Normalcdf Calculator
Normalcdf Calculator works out Probability and Z score of x from Mean (mu), Standard deviation (sigma) and Value x (or the lower bound a). The default figures on this page give probability of 0.950004; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
4 steps
- X ~ Normal(mu = 0, sigma = 1), so the variance is 1.
- z(lower) = (-1.96 - 0) / 1 = -1.96; z(upper) = (1.96 - 0) / 1 = 1.96
- P(a < X < b) = Phi(1.96) - Phi(-1.96) = 0.9750021049 - 0.0249978951 = 0.9500042097
- The two tails together hold the remaining 0.0499957903.
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 normalcdf 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 normalcdf Calculator
Enter Mean (mu), Standard deviation (sigma), Value x (or the lower bound a) and Upper bound b. 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), Value x (or the lower bound a), Upper bound b and Confidence level.
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
Where is normalcdf on TI-84?
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. 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 0.950004. Put your own figures in their place and the working updates beside the result.
What does normalcdf mean?
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. On this page it is computed from Mean (mu), Standard deviation (sigma) and Value x (or the lower bound a) and shown as Probability, Z score of x and Z score of the upper bound.
What information do I need for the normalcdf Calculator?
You need Mean (mu), Standard deviation (sigma), Value x (or the lower bound a), Upper bound b and Confidence level. The page opens with default figures that give probability of 0.950004, so you can see the scale before entering your own.
What does this normalcdf 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.
Can I check how the normalcdf Calculator result was calculated?
Yes. The “How this is worked out” section shows the calculation steps, and the sources section identifies the references used for the method.