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 |