Normal CDF Calculator

Last reviewed: June 2026 by the InvNorm Calculator Editorial Team. Report an issue

Use this Normal CDF Calculator to find cumulative probabilities for any normal distribution. The cumulative distribution function (CDF) returns the probability that a normally distributed random variable falls at or below a given value. Select from three calculation modes to find left-tail, right-tail, or interval probabilities.

What Is the Normal CDF?

The cumulative distribution function (CDF) of a normal distribution gives the probability that a random variable X takes a value less than or equal to a specified number x. Written mathematically, the CDF is defined as:

Φ(x) = P(X ≤ x) = ∫_-∞^x f(t) dt

where f(t) is the normal probability density function. The CDF ranges from 0 to 1 and is a monotonically increasing function. For a standard normal distribution (mean 0, standard deviation 1), the CDF at x = 0 equals exactly 0.5, reflecting the symmetry of the bell curve around its center.

Unlike the PDF, which gives the height of the curve at a point, the CDF gives the accumulated area under the curve from negative infinity up to x. This area interpretation makes the CDF essential for answering probability questions such as "what is the chance a value falls below this threshold?" or "what percentage of observations exceed this cutoff?"

Normal CDF Formula

For a general normal distribution with mean μ and standard deviation σ, the CDF is computed by first standardizing the value to a Z score:

z = (x − μ) / σ

Then the cumulative probability is:

P(X ≤ x) = Φ(z) = ½[1 + erf(z / √2)]

For a right-tail probability:

P(X ≥ x) = 1 − Φ(z)

For the probability that X falls between two values a and b:

P(a ≤ X ≤ b) = Φ((b − μ) / σ) − Φ((a − μ) / σ)

Worked Example

Suppose test scores follow a normal distribution with mean μ = 100 and standard deviation σ = 15. What is the probability that a randomly selected student scores 120 or below?

1. Standardize: z = (120 − 100) / 15 = 20 / 15 = 1.3333
2. Look up CDF: Φ(1.3333) ≈ 0.9088
3. Interpret: There is approximately a 90.88% probability that a student scores 120 or below.

To find the probability of scoring between 90 and 120, calculate Φ((120 − 100)/15) − Φ((90 − 100)/15) = Φ(1.3333) − Φ(−0.6667) ≈ 0.9088 − 0.2525 = 0.6563, or about 65.63%.

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