What are the steps to find the lower and upper quartiles of a data set?

A) Find the size of the data set. Denote this by #n#.

B) Arrange the data in ascending order (low to high). This will be #{x_1, x_2, x_3, ..., x_n}.#

C) Find the position numbers that correspond to the 25th and 75th percentiles (a.k.a. the lower and upper quartiles).

The #p^"th"# percentile of a data set is an element (derived from the data set) that is greater than #p%# of the data set.

To find the #p^"th"# percentile, we treat the data set as having #n+1# elements. Then, we find the position that is #p%# of the way through this set.

Example: The lower quartile is 25% of the way through our set of #n+1# elements. Meaning, it is at position #25%(n+1)#.

D) If the position of a percentile is an integer, then the percentile is the element in that position. If the position is not an integer, then we derive the best guess for what element would be at that fractional position.

Example: A data set has #n=9# elements. The median (a.k.a. the 50th percentile) is at position

#50%(9+1)#
# = 50% (10)#
# = 5#

Thus, the median of our data set is its 5th element. The lower quartile, however, is at position

#25%(9+1)#
#=25%(10)#
#=2.5#

Thus, the lower quartile of our data set is the element that would be at position 2.5, if such a position existed. Since it doesn't, we calculate the best guess for what would be at that position.

2.5 is halfway between 2 and 3. Our lower quartile is, correspondingly, halfway between elements 2 and 3.

It's easy to see this lower quartile will be the average of the 2nd and 3rd elements. In general, if the position of the #p^"th"# percentile is #"w.dd"#, then the #p^"th"# percentile is #dd%# along the way between the elements at positions #w# and #w+1#.

Mathematically: #p^"th" " percentile" = x_w+dd(x_(w+1) - x_w)# *

Full example:

Find #Q_1# and #Q_3# for the data set #{4,5,3,4,10,1,7,9}.#

There are #n = 8# elements.

The ordered data set is #{1, 3, 4, 4, 5, 7, 9, 10}.#

The position of #Q_1# is #25%(8+1) = 25%(9) = 2.25.#
The position of #Q_3# is #75%(8+1) = 25%(9) = 6.75.#

#Q_1# is 25% along the way between elements 2 and 3.
#Q_3# is 75% along the way between elements 6 and 7.

#Q_1= x_2 + 25%(x_3-x_2)#
#color(white)(Q_1)= 3 + 25%(4-3)#
#color(white)(Q_1)= 3 + 25%(1)#
#color(white)(Q_1)= 3 + 0.25#
#color(white)(Q_1)= 3.25#

#Q_3= x_6 + 75%(x_7-x_6)#
#color(white)(Q_1)= 7 + 75%(9-7)#
#color(white)(Q_1)= 7 + 75%(2)#
#color(white)(Q_1)= 7 + 1.5#
#color(white)(Q_1)= 8.5#