Normal distribution - Values
In this lecture we discuss how to compute the values of the normal distribution function, using normal distribution tables or computer programs (in particular Matlab or Excel). For an introduction to the normal distribution, see the lecture entitled Normal distribution.
Values of a standard normal distribution
Let
be a standard normal random variable (i.e. a normal random variable with zero
mean and unit variance) and denote its
distribution function by
![[eq1]](http://images1.statlect.com/normal_distribution_values__2.png)
As we have discussed in the lecture entitled Normal
distribution, there is no simple analytical expression for
and its values are usually looked up in a table or computed with a computer
algorithm.
This lecture tackles the practical problem of computing (numerically) the
values of
when
is a specific number (e.g.
).
We can confine our attention to standard normal random variables: as shown in
the next section, if we know how to compute the values of a standard normal
distribution, we also know how to compute the values of any other normal
distribution.
Frequently used values
Some values of the normal distribution function are used very frequently and
people usually learn them by
heart:![[eq5]](http://images2.statlect.com/normal_distribution_values__7.png)
Note also
that:![[eq6]](http://images1.statlect.com/normal_distribution_values__8.png)
which
is due to the symmetry around
of the normal density and is often used in calculations.
Values of a non-standard normal distribution
Let
be a normal random variable with mean
and variance
and denote its distribution function by
![[eq7]](http://images2.statlect.com/normal_distribution_values__13.png)
.
Remember that any normal random variable
with mean
and variance
can be written
as:where
is a standard normal random variable.
Using this fact, we obtain the following relation between the distribution
function of a standard normal random variable
and the distribution function of any other normal random variable
:![[eq9]](http://images1.statlect.com/normal_distribution_values__21.png)
Therefore, if we know how to compute the values of a standard normal
distribution, we also know how to compute the values of a normal distribution
with mean
and variance
.
Example_
If we need to compute the value
of a normal random variable
with mean
and variance
,
we can compute it using the distribution function of a standard normal random
variable
:
![[eq11]](http://images2.statlect.com/normal_distribution_values__29.png)
Normal distribution tables
In the past, when computers were not widely available, people used to look up
the values of
in normal distribution tables.
A normal distribution table is a table where
is tabulated for several values of
.
A normal distribution table looks something like
this:
For example, to compute
we need to search for the row corresponding to the value
.
We
find:![[eq15]](http://images1.statlect.com/normal_distribution_values__35.png)
If we are searching for a value of
that is not tabulated, we can compute an approximation of
by interpolating the two values that are closest to
.
For
example:![[eq17]](http://images2.statlect.com/normal_distribution_values__39.png)
Normal distribution values in Excel
To compute the values of the normal distribution function
,
we can use the built-in Excel function NORM.S.DIST().
For example, if we need to compute
and the value
is stored in cell A1, we can type in another cell:
=NORM.S.DIST(A1)
Normal distribution values in Matlab
To compute the values of the normal distribution function
,
we can use the Matlab function normcdf(). For example,
if we need to compute
,
we can input the following command:
normcdf(0.5)
Solved exercises
At the end of the lecture entitled Normal distribution, you can find some solved exercises that also require the computation of normal distribution values.