Calculate probabilities for the Normal Distribution using the Cumulative Distribution Function (CDF). Enter your mean (μ), standard deviation (σ), and an x value, then choose your probability type — left-tail P(X < x), right-tail P(X > x), or two-tailed — to get the exact cumulative probability and Z-score for your data point.
Cumulative Probability
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Z-Score
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Complement Probability
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Probability (%)
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The Cumulative Distribution Function (CDF) gives the probability that a random variable X takes a value less than or equal to a specified value x. For the normal distribution, CDF(x) = P(X ≤ x), which represents the area under the normal curve to the left of x. It always returns a value between 0 and 1.
P(X < x) is the left-tail probability — the chance that a randomly drawn value falls below x. P(X > x) is the right-tail (or survival) probability — the chance it exceeds x. Since the total probability is 1, P(X > x) = 1 − P(X < x). For a standard normal distribution, P(X < 1.96) ≈ 0.975 and P(X > 1.96) ≈ 0.025.
A Z-score standardizes your x value relative to the distribution's mean and standard deviation: Z = (x − μ) / σ. It tells you how many standard deviations x is from the mean. Once you have the Z-score, you can look up or compute the CDF using the standard normal distribution, which has μ = 0 and σ = 1.
You need three values: the population mean (μ), the population standard deviation (σ, which must be positive), and the observed value x. You also choose the probability type — left-tail, right-tail, two-tailed, or middle — to get the specific cumulative probability you need.
The two-tailed probability 2·P(X > |x|) measures the probability of observing a value at least as extreme as x in either direction from the mean. It's commonly used in hypothesis testing to determine significance without specifying direction, such as in a two-sided t-test or z-test.
The standard normal distribution is a special case of the normal distribution with mean μ = 0 and standard deviation σ = 1. It is denoted N(0, 1). Any normal distribution can be converted to the standard normal using the Z-score formula, which is why this calculator works for any normal distribution.
This calculator uses the error function (erf) approximation to compute the normal CDF, which is accurate to many decimal places for most practical inputs. Results are displayed to 6 decimal places. For extreme Z-scores (e.g. beyond ±8), results approach 0 or 1 and numerical precision may vary slightly.
Use the CDF when you want the probability of a value falling within a range or below/above a threshold. Use the probability density function (PDF) when you want the relative likelihood at a specific point. In practice, the CDF is far more useful for hypothesis testing, confidence intervals, and quality control applications.