# Normal Distribution CDF Calculator

> Find the area under any normal curve: P(X ≤ x), P(X ≥ x), between two values or outside them, for your mean and standard deviation.

Category: Statistics & probability · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/normal-distribution-cdf-calculator

Enter a normal distribution’s mean and standard deviation and the values you care about to get the probability below, above, between or outside them.

Also called: Normal CDF calculator, Bell curve probability calculator.

## Inputs
- Mean (μ) (default 100)
- Standard deviation (σ) (default 15)
- Probability of (options: Below x: P(X ≤ x), Above x: P(X ≥ x), Between a and b, Outside a and b (both tails); default between)
- Value (x) (default 130)
- Lower value (a) (default 85)
- Upper value (b) (default 115)

## Results
- Probability
- As a percentage
- Probability, full precision
- Standardised value(s)
- Event

## Formula

```
z = (x − μ) ÷ σ;  P(X ≤ x) = Φ(z) = ½·erfc(−z ÷ √2);  P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
```

Where:
- μ, σ: mean and standard deviation
- Φ: standard normal cumulative distribution function
- erfc: complementary error function, computed to near machine precision (also far into the tails)

## Worked example
1. z for 85 = (85 − 100) ÷ 15 = −1; z for 115 = +1.
2. Φ(1) = 0.841345 and Φ(−1) = 0.158655.
3. P(85 ≤ X ≤ 115) = 0.841345 − 0.158655 = 0.682689.

**Result:** For scores with mean 100 and standard deviation 15, 68.27% fall between 85 and 115.

## What the result means

For a continuous distribution, “below” and “at or below” are the same: the probability of hitting a value exactly is zero. The answer is only as good as the normal model, so check a histogram first if the data could be skewed or have heavy tails.

## Assumptions and limits
- The quantity follows a normal distribution with the mean and standard deviation entered.
- Bounded or skewed data (times, incomes, counts near zero) are often poorly described by a normal curve, especially in the tails.

## Questions

### How is this different from the z-score calculator?

The z-score calculator works from one value and symmetric bands around the mean. This one handles any interval between two values and either tail.

### Why does the empirical rule say 68%?

Because the area between z = −1 and z = 1 is 0.6827. Between ±2 it is 0.9545, and between ±3 it is 0.9973.

## Related calculators
- [Normal Distribution Quantile Calculator](https://calculators.geekmidst.com/calculators/normal-distribution-quantile-calculator)
- [Z-score Calculator](https://calculators.geekmidst.com/calculators/z-score-calculator)
- [P-value Calculator](https://calculators.geekmidst.com/calculators/p-value-calculator)
- [Standard Deviation Calculator](https://calculators.geekmidst.com/calculators/standard-deviation-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/normal-distribution-cdf-calculator