# Important z-scores

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

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.

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%.
return to index

Questions?