Normal Distribution Calculator
Adjust the mean and standard deviation, choose a probability mode, and watch the shaded area change in real-time. Read off z-scores and exact probabilities instantly.
Controls
Results
| P | — |
| z-score | — |
| Percent | — |
The Math Behind It
Normal Distribution
- PDF: f(x) = (1/σ√2π) e-(x-μ)²/2σ²
- The bell curve is symmetric about the mean μ
- 68% of data falls within μ ± 1σ
- 95% within μ ± 2σ, 99.7% within μ ± 3σ
Z-Score
- The z-score converts any normal to the standard normal: z = (x - μ) / σ
- A z-score of 1.96 corresponds to the 97.5th percentile
- Used for hypothesis testing, confidence intervals, and comparing across distributions
Try This
- Set μ=0, σ=1 for the standard normal — check that P(X<1.96) ≈ 0.975
- Try P(X > 0) — always 0.5 regardless of σ
- Use Between mode to find P(-1 < X < 1) ≈ 0.6827
- Increase σ — the bell flattens but total area stays 1
- Shift μ — the entire curve moves left or right
Common Values
- z = 1.645 → 90% one-tail confidence
- z = 1.960 → 95% two-tail / 97.5% one-tail
- z = 2.576 → 99% two-tail confidence
You're crushing it!