Normal Distribution CDF Calculator

How to use this calculator

1. Enter or choose mean (μ).
2. Enter or choose standard deviation (σ).
3. Enter or choose probability of.
4. The probability updates as you type. Use Copy result or Share to keep it.

Method

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

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.

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.