# Important z-scores

Understanding what the normal distribution means is enhanced by being familiar with a few z-scores and their associated areas.

## 68-95 rule

It is readily calculated that 68% (.6826) of normally distributed data is within one standard deviation of the mean (between -1 and 1). Similarly, 95% (.9544) is within two standard deviation units of the mean, and 99.7% (.9974) is within three standard deviation units of the mean.

## Quartiles

It is readily calculated that for the standard normal distribution the first quartile is -.67 (using .2514 for .25) and the third quartile is .67. This means that for normally distributed data, one-half of the data is within 2/3 of a standard deviation unit of the mean.

## Outliers

One definition of outliers is data that are more than 1.5 times the inter-quartile range before Q1 or after Q3. Since the quartiles for the standard normal distribution are +/-.67, the IQR = 1.34, hence 1.5 times 1.34 = 2.01, and outliers are less than -2.68 or greater than 2.68. Hence for normally distributed data, the probability of being an outlier is 2 times .0037 = .0074. This is less than 1%.