Explore the bell curve, drag markers to calculate probabilities, and learn how to convert any raw score into a standardized z-score. Master the empirical rule and compare scores across different scales.
Drag the gold and orange markers to select a range and see the probability
Drag markers to explore probabilities
For any normal distribution, ~68% of data falls within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ.
Calculate exact probabilities from z-score boundaries
Area
68.27%
Convert a raw score to a z-score
Formula
Your z-score is
1.00
Slightly above average
115 is 1.00 standard deviation above the mean.
Compare scores from different tests using z-scores
Test A
Test B
Standard Normal: Both Scores
Score A: z = 1.50 | Score B: z = 2.00
Score B performs better: higher z-score means further above the mean.
Key Concepts
Normal Distribution
A symmetric, bell-shaped distribution defined entirely by its mean (μ) and standard deviation (σ).
Z-Score
A standardized score: z = (x − μ) / σ. Tells you how many SDs a value is from the mean.
Standard Normal
A normal distribution with μ = 0 and σ = 1. Any normal distribution can be transformed to this.
Empirical Rule
68% of data lies within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean.