Normal Distribution Calculator
Our free statistics calculator solves normal distribution problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
z = (x - μ) / σ | P(X ≤ x) = Φ(z), the standard normal CDF
The z-score converts a value x from a normal distribution with mean μ and standard deviation σ into the equivalent position on the standard normal distribution (mean 0, standard deviation 1). The cumulative distribution function Φ(z) then gives the probability of observing a value at or below x.
Worked Examples
Example 1: IQ score percentile
Problem:IQ scores are normally distributed with mean 100 and standard deviation 15. What is the probability a randomly selected person scores below 115?
Solution:z = (115 − 100) / 15 = 1.00. Looking up (or computing) the standard normal CDF at z = 1 gives P(X ≤ 115) ≈ 0.8413.
Result:≈ 84.13% of people score below 115 (an IQ of 115 is the 84th percentile)
Example 2: Manufacturing tolerance check
Problem:A machined part has target length 50 mm with standard deviation 0.2 mm. What fraction of parts exceed 50.4 mm?
Solution:z = (50.4 − 50) / 0.2 = 2.00. P(X ≤ 50.4) ≈ 0.9772, so P(X > 50.4) = 1 − 0.9772 = 0.0228.
Result:≈ 2.28% of parts exceed 50.4 mm
Frequently Asked Questions
What is the normal distribution and why is it so important in statistics?
The normal distribution (also called the Gaussian distribution or 'bell curve') is a symmetric, continuous probability distribution defined entirely by its mean (μ) and standard deviation (σ). It is central to statistics because the Central Limit Theorem shows that the average of many independent random samples tends toward a normal distribution regardless of the shape of the underlying data — making it the natural model for measurement error, heights, test scores, and countless natural and social phenomena.
How is the z-score related to the standard normal distribution?
The z-score, z = (x − μ)/σ, converts any normal distribution into the standard normal distribution, which has mean 0 and standard deviation 1. This standardization lets a single set of probability tables (or a single CDF formula) work for every possible normal distribution, since every value can be re-expressed as 'how many standard deviations from its own mean.'
How is the normal distribution used in hypothesis testing and quality control?
In hypothesis testing, a computed z-score beyond ±1.96 corresponds to a two-tailed p-value below 0.05, the conventional significance threshold. In manufacturing, Six Sigma quality control assumes normally distributed process variation and targets defect rates below 3.4 per million by keeping the process mean at least six standard deviations from the nearest specification limit.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy