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### 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
```
Associated Topics:
Middle School Ratio and Proportion
Middle School Statistics

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