Normal Distribution Calculator
Our free statistics calculator solves normal distribution problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Normal Distribution Calculator
Calculator
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Formula: z = (x - μ) / σ | P(X ≤ x) = Φ(z), the standard normal CDF
Worked example — ≈ 84.13% of people score below 115 (an IQ of 115 is the 84th percentile)
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.
What do the PDF and CDF values returned by Normal Distribution Calculator mean?
The PDF (probability density function) gives the relative likelihood of the variable landing exactly at value x — useful for comparing density at different points but not itself a probability. The CDF (cumulative distribution function), P(X ≤ x), gives the actual probability that a randomly drawn value from the distribution is less than or equal to x — this is what you look up in a standard normal table or use for percentile calculations.
What is the 68-95-99.7 empirical rule?
For any normal distribution, approximately 68% of values fall within 1 standard deviation of the mean, 95% fall within 2 standard deviations, and 99.7% fall within 3 standard deviations. This rule of thumb lets you quickly estimate probabilities without a calculator once you know the mean and standard deviation.
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 do I find the probability that a value falls between two numbers, not just above or below one?
Compute the CDF at each boundary separately and subtract: P(a < X < b) = P(X ≤ b) − P(X ≤ a). For example, to find the probability a value falls between the mean and 1 standard deviation above it, compute CDF at (μ+σ) and subtract CDF at μ (which is always exactly 0.5 for a normal distribution).
What real-world quantities are approximately normally distributed?
Adult human height, standardized test scores (SAT, IQ tests are explicitly designed to be normal), measurement errors in scientific instruments, blood pressure readings across a population, and manufacturing tolerances in quality control are all commonly modeled as normal. Note that many real-world quantities — income, city population, word frequency — are NOT normal and instead follow skewed distributions like the log-normal or power law.
Why can't Normal Distribution Calculator give an exact closed-form CDF value?
The normal distribution's CDF integral has no elementary closed-form antiderivative — it must be expressed using the error function (erf), which itself requires numerical approximation. Normal Distribution Calculator uses the Abramowitz and Stegun polynomial approximation (accurate to about 7 decimal places), the same technique used in most statistical software and textbooks before high-precision numerical libraries became standard.
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 for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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