Normal Distribution

The normal distribution is specified by a mean and positive standard deviation. It places approximately 68.27 percent of probability within one standard deviation and 95.45 percent within two.

For a normal variable, subtracting the mean and dividing by the standard deviation gives a standard normal variable. This allows probability questions to be expressed in standard-deviation units. Approximately 68.27 percent of the distribution lies within one standard deviation of the mean and 95.45 percent within two; these are model probabilities, not guarantees for every sample.

The same arithmetic can produce standardized values for nonnormal data, but normal probability tables then need separate justification. Mean and standard deviation alone do not identify a distribution. Check the relevant quantity, tails, and sampling assumptions, and distinguish normality of observations from an approximation for an estimator.

When to use it

When analyzing data; when building predictive models; when evaluating statistical claims; when quantifying uncertainty.

How it can help

Identify the variable, assess the approximation, standardize consistently, and check the tail or interval relevant to the decision. Account for uncertainty in estimated parameters.