# Questionable statistics formula?

Ok so I started this debate with my teacher. It is about this formula for finding the Quartiles of grouped data. Let's take a look at this data:

With ungrouped data, 1 2 3 4 5 6 7 8 9 11 12, for example, we solve for Q1 as (at least that's what she taught us):
Q1 = 1(12) / 4
Q1 = 3
Q1 = [ 3rd + 4th ] / 2
Q1 = [3 + 4] / 2
Q1 = 3.5

which makes sense, with this formula, Q4 is not possible:
Q4 = 4(12) /4
Q4 = 12
Q4 = [12th + 13th] / 2
Q4 = doesn't exist

If kn / 4 results in a whole number, get the value between the (kn / 4)th and (kn / 4 + 1)th term. Else, if it is a decimal, round it up to the nearest whole number.

With grouped data:

Class Interval f <F
17-24 3 3
25-32 9 12
33-40 10 22
41-48 18 40
49-56 9 49
57-64 6 55
65-72 5 60
-----
n=60

She uses the formula,

Qk = L + [ (( kn / 4 ) - <F) / f ] * s,

where Qk is the kth quartile, 'L' is the lower boudary of the class of Qk, 'n' is the total frequency, '<F' is the cumulative frequency below L, 's' for the size of each class, 'f' for
the frequency of the class of Qk.

My grounds:
How the formula is evaluated is wrong because:
• Data, grouped or not, can only have 3 Quartiles. Quartiles by definition are those 3 values that together divide the data into 4 equal parts. So there can't be a 4th one.

Solving for Q4:

Q4 = 64.5 + [ ( ( 4 * 60 / 4 ) - 55 ) / 5 ] * 8
Q4 = 64.5 + [ ( 60 - 55 ) / 5 ] * 8
Q4 = 64.5 + [ 5 / 5 ] * 8
Q4 = 64.5 + 8
Q4 = 72.5

72.5 is the upper limit of the 65 - 72 class, therefore Q4 exists which contradicts the definition of quartiles. So this approach is wrong.

My proposal:

Leave the formula as is but evaluate it differently.
kn / 4 is the part that tells us in which class Qk lies in.
kn / 4 should be consistent with how we get Qk with ungrouped data since data, grouped or not, doesn't change the definition of quartiles.
so if kn / 4 is a whole number, get [(kn/4)th + (kn/4 + 1)th] / 2
else round it to the nearest whole number

So solving for Q4:
this is Q4's position:

Q4 = 4 ( 60 ) / 4
Q4 = [60th + 61st] / 2 -> 61st data doesn't exist, therefore Q4 doesn't exist.
pretending 61st data exists....
Q4 = 60.5

So:
Q4 = 64.5 + [ (60.5 - 55) / 5 ] * 8
Q4 = 64.5 + [ 5.5 / 5 ] * 8
Q4 = 64.5 + (1.1) * 8
Q4 = 64.5 + 8.8
Q4 = 73.3

which agrees with the definition of quartiles. There is no Q4 with this evaluation, which is True.

She insists that references ( books ) are more reliable than this "proof" and that we should follow their formulas.
I don't know what to do when exams come asking for quartiles. I insist in using what I believe is right, though.

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Stephen Tashi