Page 1 of 3
What is a normal distribution?
Normal distributions are a family of distributions that
have the same general shape. They are symmetric with
scores more concentrated in the middle than in the tails.
Normal distributions are sometimes described as bell
shaped. Examples of normal distributions are shown to
the right. Notice that they differ in how spread out they
are. The area under each curve is the same. The height of
a normal distribution can be specified mathematically in
terms of two parameters: the mean (m) and the standard
deviation (s).
Area under a portion of the normal curve
If a test is normally distributed with a mean of 60 and a standard deviation of 10, what
proportion of the scores are above 85? This problem is very similar to figuring out the percentile
rank of a person scoring 85. The first step is to figure out the proportion of scores less than or
equal to 85. This is done by figuring out how many standard deviations above the mean 85 is.
Since 85 is 85-60 = 25 points above the mean and since the standard deviation is 10, a score of
85 is 25/10 = 2.5 standard deviations above the mean. Or, in terms of the formula,
A z table can be used to calculate that .9938 of the scores are less
than or equal to a score 2.5 standard deviations above the mean. It
follows that only 1-.9938 = .0062 of the scores are above a score
2.5 standard deviations above the mean. Therefore, only .0062 of
the scores are above 85.
Suppose you wanted to know the proportion of students receiving scores between 70 and 80. The
approach is to figure out the proportion of students scoring below 80 and the proportion below
70. The difference between the two proportions is the proportion scoring between 70 and 80.
First, the calculation of the proportion below 80. Since 80 is 20 points above the mean and the
standard deviation is 10, 80 is 2 standard deviations above the mean.
