# What is the normal distribution area?

Computes the probability that a normal random variable falls between two bounds, below one, or above one, with the tails outside and the z-scores.

- Page: https://www.acalculator.org/statistics/normal-distribution-calculator
- JSON spec: https://www.acalculator.org/statistics/normal-distribution-calculator.json
- Version: 8d1f09c775da

## Default answer

Example with the default inputs (Mean (μ) 0, Standard deviation (σ) 1, Lower bound (a) -1, Upper bound (b) 1): For a normal distribution with mean 0 and standard deviation 1, P(a < X < b) = 0.682689.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mean | Mean (μ) | The mean of the normal distribution. |
| sd | Standard deviation (σ) | The standard deviation of the normal distribution, at least 0.000000000001. |
| a | Lower bound (a) | The low end of the range. Leave empty for no lower end: P(X < b). |
| b | Upper bound (b) | The high end of the range. Leave empty for no upper end: P(X > a). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| probability | P(a < X < b) | The probability that X falls between the bounds; an empty bound is open (minus or plus infinity). |
| outside | Outside the bounds | The probability that X falls below a or above b: 1 − P(a < X < b). |
| belowLower | P(X < a) | The probability below the lower bound. Shown when there is a lower bound. |
| aboveUpper | P(X > b) | The probability above the upper bound. Shown when there is an upper bound. |
| zLower | z-score of a | How many standard deviations the lower bound is from the mean: (a − μ) ÷ σ. |
| zUpper | z-score of b | How many standard deviations the upper bound is from the mean: (b − μ) ÷ σ. |

## Method

z = (x − μ) ÷ σ; P(a < X < b) = Φ(z(b)) − Φ(z(a)), where Φ is the standard normal cumulative distribution function; an empty bound is −∞ or +∞.

## Assumptions

- X follows a normal distribution with the mean and standard deviation you enter.
- Each area is worked out from the tail it lies in, not by subtracting from 1, so tiny probabilities keep their precision.

## Worked examples

1. mean = 0, sd = 1, a = -1, b = 1 gives probability = 0.682689, outside = 0.317311, belowLower = 0.158655. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm.
2. mean = 0, sd = 1, a = -1.96, b = 1.96 gives probability = 0.950004, outside = 0.049996. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm (table: Φ(1.96) = 0.97500).
3. mean = 100, sd = 15, b = 130 gives probability = 0.97725, aboveUpper = 0.02275, zUpper = 2. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm.
4. mean = 5, sd = 2, a = 7 gives probability = 0.158655, zLower = 1. Source: OpenStax, Introductory Statistics 2e, §6.2 Using the Normal Distribution. https://openstax.org/books/introductory-statistics-2e/pages/6-2-using-the-normal-distribution.
5. mean = 0, sd = 1, a = 10 gives probability = 0. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm.
6. mean = 0, sd = 1, a = 8, b = 9 gives probability = 0. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm.

## FAQ

### How do I find a probability from a normal distribution?

Turn each bound into a z-score, z = (x − μ) ÷ σ, then read the area from the standard normal cumulative distribution Φ. The probability between a and b is Φ(z(b)) − Φ(z(a)). With mean 100 and standard deviation 15, the chance of a value below 130 is Φ(2) = 0.97725.

### How do I get P(X < b) or P(X > a)?

Leave the other bound empty. With only an upper bound b, the calculator gives P(X < b); with only a lower bound a, it gives P(X > a). The area outside the bounds is shown too.

### What is the empirical rule?

For any normal distribution, about 68.27% of values lie within 1 standard deviation of the mean, 95.45% within 2, and 99.73% within 3. Enter bounds of μ − σ and μ + σ to see the first: 0.682689.

### Is P(X < b) the same as P(X ≤ b)?

Yes. A normal distribution is continuous, so any single value has probability 0, and it makes no difference whether the bounds are included.

### Why does the calculator show such tiny probabilities?

Far from the mean the tails are very thin but not zero: P(Z > 10) is about 7.6 × 10⁻²⁴. The calculator works out each tail directly instead of subtracting from 1, so these small areas keep their precision.

### How do I find the value for a given probability?

That is the inverse question. The z-score calculator finds the z for a percentile, and the value is then x = μ + z × σ. The critical value calculator gives the z or t that cuts off a tail of a given size.

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm
- OpenStax, Introductory Statistics 2e, §6.2 Using the Normal Distribution. https://openstax.org/books/introductory-statistics-2e/pages/6-2-using-the-normal-distribution