Standard Normal Distribution Calculator
Find cumulative and upper-tail probabilities from a z score.
Standard Normal Probability
The standard normal distribution has mean 0 and standard deviation 1. This calculator returns P(Z ≤ z) and P(Z > z) for the entered z score.
Why It Is Useful
Z probabilities are widely used in hypothesis testing, confidence intervals and probability calculations.
Using the Standard Normal Distribution
The standard normal distribution has mean 0 and standard deviation 1. It lets you convert z scores into cumulative probabilities and compare observations on a common standardized scale.
Standard Normal Example
A z score of 1 means the observation is one standard deviation above the mean. The cumulative probability to the left is about 0.8413 under the standard normal curve.
How to Interpret Z Probabilities
Positive z scores lie above the mean and negative z scores lie below it. Tail probabilities are commonly used in hypothesis tests, confidence intervals and probability calculations.
Frequently Asked Questions
What is the mean of the standard normal distribution?
Zero.
What is its standard deviation?
One.
Is every normal distribution standard normal?
No. Other normal distributions can be standardized by converting values to z scores.
Methodology and Practical Limits
Standard-normal probabilities use z=(x−μ)/σ and the N(0,1) cumulative distribution. Tail direction matters: left-tail, right-tail and between-values questions require different combinations of cumulative probabilities.