Questionable statistics formula?

[13ugwh!z]

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 numberSo 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.

 

[Stephen Tashi]

You shouldn't call what you are writing "formulas". If you were writing mathematical formulas, you couldn't have the same variable Q1 equal to 3 and then equal to 3.5. What you are writing is algorithms, where that sort of thing is allowed to happen.

The fact that the teacher's formula produces an answer for Q4 does not imply it is an incorrect formula. Formulas in a given field of science (e.g. F = MA, in classical physics) have associated verbal descriptions of the situations to which they apply. The fact that they produce answers when numbers are used that can't describe one of those situations (e.g. M = sqrt(-1) A = sqrt(-5) ) doesn't show the formulas are incorrect.

If there is an argument for your algorithm being better, it would be that it helps people remember than Q4 is not defined. (I haven't checked your algorithm.)

 

[13ugwh!z]

Thanks for the correction! And I concede to my teacher although, we didn't really had a debate. With her algorithm Q0 and Q4 exist but they don't divide the data, instead they serve as boundaries for the whole data. With my algorithm, Q4 would extend out of the true boundary of the data, which is wrong, plus, Q0 would be shifted to the right making a division in the first 25% of the data (from the lowest boundary to Q0 and from Q0 to Q1). She didn't tell me this, I just realized it.