Normal Distribution Calculator
Find the cumulative probability P(X ≤ x) for a normal distribution given any mean, standard deviation, and x value. Useful for statistics students, researchers, and engineers working with continuous random variables.
About this calculator
The normal distribution is a symmetric bell-shaped curve defined by its mean μ and standard deviation σ. To find the probability that a value falls below x, first compute the z-score: z = (x − μ) / σ. The cumulative probability P(X ≤ x) is then the area under the normal curve to the left of x, found via the cumulative distribution function (CDF). This calculator uses an accurate approximation of the CDF based on the error function: P ≈ 0.5 × (1 + erf(z / √2)). About 68% of values fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ (the empirical rule). Normal distributions describe heights, test scores, measurement errors, and countless natural phenomena.
How to use
Suppose exam scores are normally distributed with μ = 70 and σ = 10, and you want to know the probability a student scores below 85. Step 1 — Compute z: z = (85 − 70) / 10 = 1.5. Step 2 — Look up or compute CDF(1.5) ≈ 0.9332. So about 93.3% of students score below 85. Enter mean = 70, standard deviation = 10, and x = 85 into the calculator to confirm P(X ≤ 85) ≈ 0.9332.
Frequently asked questions
How do I find the probability that X falls between two values in a normal distribution?
To find P(a ≤ X ≤ b), compute the CDF at both endpoints and subtract: P(a ≤ X ≤ b) = CDF(b) − CDF(a). For example, with μ = 70 and σ = 10, P(60 ≤ X ≤ 80) = CDF(80) − CDF(60) ≈ 0.8413 − 0.1587 = 0.6827, which is exactly the 68% rule. Run the calculator twice, once for each bound, and subtract the results. This technique works for any interval.
What is a z-score and why is it important for normal distribution calculations?
A z-score measures how many standard deviations a value x lies from the mean: z = (x − μ) / σ. It standardizes any normal distribution onto a common scale with μ = 0 and σ = 1, called the standard normal distribution. This lets you use a single set of probability tables or formulas regardless of the original mean and spread. Z-scores above 2 or below −2 are often considered statistically unusual, and z-scores are foundational in hypothesis testing and confidence intervals.
When is the normal distribution not appropriate to use for probability calculations?
The normal distribution assumes data is continuous, symmetric, and bell-shaped — assumptions that fail in many real situations. It is inappropriate for data that is heavily skewed, such as income distributions or wait times, which are better modeled by log-normal or exponential distributions. It also fails for discrete count data (use binomial or Poisson) or data with hard boundaries, like percentages near 0 or 100. Always check a histogram or normality test (e.g., Shapiro-Wilk) before assuming normality.