Add Up Data First, Then Take the Average? Or Take the Averages First, Then Add?

Date: 01/20/2010 at 12:36:21
From: brenda 
Subject: Confused when trying to find the sum of a ratio

Hi,

I have a database with data on 100 companies.  I have a column for 
sales and a column for employees.  I want to work out total sales per 
employee.

However, I am confused about whether I should sum the sales for all 
100 companies first, like so ...

   sales for firm 1
 + sales for firm 2
 + sales for firm 3
 + ... etc.
   
... and then divide by the sum of the employees for all 100 
companies ...

   employees for firm 1
 + employees for firm 2
 + employees for firm 3
 + ... etc.
   
... or whether I should work out sales per employee for each 
individual company first, like so ...

   sales for firm 1 / employees for firm 1
   sales for firm 2 / employees for firm 2
   sales for firm 3 / employees for firm 3
   ... etc.
   
... and then total all those averages.

I thought I would get the same answer doing it either way, but this 
is not the case, and I'm not sure which method I should be using.

Also, if I do the latter method and then take the average of that, 
what does that figure mean?

Date: 01/20/2010 at 13:55:06
From: Doctor Peterson
Subject: Re: Confused when trying to find the sum of a ratio

Hi, Brenda.

These are just two different calculations; you have to decide which 
one is what you want.  Since you phrase your intention as "TOTAL 
sales per employee," without mention of companies, I presume the 
number you want is the total sales, divided by the total number of 
employees.  That calculation pays attention to each employee equally.

If, instead, you want to focus attention on each company equally, you 
would find the sales per employee for each company, and then average 
those.

To see what is going on, take an extreme example.  Say we have two 
companies, one with 10 employees and the other with 100; and each has 
sales of $1000. Here are the two calculations, in table form:

  company    sales    employees       sales/emp
  -------    -----    ---------       ---------
     A       $1000        10             $100      }
     B       $1000       100              $10      } average: $55
   -----     -----       ---             ----
   Total     $2000       110     average: $18.18

The average sales per employee overall is

  1000 + 1000   2000
  ----------- = ---- = 18.18
    10 + 100     110

The average per company is

  100 + 10   110
  -------- = --- = 55
      2       2

That $55 per company average is bigger because both companies are 
treated equally, giving the small company (with its larger sales per 
employee) as much influence, or "weight," as the other.  The average of 
sales per employee, $18.18, gives every employee the same weight, 
making the larger company (with its smaller sales per employee) more 
influential.

These are two different weighted averages, with different 
weightings.  Both calculations might make sense for different 
purposes.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/21/2010 at 05:19:00
From: brenda 
Subject: Thank you (Confused when trying to find the sum of a ratio)

Thank you so much for your answer.  It has really helped clarify 
things for me.

I didn't expect such a prompt reply, either.  It's very much 
appreciated.

Regards
Brenda