Normal Distribution Calculator
Interactive normal distribution calculator. Drag mean and standard deviation sliders to see the bell curve, shaded probability, z-score and percentile update live.
Set μ and σ, then read the probability, z-score and percentile together
Set the mean and the standard deviation, then read the probability, the z-score and the percentile from one screen. The defaults are the IQ scale (μ = 100, σ = 15).
P(X ≤ 100) = 0.500000
Parameters
What to calculate
Type in either box — the other one is solved for you.
Result
Related probabilities
Distribution summary
Empirical rule (68–95–99.7)
| Region | Range | P |
|---|---|---|
| Within ±1σ | 85 – 115 | 0.682689 |
| Within ±2σ | 70 – 130 | 0.954500 |
| Within ±3σ | 55 – 145 | 0.997300 |
Critical values
Click a row to jump the graph to that value.
| Significance level α | One-tailed | Two-tailed |
|---|---|---|
| 0.1 | ||
| 0.05 | ||
| 0.025 | ||
| 0.01 | ||
| 0.005 | ||
| 0.001 |
Same calculation in code
from scipy.stats import norm norm.cdf(100, loc=100, scale=15) # 0.5
The ±1σ, ±2σ and ±3σ buttons shade the empirical-rule regions instantly — handy for checking homework answers.
How to use
- Enter the mean (μ) and standard deviation (σ), or pick a preset such as IQ scores.
- Type the value you are interested in — the probability and z-score appear immediately.
- Switch the mode to get a left tail, a right tail or the probability between two values.
- Use the 68–95–99.7 buttons to shade the ±1σ, ±2σ and ±3σ regions.
FAQ
How do I find a probability in a normal distribution?
Standardise the value with z = (x − μ)/σ and read the standard normal CDF Φ(z). For example with μ = 100 and σ = 15, P(X > 130) = 1 − Φ(2) = 0.02275. This calculator does the standardising and the lookup in one step.
What is the 68–95–99.7 rule?
It is the empirical rule: about 68% of the values fall within ±1σ of the mean, 95% within ±2σ and 99.7% within ±3σ. The exact figures are 0.682689, 0.954500 and 0.997300.
What happens if the standard deviation is 0?
You get a degenerate distribution with all the mass at the mean. That is not an error: the calculator draws a vertical line at μ and returns P(X ≤ x) as either 0 or 1.
What if I only have sample data, not μ and σ?
Use the sample mean and sample standard deviation instead. With fewer than about 30 observations and an unknown population σ, the t distribution with n − 1 degrees of freedom is the safer choice.
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