Normal Distribution Calculator
The probability below, above, between or outside values of any normal distribution, or the value at a given probability.
Result
Result: 0.682689
Exact to the precision shown, for a normal distribution.
- What this is
- P(85 < X < 115), as a proportion from 0 to 1
- As a percentage
- 68.2689%
- Z-score
- -1
- Z-score of the upper value
- 1
https://calcgallery.com/statistics/normal-distribution-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Normal Distribution Calculator
- Choose what to find. A probability below, above, between or outside values, or the value at a given probability.
- Enter the distribution's mean and standard deviation, then the value or values.
- Read the result. A probability is shown as a proportion and as a percentage, with the z-scores of the values used.
The formulas
z = (x − μ) ÷ σP(X < x) = Φ(z)P(a < X < b) = Φ(z_b) − Φ(z_a)the value at probability p: x = μ + σ × Φ⁻¹(p)Φ is the cumulative distribution of the standard normal, the area under the bell curve to the left of z. Every normal distribution is the standard one stretched by σ and shifted by μ, which is why converting to z-scores is all that is needed.
Four regions and the inverse
- Below a value is the area to its left, the value’s percentile.
- Above a value is the area to its right, 1 minus the area below.
- Between two values is the area from the lower to the upper value.
- Outside two values is the two tails together, 1 minus the area between.
- The value at a probability runs the other way. Given the share of the distribution below, it finds the value.
The page refuses an upper value below the lower value rather than reporting a negative probability.
Precision
The normal curve is calculated to around 15 significant figures, far beyond what the result shows. Differences between two areas far out in the same tail lose some of that precision, but not enough to affect the figures displayed.
With your numbers
- z = (x − μ) ÷ σ = (x − 100) ÷ 15
- z for 85 = -1; z for 115 = 1
- Probability = 68.2689%
Quick reference
Exact to the precision shown, for a normal distribution.
| Example | Result |
|---|---|
| 85 to 115, mean 100, SD 15 | 0.682689 |
| Below 100, mean 100, SD 15 | 0.5 |
| Above 130, mean 100, SD 15 | 0.02275 |
| Above 1.96, standard normal | 0.024998 |
| Outside ±2, standard normal | 0.0455 |
| 90th percentile, mean 100, SD 15 | 119.223273 |
| 97.5th percentile, standard normal | 1.959964 |
Common questions about the Normal Distribution Calculator
- What is a normal distribution?
- The symmetric, bell-shaped distribution that many measurements roughly follow: heights, test scores, measurement errors. It is fixed by two numbers, its mean, where the peak sits, and its standard deviation, how wide the bell is. About 68% of it lies within one standard deviation of the mean and about 95% within two.
- How do I find the value at the 90th percentile?
- Choose "The value at a probability" and enter 90%. The page finds the z-score with 90% of the standard normal curve below it, about 1.2816, and converts it back with x = μ + σz. For IQ scores with a mean of 100 and a standard deviation of 15, that is about 119.2.
- Why is the probability of an exact value zero?
- Because a continuous distribution spreads its probability over infinitely many possible values, so any single exact value gets none. Probabilities are for ranges: below, above, between or outside values. The density shown for a single value is a height on the curve, not a probability.
- Is this the same as a z-table?
- It gives the same numbers a z-table would, for any mean and standard deviation and to more decimal places, without rounding the z-score to two decimals first. The page calculates the normal curve directly rather than reading a printed table.