The normal distribution — the bell curve — describes everything from manufacturing tolerances to the sampling error in a poll. Once you know its mean and standard deviation, any probability question about it comes down to areas under the curve. This calculator finds those areas for values below, above, between or outside your cutoffs, and also works in reverse to find the value that matches a probability.
How to use the normal distribution calculator
- Enter the mean (μ) and standard deviation (σ) of the distribution.
- Choose the probability to find: less than a value, greater than a value, between two values, outside two values, or the inverse problem.
- Enter the value or values (or the probability for the inverse option).
- Read the probability on the tape. The curve below shades the area, and the steps show the z-scores used.
Normal distribution formulas
The density of a normal distribution with mean μ and standard deviation σ is
Probabilities are areas under this curve. Standardizing with z = (x − μ) ÷ σ reduces every question to the standard normal CDF Φ:
For the inverse problem the calculator solves Φ(z) = p and returns x = μ + zσ.
Worked example
IQ-style scores are scaled to a mean of 100 and a standard deviation of 15. What share of people score between 85 and 130?
- Standardize: zₐ = (85 − 100) ÷ 15 = −1 and zᵦ = (130 − 100) ÷ 15 = 2.
- Look up the left-tail areas: Φ(2) ≈ 0.977250 and Φ(−1) ≈ 0.158655.
- Subtract: 0.977250 − 0.158655 = 0.818595, or about 81.9%.
Going the other way, the score that 90% of people fall below solves Φ(z) = 0.9, so z ≈ 1.2816 and x = 100 + 1.2816 × 15 ≈ 119.2.
The empirical rule
| Interval | Share of values | Outside the interval |
|---|---|---|
| μ ± 1σ | 68.27% | 31.73% |
| μ ± 2σ | 95.45% | 4.55% |
| μ ± 3σ | 99.73% | 0.27% |
These percentages hold for every normal distribution, whatever its mean and spread. In the IQ example, about 95% of people score between 70 and 130.
Practical applications
Quality control
A machine fills bottles with a mean of 500 mL and a standard deviation of 4 mL. Bottles under 490 mL fail inspection. P(X < 490) = Φ(−2.5) ≈ 0.0062, so about 6 bottles in 1,000 fail. Choose the “outside two values” option to count rejects when there is both a lower and an upper limit.
Setting cutoffs
Use the inverse option to find thresholds: the top 5% of a test, a safety stock level that covers 95% of demand, or a tolerance limit. Remember that a one-sided 95% cutoff uses z ≈ 1.645, while a two-sided 95% interval uses z ≈ 1.96.
Sampling distributions
By the central limit theorem, averages of large samples are approximately normal even when individual values are not. That is why confidence intervals and many hypothesis tests rely on the normal curve with the standard error as σ.
Probabilities are computed with a double-precision algorithm, so far-tail values such as P(X > μ + 6σ) ≈ 9.9 × 10⁻¹⁰ are shown accurately instead of rounding to zero.
To work with a single standardized score, including two-tailed p-values, use the z-score calculator. If your data are not bell-shaped, the percentile calculator finds cutoffs directly from the raw values.
Frequently asked questions
How do I find the probability between two values?
Convert both values to z-scores and subtract the left-tail areas: P(a < X < b) = Φ(zb) − Φ(za). For a mean of 100 and SD of 15, P(85 < X < 130) = Φ(2) − Φ(−1) ≈ 0.977250 − 0.158655 = 0.818595.
Is P(X < a) the same as P(X ≤ a)?
Yes, for a continuous distribution like the normal. The probability of landing on any single exact value is zero, so including or excluding the endpoint makes no difference.
What is the inverse normal option for?
It works backward from a probability to a value. Asking for P(X < x) = 0.9 with mean 100 and SD 15 returns x ≈ 119.22, the 90th percentile — useful for setting cutoffs, tolerances and qualifying scores.
What is the empirical rule?
For any normal distribution, about 68.27% of values fall within one standard deviation of the mean, 95.45% within two and 99.73% within three. The table under the calculator lists those intervals for your mean and SD.
When is the normal distribution a reasonable model?
When data are continuous, roughly symmetric and bell-shaped, such as measurement errors, many physical dimensions and averages of large samples. It is a poor fit for strongly skewed data such as incomes or waiting times.