CDF PDF Calculator
PDF and CDF Calculator works out Probability and Z score of x from Mean (mu), Standard deviation (sigma) and What to calculate. The default figures on this page give probability of 0.975002; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
5 steps
- X ~ Normal(mu = 0, sigma = 1), so the variance is 1.
- z = (1.96 - 0) / 1 = 1.96
- P(X < 1.96) = Phi(1.96) = 0.9750021049
- P(X > 1.96) = 1 - Phi(1.96) = 0.0249978951
- That places 1.96 at the 97.5002th percentile.
Confidence interval around the meantable
One-sigma intervaltable
Two-sigma intervaltable
Three-sigma intervaltable
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
3 sources
What the PDF and CDF Calculator calculates
Evaluates the normal (Gaussian) distribution: the density f(x), the cumulative probabilities to the left, to the right and between two values, the inverse CDF, z scores, the 68-95-99.7 empirical-rule bands and two-sided critical values for any mean and standard deviation.
How to use this PDF and CDF Calculator
Enter Mean (mu), Standard deviation (sigma), What to calculate and Value x (or the lower bound a). Probability updates as you type. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Probability, Z score of x and Z score of the upper bound.
The method follows NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution, Abramowitz, M. & Stegun, I.A. (1964), Handbook of Mathematical Functions, chapter 26 and Pukelsheim, F. (1994), "The Three Sigma Rule", The American Statistician 48(2), 88-91.
What you can compare
Change one input at a time to see how it affects Probability, Z score of x and Z score of the upper bound. The inputs you can vary are Mean (mu), Standard deviation (sigma), What to calculate, Value x (or the lower bound a), Upper bound b and Cumulative probability (area to the left).
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
How do I convert a CDF to a PDF?
Evaluates the normal (Gaussian) distribution: the density f(x), the cumulative probabilities to the left, to the right and between two values, the inverse CDF, z scores. Open “How this is worked out” beside the result to see each step for your own figures.
How to calculate normal CDF on calculator?
It needs Mean (mu), Standard deviation (sigma), What to calculate, Value x (or the lower bound a), Upper bound b and Cumulative probability (area to the left). Nothing else is required, and the figures stay in your browser.
What is the difference between CDF and PDF?
On the figures this page opens with — mean (mu) 0, standard deviation (sigma) 1 and value x (or the lower bound a) 1.96 — the probability is 0.975002. No single number fits everyone: change any figure above and the answer updates as you type, with each step of the arithmetic beside it.
How to calculate CDF formula?
The result gives Probability, Z score of x, Z score of the upper bound, Density f(x), P(X < x) and P(X > x), each labelled with its unit so the figure can be read on its own.
What information do I need for the PDF and CDF Calculator?
You need Mean (mu), Standard deviation (sigma), What to calculate, Value x (or the lower bound a), Upper bound b, Cumulative probability (area to the left) and Confidence level. The page opens with default figures that give probability of 0.975002, so you can see the scale before entering your own.
What does this PDF and CDF Calculator show?
It shows Probability, Z score of x, Z score of the upper bound, Density f(x), P(X < x), P(X > x), P(a < X < b) and P(X < a or X > b), with the working and any warnings for your entries.