Normal Distribution

Frequency Distributions

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Density Function •

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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 

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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. 5

Normal Distribution 

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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

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-1

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1

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C1

0 .15

0 .10

) z f( 0 .05

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0 .00 -3

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Z

1

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Rules for normal distribution

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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 

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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 

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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

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The percentage of observations that are less than 1 

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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

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The percentage of observations that are less than -1 

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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 

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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