Normal Distribution Probability: formula
Many real datasets are not normal: incomes, times and counts are often skewed. The normal model is an approximation, and probabilities in the tails are especially sensitive to it. Estimating the mean and deviation from small samples adds uncertainty.
Results are probabilities of ranges, not of exact values.
Normal Distribution Probability: worked example
| Mean (μ) | 100 |
|---|---|
| Standard deviation (σ) | 15 |
| Lower value (a) | 85 |
| Upper value (b) | 115 |
| Probability between a and b | 68.3% |
| Probability below a | 15.9% |
| Probability above b | 15.9% |
Enter the mean and standard deviation of the normal distribution, and the lower and upper limits of the range whose probability you want. The calculator standardizes each limit to a z-score and finds the area between them under the bell curve. The standard deviation must be positive.
Use realistic parameters from data, such as an IQ with mean 100 and deviation 15.
Normal Distribution Probability: changing Mean (μ)
| Mean (μ) | Probability between a and b | Probability below a | Probability above b |
|---|---|---|---|
| 80 | 36.0% | 63.1% | 0.982% |
| 90 | 58.3% | 36.9% | 4.8% |
| 100 (example) | 68.3% | 15.9% | 15.9% |
| 110 | 58.3% | 4.8% | 36.9% |
| 120 | 36.0% | 0.982% | 63.1% |
Normal Distribution Probability: compare input ranges
| Input | Tested from | To | Probability between a and b: low input | Probability between a and b: high input | Swing |
|---|---|---|---|---|---|
| Upper value (b) | 104 | 127 | 44.6% | 80.5% | 35.9% (53%) |
| Lower value (a) | 77 | 94 | 77.9% | 49.7% | 28.2% (41%) |
| Standard deviation (σ) | 14 | 17 | 71.6% | 62.2% | 9.4% (14%) |
| Mean (μ) | 90 | 110 | 58.3% | 58.3% | 0.000% |
Normal Distribution Probability: individual input comparisons
| Input | Value | Probability between a and b | Probability below a | Probability above b |
|---|---|---|---|---|
| Mean (μ) | 80 | 36.0% | 63.1% | 0.982% |
| 90 | 58.3% | 36.9% | 4.8% | |
| 100 | 68.3% | 15.9% | 15.9% | |
| 110 | 58.3% | 4.8% | 36.9% | |
| 120 | 36.0% | 0.982% | 63.1% | |
| Standard deviation (σ) | 12 | 78.9% | 10.6% | 10.6% |
| 14 | 71.6% | 14.2% | 14.2% | |
| 15 | 68.3% | 15.9% | 15.9% | |
| 17 | 62.2% | 18.9% | 18.9% | |
| 18 | 59.5% | 20.2% | 20.2% | |
| Lower value (a) | 68 | 82.5% | 1.6% | 15.9% |
| 77 | 77.9% | 6.3% | 15.9% | |
| 85 | 68.3% | 15.9% | 15.9% | |
| 94 | 49.7% | 34.5% | 15.9% | |
| 102 | 28.8% | 55.3% | 15.9% | |
| Upper value (b) | 92 | 13.8% | 15.9% | 70.3% |
| 104 | 44.6% | 15.9% | 39.5% | |
| 115 | 68.3% | 15.9% | 15.9% | |
| 127 | 80.5% | 15.9% | 3.6% | |
| 138 | 83.6% | 15.9% | 0.565% |
Normal Distribution Probability: combined input changes
| Mean (μ) / Standard deviation (σ) | 12 | 15 | 18 |
|---|---|---|---|
| 80 | 33.7% | 36.0% | 36.5% |
| 90 | 64.3% | 58.3% | 52.7% |
| 100 | 78.9% | 68.3% | 59.5% |
| 110 | 64.3% | 58.3% | 52.7% |
| 120 | 33.7% | 36.0% | 36.5% |
Normal Distribution Probability: which assumptions move the result?
Normal Distribution Probability: values and definitions
| Quantity and definition | Value used |
|---|---|
| Mean (μ) — The center of the distribution. | 100 |
| Standard deviation (σ) — The spread of the distribution. It must be above zero. | 15 |
| Lower value (a) — The bottom of the range you are asking about. | 85 |
| Upper value (b) — The top of the range you are asking about. | 115 |
| Probability between a and b | 68.3% |
| Probability below a | 15.9% |
| Probability above b | 15.9% |
Normal Distribution Probability: half and double Upper value (b)
| Upper value (b) | Probability between a and b |
|---|---|
| 57.5 | 0.000% |
| 115 | 68.3% |
| 230 | 84.1% |
Common error: Entering the variance as the standard deviation. Variance is the square, so a spread of 15 entered as 225 stretches the band far wider than the data.
Statistics & Probability calculators: P-Value from Z-Score Calculator, Expected Value Calculator, Bayes' Theorem Calculator, Mean & Median Calculator, Standard Deviation Calculator, Range & Midrange Calculator.
A bell curve turns a mean and a spread into the share of results in any band. About 68 percent of values sit within one standard deviation of the mean, which for a mean of 100 and a spread of 15 is the range from 85 to 115. The two tail figures show what is left outside, split evenly here because the band is centered.
Confirm the data is roughly bell-shaped. Income, wait times and many counts are skewed, and the curve misstates their tails badly.
Standardize values on the z-score calculator, get p-values on the p-value page, and build ranges on the confidence interval calculator. The binomial probability calculator covers counts, and the standard deviation page gives the spread. Sample size planning uses the same distribution.
Is my data normally distributed?
Check a histogram or a normal probability plot. Many natural measurements are close to normal, but incomes and wait times often are not.
Why is the probability of a single exact value zero?
Because a continuous distribution spreads probability over a range of values, so only intervals have nonzero probability and a single exact value has none.
What is the 68-95-99.7 rule?
For a normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three.