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Frequency Distributions

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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.

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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:

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The smaller the S.D., the narrower the curve

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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.

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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.

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A normal distribution and its envelope

3210-1-2-3

100

50

0

C1

Freq

uenc

y

3210-1-2-3

0.15

0.10

0.05

0.00

Z

f(z)

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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.

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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?

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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 than -z.

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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

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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.

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The graphical representation

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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 or positive.

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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.

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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.

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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.

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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.

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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

P<50%.

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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.

min69902584.025

9084.0

x

x

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The graph

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Verify Normal distribution

To see whether a distribution is normal or not:Store the data in a column, say C1.Use Minitab