Interactive normal distribution: area, probability and z-score

The relationship to understand

z=(x−μ)/σ; Φ(z)=P(Z≤z), with Z∼N(0,1). Here σ is the standard deviation.

Try it and interpret

At z=1.96 the area to the left is approximately 0.975. The central interval from −1.96 to +1.96 contains about 95% of a standard normal distribution. Try the inverse calculation from a probability.

Assumptions and limitations

The calculation assumes the selected normal distribution; it does not establish that measurements are normally distributed. A population interval is not a confidence interval for a mean.

Go further with the course

This page focuses on one concept. The full chapter (French) contains the reasoning, further examples, slides and FAQ. Demonstration values are computed or constructed within the stated model; they are not new research findings.