Bell Curve visualization showing the area under the Normal Distribution.
About
In statistics, a Z-Score (or standard score) tells you exactly how many standard deviations a data point is from the mean. It is the gold standard for comparing apples to oranges - such as comparing test scores from different exams or height distributions across different populations.
This tool does more than just crunch numbers; it visualizes the concept. By generating a dynamic Bell Curve and shading the area corresponding to your probability (P-value), it bridges the gap between abstract statistical formulas and geometric understanding. It uses high-precision integration of the Probability Density Function (PDF) to provide accurate left-tailed, right-tailed, and two-tailed probabilities.
Formulas
The Z-Score transforms a raw score x into a standardized unit using the population mean μ and standard deviation σ.
The Probability Density Function (PDF) for the standard normal distribution is:
Reference Data
| Z-Score | P-Value (Left-Tail) | Interpretation |
|---|---|---|
| -3.00 | 0.0013 | Bottom 0.13% (Extremely Low) |
| -2.00 | 0.0228 | Bottom 2.28% (Significantly Low) |
| -1.00 | 0.1587 | Bottom 15.9% (Below Average) |
| 0.00 | 0.5000 | Exact Mean (Average) |
| +1.00 | 0.8413 | Top 15.9% (Above Average) |
| +2.00 | 0.9772 | Top 2.28% (Significantly High) |
| +3.00 | 0.9987 | Top 0.13% (Extremely High) |
| +4.00 | ~0.9999 | Statistical Outlier |