The normal distribution describes a great many measured quantities: heights, measurement errors, exam marks, manufacturing tolerances. This page integrates the bell curve between two bounds by converting each into a z score and taking the difference of the cumulative distribution.
The empirical rule is worth carrying in your head: about 68 per cent of values fall within one standard deviation of the mean, 95 per cent within two, and 99.7 per cent within three. Those figures come straight from this calculation and explain why a three-sigma event is treated as remarkable. The engine uses a high-accuracy rational approximation to the error function, so the results match published tables to several decimal places.
The probability within a range of the mean table follows the calculator: 8 rows spanning ±0.5σ through ±4σ, there so the result can be cross-checked against a value you already trust. It loads pre-filled, with mean at 100, and the answer updates on every keystroke after that. The result is also read against 4 bands, from narrow slice at the low end to nearly everything at the high end, each with a note on what that range means in practice. Coin Flip Probability Calculator and Binomial Probability Calculator are the nearest neighbours to this one.
The formula
μ- Mean of the distribution
σ- Standard deviation
a, b- The lower and upper bounds
Φ- Cumulative normal distribution
How it works, step by step
- Enter the mean and standard deviation.
- Enter the lower and upper bounds of the range.
- The gauge shows the probability of falling inside the range.
- The tail probabilities and z scores appear below.
Worked examples
IQ between 85 and 115
With a mean of 100 and a deviation of 15, those bounds are exactly one deviation either side, so the probability is 68.2689% — the familiar 68 per cent figure.
A one-sided tail
The chance of exceeding 130 with the same parameters is 2.2750%, since 130 sits two deviations above the mean.
Checking the ±2σ row
The reference table lists Range = ±2σ, giving Inside: 95.4500%, Outside: 4.5500%, One tail: 2.2750%. Entering that above should reproduce the same figures — if it does not, the input is being read differently to the way the table assumes.
How to read your score
Frequently asked questions
How accurate is the approximation used?
It is the Zelen and Severo rational approximation, accurate to better than one part in ten million, which is well beyond the precision of any published table.
What is the empirical rule?
That roughly 68, 95 and 99.7 per cent of values fall within one, two and three standard deviations of the mean. This calculator reproduces those figures exactly.
Can I get a one-sided probability?
Yes. Set one bound far beyond the plausible range and read the result, or use the tail probabilities reported in the extras.
How do I know my data is normal?
Plot a histogram and a quantile-quantile plot. Skewed data, bounded data and data with heavy tails will all give misleading answers here.
What do the bands under the probability in range mean?
They label the figure so it can be read without a reference point of your own: Narrow slice from 0 to 20; Moderate slice from 20 to 60; Most of the distribution from 60 to 90; Nearly everything from 90 upwards. The band is a reading aid only — the number itself is exact, and a value sitting just over a boundary is not meaningfully different from one just under it.
What formula does the Normal Distribution Calculator use?
It computes P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ), where μ is mean of the distribution; σ is standard deviation; a, b is the lower and upper bounds; Φ is cumulative normal distribution. The same expression is printed on the page above the inputs, so the result can be reproduced by hand or in a spreadsheet.
The empirical rule
| Range | Inside | Outside | One tail |
|---|---|---|---|
| ±0.5σ | 38.2925% | 61.7075% | 30.8538% |
| ±1σ | 68.2689% | 31.7311% | 15.8655% |
| ±1.645σ | 90.0030% | 9.9970% | 4.9985% |
| ±1.96σ | 95.0004% | 4.9996% | 2.4998% |
| ±2σ | 95.4500% | 4.5500% | 2.2750% |
| ±2.576σ | 99.0005% | 0.9995% | 0.4998% |
| ±3σ | 99.7300% | 0.2700% | 0.1350% |
| ±4σ | 99.9937% | 0.0063% | 0.0032% |
A range of ±1.96σ captures exactly 95 per cent, which is why that value appears in confidence intervals.