Normal Distribution Calculator
Find the probability that a normal variable is below, above, or between values.
15.87% of the distribution
The normal distribution gives the probability a value falls in a range by standardizing the bound to z = (x − μ) ÷ σ and reading the area under the curve. For IQ scores (μ = 100, σ = 15), P(X < 130) = Φ(2.0) = 0.9772, about 97.7%.
Probabilities under the bell curve
The normal distribution describes data that clusters symmetrically around a mean — heights, test scores, measurement error. To find the chance a value falls in some range, this tool standardizes your bounds into z-scores with z = (x − μ) ÷ σ, then reads the area under the standard normal curve. The total area is always 1, so every probability is just the slice of that area you care about.
Φ is the standard normal cumulative area; P(X > a) = 1 − Φ
Worked example
IQ scores with mean 100 and standard deviation 15. What is P(X < 130)?
- 1 Standardize the bound. z = (130 − 100) ÷ 15 = 30 ÷ 15 = 2.0.
- 2 Look up the area to the left. Φ(2.0) = 0.9772, so P(X < 130) ≈ 97.7%.
- 3 Take the complement for the upper tail. P(X > 130) = 1 − 0.9772 = 0.0228, about 2.3%.
The 68-95-99.7 rule
Proportion of a normal distribution within k standard deviations of the mean.
| Range | z-bounds | Probability |
|---|---|---|
| μ ± 1σ | −1 to +1 | 68.3% |
| μ ± 2σ | −2 to +2 | 95.4% |
| μ ± 1.96σ | −1.96 to +1.96 | 95.0% |
| μ ± 3σ | −3 to +3 | 99.7% |
Reading the three questions it answers
Less than. P(X < a) is the area to the left of a — the cumulative probability Φ(z). This is the percentile of the value a.
Greater than. P(X > a) is the right-tail area, found as 1 − Φ(z). Because the curve is symmetric, P(X > μ) is always exactly 0.5.
Between two values. P(a < X < b) is the difference of the two cumulative areas, Φ(z_b) − Φ(z_a). For symmetric bounds like μ ± 1σ this gives the familiar 68%.
When the normal model fits. The bell curve assumes symmetry and thin tails. Skewed data, hard limits, or heavy outliers can break it, so plot a histogram before trusting these probabilities.