NORMAL DISTRIBUTIONS Properties
• Symmetrical
•Bell shape
• Unimodal (mountain with a single peak)
0 1 A 0They are described by giving the mean and standard deviation N (µ, )
This week we have learnt how to perform computations with the normal distribution. We are
going to review now how to compute proportions of cases within given intervals of the normal
distribution, and how to do some other computations with the standard normal distribution.
As we saw last week, sometimes when we have a lot of observations the distribution becomes
regular and we can describe it with a smooth curve (the density curve). A very important class
of density curves are symmetrical, with one peak and with the form of a bell. We call
these normal curves and the distribution behind it normal distribution. This type of distribution
is quite frequent in real data, such as the height of a group of people of the same sex.
All normal distributions share a set of properties. One of these properties is known as the
68-95-99.7% rule, and refers to the following:
In any normal distribution with mean and standard deviation ,
• 68% of the cases fall in the interval .
• 95% of the cases fall in the interval .
• 99.7% of the cases fall in the interval .
How can we find a proportion given a value?
With this rule we can compute the proportion of cases for some intervals derived from the
different parts of the rules.
For example, suppose that we know that the distribution of blood colesterol level in 14 year old
kids has a mean equal to 170 mg/dl and a standard deviation of 30 mg/dl. Knowing that the
distribution is approximately normal, what is the percentage of 14 year old kids who have more
than 230 mg/dl of colesterol?
Draw the distribution and find the different points given by the 68-96-99.7% rule. Notice that
230 is exactly the mean plus two times the standard deviation. Therefore we can use the rule to
answer to the following question.
What happens if we want to know the percentage of 14 year old kids with a cholesterol level
larger than 240? This number does not coincide with any of the ones indicated by the
68-95-99.7% rule. To answer this question we need the standard normal distribution, and follow
the steps that we studied in class:
1) State the problem in terms of the observed variable x.
x: blood cholesterol level
x has a normal distribution N(170,30) and we want to know which proportion has x>240 , in
other words we are looking for the area under the curve to the right of 240
Week 3 and 4: Normal distribution
