Question 1205166
mean is 63.6
standard deviation is 2.5


at quartile 3, 75% of the values are less than that.


z-score formula is z = (x-m)/s


z is the z-score
x is the raw score
m is the raw mean
s is the stqandard deviation


formula becomes z = (x - 63.6) / 2.5


z-score with 75% of the area under the normal distribution curve to the left of it is equal to .67449 rounded to 5 decimal places.


when z = that, z-score formula becomes .67449 = (x - 63.6) / 2.5
solve for x to get x = 2.5 * .67449 + 63.6 = 65.286225.
round to 2 decimal places to get 65.29.


65.29 is the score that is at the 75th percentile of the data set which is at quartile 3.


here's what it looks like on a normal distribution calculator.
first display is with z-scores.
second display is with raw scores.


<img src = "http://theo.x10hosting.com/2023/122301.jpg">


<img src = "http://theo.x10hosting.com/2023/122302.jpg">