Percentage of a normalized value

[OctetHat]

Let's say we have N cookies. We are giving the cookies to kids in different schools in different states. Each school has a fixed number of kids. We want a final value of cookies per kid for each state.

Once we have that value, Cs, "cookies per kid" for each state, what is the interpretation of a percentage of total? For example, if we added up all 50 states value (C1 + C2 + C3 + ...) then computed the fractional 'contribution' of each state (C1/(C1 + C2 + C3 + ...) what exactly would that mean?

I argue that it's meaningless, but I'm not sure I can articulate why. Perhaps I am wrong. The premise here is that I have created a normalized count of something and the "percentage contribution of each state" has been requested by management.

The normalization, however, is necessary, since the number of kids from state to state varies and we don't want a high volume of schools in a particular state to skew the numbers.

 

[Stephen Tashi]

if we added up all 50 states value (C1 + C2 + C3 + ...) then computed the fractional 'contribution' of each state (C1/(C1 + C2 + C3 + ...) what exactly would that mean?

The meaning of statistics computed from data depends upon what decisions are made using those statistics - or how they are used in computations of other statistics that are used to make decisions.

I argue that it's meaningless, but I'm not sure I can articulate why.

You can argue that the statistic is not useful for making certain decisions, but to say it is "meaningless" without having a purpose in mind is impossible to prove.

the "percentage contribution of each state" has been requested by management.

Not being management, you can only observe the decision making process and, if allowed, comment on it. For example, perhaps you will attend a talk where someone gives a slide presentation showing pie charts of the data. You can comment on claims the presenter makes or inferences members of the audience make. If you give examples of such inferences, forum members can comment on whether they are valid.

 

[OctetHat]

The reason is because the client wants to see percentages... there's no actual logical story they have in mind, which was explicitly stated by management. Just making the client happy...

Can you give me an interpretation of the percentage of cookies per kid when per kid is not a constant across states? I really don't need career advice, just looking for a meaningful interpretation.

 

[plasmon_shmasmon]

I agree with you, that it's kind of meaningless. For example, let's consider what the statistic S1 = C1/(C1+C2+C3) means for 3 states and N=100 cookies (and correct me if I'm not understanding your notation). If the contribution is 5 cookies per kid for state1 (C1=5), 8 per kid for state2 (C2=8), and 7 per kid for state3 (C3=7), then S1 = 0.25. So 25% of what? If state1 has only 1 kid, then the kid is utilizing 5/100 = 5% of the total cookies. So I don't know what the statistic S1 tells you.

What exactly do you want to know about the system?

 

[plasmon_shmasmon]

It might be more meaningful to compare Cs with the average. So in the above example the average number of cookies per kid per state is 6.67, so clearly C1 is below average. They are (6.67-5)/6.67 = 0.25 ---> 25% below average (by coincidence I think). So 75% of 6.667 is 5.

 

[plasmon_shmasmon]

To make a final point (probably not final), we can write the percent difference between Cs and the average as: |C - nCs|/C, where C = C1 + C2 + C3, and n is the number of states (not cookies). So this is kind of like your original ratio.

 

[OctetHat]

I agree with you, that it's kind of meaningless. For example, let's consider what the statistic S1 = C1/(C1+C2+C3) means for 3 states and N=100 cookies (and correct me if I'm not understanding your notation). If the contribution is 5 cookies per kid for state1 (C1=5), 8 per kid for state2 (C2=8), and 7 per kid for state3 (C3=7), then S1 = 0.25. So 25% of what? If state1 has only 1 kid, then the kid is utilizing 5/100 = 5% of the total cookies. So I don't know what the statistic S1 tells you.

Yeah, that's one of the scenarios I ran through too.

What exactly do you want to know about the system?

I already have measures for the things I want to know. This is just management trying to please clients that don't know what they want and I'm informing them not to set this precedent. I just wanted to be sure that there was no standard meaningful interpretation here. I'm the subject matter expert in this scenario, so the only people that are going to prove me wrong are other scientists/mathies in the STEM community (such as yourselves). So that's why I turned to you as my peer group. To make sure my assertion that it's meaningless was mutual.

Honestly I think this is a case of: