Frequency Distributions
1
Density Function
•
•
We’ve discussed frequency distributions.
Now we discuss a variation, which is called a
density function.
A density function shows the percentage of
observations of a variable being in an interval
between two values—a question asked
frequently in business, as displayed below.
2
The Percentages


The total area under the curve is the percentage of observations
that are greater than minus infinity but less than infinity. It is
therefore 1 or 100%.
The percentage of observations that are less than x2 but larger
than x1:
3
The smaller the S.D., the
narrower the curve
4
Normal Distribution
We now discuss a specific distribution which is
called the normal distribution. The reasons
for paying special attention to this distribution
are:
 It is commonly seen in practice.
 It is extremely useful in theoretical analysis.
 Knowing how normal distribution is handled
will help you understand how other
distributions are handled.
5
Normal Distribution



It is bell-shaped and symmetrical with respect
to its mean.
It is completely characterized by its mean and
standard deviation.
It arises when measurements are the
summation of a large number of independent
sources of variation.
6
A normal distribution and its
envelope
Frequency
100
50
0
-3
-2
-1
0
1
2
3
C1
0 .15
0 .10
)
z
f(
0 .05
7
0 .00
-3
-2
-1
0
Z
1
2
3
Rules for normal distribution



If the distribution is normal,
Precisely 68% of the observations will be
within plus and minus one standard deviation
from he mean.
95% observations will be within two standard
deviation of the mean.
99.7% observations will be within three
standard deviations of the mean.
8
Computing percentages



The less-than problem. We ask: what is the
percentage of observations that are less than a
specific value, say 2.0?
The greater-than problem. We ask: what is the
percentage of observations that are greater than a
specific value, say 1.5?
The in-between problem. We ask: what is the
percentage of observations that are greater than a
specific value, say 1.5, but less than another value,
say 2.0?
9
Computing percentages
-- Standard normal distribution



As we will see shortly, by introducing the
standard normal distribution, we only need
one table to calculate percentages.
A standard normal distribution has a zero
mean and a standard deviation of 1.
A Normal table provides the percentage of
observations of a standard normal distribution
that are less than a specific value z but larger
10
than -z.
The Normal Table
The normal table shows the percentage of
observations of a standard normal distribution that
are less than a specific value z but larger than -z.
Assume z=2. Graphically, we have
11
The percentage of observations that are
less than 1




Find the area between -1 and 1 from the
normal table. It is 68.27%.
68.27% divided by 2 is the dark area 34.14%.
One half of the area under the curve (the
area to the left of the center) is 50%.
The sum of 34.14% and 50% is 84.14%
which is the percentage of observations that
are less than 1.
12
The graphical representation
13
The percentage of observations that are
less than -1



This problem is similar to the above problem.
Use a graph to find the solution procedure.
The difference of 34.14% and 50% is 15.86%
which is the percentage of observations that
are less than -1.
By now you should be able to find the
percentage of observations that are less than
some arbitrary z which can be either negative
14
or positive.
Other problems


A greater-than problem can be converted into
a less-than problem. That is, the percentage
of observations that are greater than 2 is
equal to 100% minus the percentage of
observations that are less than 2.
An in-between problem can be converted into
two less-than problems.
15
The less-than problem for
general normal distribution
We now consider a general normal
distribution and Compute the percentage of
observations that are less than a certain
value, say x.
 Calculate z=(x-mean)/Std.Dev.
 Find the percentage of observations that
are less than z in a standard normal
distribution.
16
The greater-than problem
Calculating the percentage of observations
that are greater than a certain value, say x.
 Solve a less-than problem first, i.e., find the
percentage of observations that are less
than x. Assume the result is P.
 The solution for the greater-than problem is
1-P.
17
The In-between problem
Calculating the percentage of observations
that are greater than a value, say x1, but less
than another value, say x2.
 Solve two less-than problems for x1 and
x2. Assume the results are P1 and P2.
 The solution for the In-between problem is
P2-P1.
18
The reverse problem
The reverse problem is to find a value (call it
x) for a given percentage (call it P) of
observations that are less than x.
 Let Q=2(P-50%) if P>50%.
 Use Q to find the corresponding z on a
Normal table.
 Solve z=(x-mean)/Std.Dev for x.
 Solve the problem by yourself when
19
P<50%.
Example: Exam time
Mean=90 min, S.D.=25 min, normal distribution
Calculate 20th percentile:
 From the graph (see below) we know that the
area is 100%-(2x20%)=60%. Therefore z=0.84.
 0 . 84 
x  90
25
x   0 . 84  25  90  69 min
20
The graph
21
Verify Normal distribution
To see whether a distribution is normal or not:
 Store the data in a column, say C1.
 Use Minitab
22