Q9 Knowledge Base

The question is, how do we do the process in reverse: finding a Raw Score when given a Percentage. For example, if we wanted to know what score you need to have to be in the top 10% of the population (otherwise stated as: what is the score that is at the 90th percentile rank?  

The mean is = 50, Standard deviation s = 10, We are converting a percentage to a raw score (90th PR).  The process remains the same for this procedure, but with the following difference.  First, we draw a normal curve, place the mean in the center, locate two standard deviations (raw scores) on either side of the mean. Next we place the Z-score for the mean (0) in the center below the mean. Locate two standard deviations (Z- scores) on either side of the mean, and then fill in the approximate percentile ranks for each whole standard deviation.

Then, in the past we have converted Zobt to raw score. But we don’t know the z-score yet for Zobt, this step will be delayed, go to the next step.  Draw a dotted line through the normal curve at the mean reminding that 50% of the scores fall above the mean, and 50% fall below.  Next, draw a dotted line through the curve approximately where Zobt will fall.  Using the approximated percentile ranks of each Z-score, get a rough guess of the location of the 90th percentile rank, so it will fall between the 85th and 98th percentile ranks. Note the Z-score will be between 1 and 2. Draw an arrow between the mean and your approximated Zobt line.  Lightly shade the area of interest, and open “The Normal Distribution,” chart which contains the proportions in the normal curve between the mean and “Zobt.” We are trying to find a Z-score (Zobt), we must first locate the exact percentage. 

a) Follow these simple guidelines for the arithmetic:

    If the percentage requested is over 50% (as in this case), subtract .50 from the requested percentage to locate the table percentage. (90th percentile rank requested, .90 - .50 = .40)

    If the percentage requested is under 50%, subtract the requested percentage from .50 to locate the table percentage (not the case here, but say the requested percentage was 35th percentile rank). 35h percentile rank requested, .50 - .35 = .15, so you would look up the nearest percentage to .1500.  

b)  In this example, the 90th percentile rank requested. Since 90% is greater than 50%:

    (.90 - .50 = .40). Therefore, 40% (or .4000) becomes the table percentage we use to find the actual nearest percentage. The closest table percentage to .4000 is .3997. Therefore Z = 1.28, as shown below from the “percentages under the normal curve” Table.