Statistics Knowledge Base - Cumulative Area Under the Normal Curve

Tuesday, May 13, 2008

Cumulative Area Under the Normal Curve

The cumulative area under a standard normal curve is the total area under a Gaussian distribution from minus infinity to a particular point along the curve (the z-score).

PROPERTIES

Some of the important properties of the Gaussian (normal) distribution include:

The normal distribution is symmetric, and as such the total area under the curve from x to infinity is exactly equal to the area under the curve from -x to minus infinity, and vice versa.
Because the standard normal distribution is a probability distribution, the total area under curve from minus infinity to infinity is exactly one.
Approximately 68.3% of normally-distributed observations lie within one standard deviation of the mean. Approximately 95.4% of normally-distributed observations lie within two standard deviations of the mean. Approximately 99.7% of normally-distributed observations lie within three standard deviations of the mean.
The use of the normal distribution in statistical contexts is intimately intertwined with the central limit theorem.

ONLINE CALCULATOR

To calculate the cumulative area under a normal curve, please click here.

FORMULAE

The formula involved in the computation of the cumulative area under a standard normal curve is detailed below.

Gaussian (normal) distribution probability density function:
Where μ is the mean and σ is the standard deviation.

REFERENCES

Patel, J.K. and Read, C.B. (1982) "Handbook of the Normal Distribution", Dekker, New York, NY.

Feller, W. (1968) "An Introduction to Probability Theory and Its Applications (3rd edition)", Vol. 1, Wiley, New York, NY.