Normal Curve Percentiles
(A worked Problem:  Procedure explained in more detail in Chapter 7)

Suppose that a personality test has:

Mean = 200
Standard Deviation = 50

What personality score is the 80th percentile?

What personality score is the 25th percentile?

To begin this problem you need to recognize that Percentile = Percent Below.

If I had been given a personality score and then had been asked to find the percent below
    I would use the following procedure:

   I would begin with the score, then convert the score to a Z score, then look up Z in table Z,
    then I would use the value in table Z to find the area below, then multiply the area below by 100
    to get the percent below.

But for this problem I am given the percentile which is the percent below so I will just have to
    work backward toward the score using Table P rather than Table Z.

    I will divide the percentile by 100 to get the area below, then use table P to find Z,
        then use the formula to convert Z to a score.

The two problems are solved below:

I.  Find the score that represents the 80th percentile.

    A.  Convert percentile to area bleow

        Area Below = 80/100 = .80

    B.  We will now use Table P to find the Z score.

        a.  Remember that every Z score will cut the normal curve into a smaller area and a larger area.
                In this case the normal curve will look like the figure below

        b.  Because the area below (.80) is larger than .50 the Z score must be positive.

        c.  We will then find .80 in the Larger Area Column in Table P

        d.  We then read the positive Z score of .8416

    C.  Next we use the following formula to find the personality score.

            X = 200 + (.8416 · 50) = 200 + 42.08 = 242.08

I.  Find the score that represents the 25th percentile.

    A.  Convert percentile to area bleow

        Area Below = 25/100 = .25

    B.  We will now use Table P to find the Z score.

        a.  Remember that every Z score will cut the normal curve into a smaller area and a larger area.
                In this case the normal curve will look like the figure below

        b.  Because the area below (.25) is smaller than .50 the Z score must be negative.

        c.  We will then find .25 in the Smaller Area Column in Table P

        d.  We then read the negative Z score of -.6745

    C.  Next we use the following formula to find the personality score.

            X = 200 + (-.6745 · 50) = 200 + (-33.725) = 166.275

Hint:  Remember to use Table P not Table Z

Hint:  Remember that if the percentile is less than .50 then the Z score will be negative.

Hint:  Remember that if the Z score is negative your resulting score will be less than the mean.

Copyright © 2004 by Mark W. Vernoy