1. 1 Frequency Distributions

2. 2 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.

3. 3 The Percentages l 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%. l The percentage of observations that are less than x2 but larger than x1:

4. 4 The smaller the S.D., the narrower the curve

5. 5 Normal Distribution We now discuss a specific distribution which is called the normal distribution. The reasons for paying special attention to this distribution are: l It is commonly seen in practice. l It is extremely useful in theoretical analysis. l Knowing how normal distribution is handled will help you understand how other distributions are handled.

6. 6 Normal Distribution l It is bell-shaped and symmetrical with respect to its mean. l It is completely characterized by its mean and standard deviation. l It arises when measurements are the summation of a large number of independent sources of variation.

7. 7 A normal distribution and its envelope

8. 8 Rules for normal distribution If the distribution is normal, l Precisely 68% of the observations will be within plus and minus one standard deviation from he mean. l 95% observations will be within two standard deviation of the mean. l 99.7% observations will be within three standard deviations of the mean.

9. 9 Computing percentages l The less-than problem. We ask: what is the percentage of observations that are less than a specific value, say 2.0? l The greater-than problem. We ask: what is the percentage of observations that are greater than a specific value, say 1.5? l 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?

10. 10 Computing percentages -- Standard normal distribution l As we will see shortly, by introducing the standard normal distribution, we only need one table to calculate percentages. l A standard normal distribution has a zero mean and a standard deviation of 1. l 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.

11. 11 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

12. 12 The percentage of observations that are less than 1 l Find the area between -1 and 1 from the normal table. It is 68.27%. l 68.27% divided by 2 is the dark area 34.14%. l One half of the area under the curve (the area to the left of the center) is 50%. l The sum of 34.14% and 50% is 84.14% which is the percentage of observations that are less than 1.

13. 13 The graphical representation

14. 14 The percentage of observations that are less than -1 l This problem is similar to the above problem. Use a graph to find the solution procedure. l The difference of 34.14% and 50% is 15.86% which is the percentage of observations that are less than -1. l 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.

15. 15 Other problems l 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. l An in-between problem can be converted into two less-than problems.

16. 16 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. l Calculate z=(x-mean)/Std.Dev. l Find the percentage of observations that are less than z in a standard normal distribution.

17. 17 The greater-than problem Calculating the percentage of observations that are greater than a certain value, say x. l Solve a less-than problem first, i.e., find the percentage of observations that are less than x. Assume the result is P. l The solution for the greater-than problem is 1-P.

18. 18 The In-between problem Calculating the percentage of observations that are greater than a value, say x1, but less than another value, say x2. l Solve two less-than problems for x1 and x2. Assume the results are P1 and P2. l The solution for the In-between problem is P2-P1.

19. 19 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. l Let Q=2(P-50%) if P>50%. l Use Q to find the corresponding z on a Normal table. l Solve z=(x-mean)/Std.Dev for x. l Solve the problem by yourself when P<50%.

20. 20 Example: Exam time Mean=90 min, S.D.=25 min, normal distribution Calculate 20 th percentile: l From the graph (see below) we know that the area is 100%-(2x20%)=60%. Therefore z=- 0.84.

21. 21 The graph

22. 22 Verify Normal distribution To see whether a distribution is normal or not: l Store the data in a column, say C1. l Use Minitab