Working backwards to find positions

Suppose we know our data follows a normal distribution with mean m = 500 and standard deviation s = 100

Percentiles in the distribution can be found by working backwards.

For example, if we wanted to find the value which is larger than 75% of all the date values, we first sketch out the problem:

then we find the number of standard deviations corresponding to 75%, and finally produce the value.

To do this, consult the tables, looking in the body of the tables for a proportion close to .75. The correponding Z-value is .67 , which indicates that our sought after value is about .67 standard deviations above the mean.

By rewriting the Z-value formula

      in the form     

we can find the value     X  =  500 + .67*100 = 567