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What Does the Median Order Value Mean?

Date: 05/17/2010 at 17:15:35
From: Nathan
Subject: Using median as an analytical tool

At my company recently, we have been using median of order value as an
important metric. My boss believes that if we meet a goal of increasing
that median order value by a specific amount -- say, by $100 -- then that
means that total sales equals the number of orders in a period multiplied
by 100.

I have been trying to convince him that -- while this would be true if the
mean increased by $100 -- it is not necessarily true of the median. He
insists that, given that everything else stays the same, the median
increasing by $100 should indeed mean total sales equals one hundred times
the number of orders.

It seems to me that the increase in the mean (as a proxy for a per-order
value for increase in total sales) could be higher than, equal to, or lower
than the increase in the median -- and even that the mean could decrease
when the median increases or vice versa. The logic for that, however, is
hard for me to explain to someone else.

Using simple examples with extreme outliers, I can explain how the mean
would change but the median would not. I have also mentioned that if the
"middle third" of orders increased by a specific amount, it would increase
the median by more than it would the mean. I think that line of reasoning
comes closer to an explanation, but doesn't really cover all relevant
ground.

Median is obviously a very tricky measurement to use as an indicator of
change, as the median itself can't be directly affected. I can work out
the logic in my own head, but I'm having a hard time explaining more than
the first sentence to someone else.

It is also completely possible that my own logic -- since it is solely in
my head -- is wrong.

So, could I possibly be wrong here? If so, why; and if not, is there a
way to explain it that does more than just repeat the definition of
"median"?

I think the big hurdle for me is that median can't be affected directly,
and this can't affect the mean (or total sales), even though affecting
some other element of orders may change all three measurements.



Date: 05/18/2010 at 00:56:24
From: Doctor Wilko
Subject: Re: Using median as an analytical tool

Hi Nathan,

Thanks for writing to Ask Dr. Math!

An increase in median order value does not indicate whether or not sales
went up overall. Median is the middle value of a set of numbers -- and
that's it. I don't have a mathematical proof to answer you decisively, but
looking at some concrete examples along your "middle thirds" line of
reasoning gives us some insight.

For example, if the figures below were your sales for two periods of time,
and you patted yourself on the back about median sales going up from the
first to the second, you might be in for some trouble with the boss!

  {2, 20, 100, 1000, 10000}  Median is 100
  {1, 10, 150, 160, 170}     Median is 150

Does having a higher median in one month over previous indicate
"success?" By itself, it just tells you the middle number is higher than
the middle number last month.  

Consider the following scenarios.

1. In month 2, total sales go up:

       Month 1   Month 2
   wk1   $ 50      $ 55 
   wk2   $ 35      $ 40 
   wk3   $100      $105 
   wk4   $ 60      $ 60 
   wk5   $ 80      $ 85 
       -----------------
   sum   $325      $345 

And average sales are higher:

       Month 1   Month 2
         $ 65      $ 69 
         
But median is identical to the month prior:
  
       Month 1   Month 2
         $ 60      $ 60 

Why? Median is just the number in the middle, and by itself doesn't show
that overall sales are up and average sale is up.

2. In month 2, total sales go down:

       Month 1   Month 2
   wk1   $ 50      $ 45 
   wk2   $ 35      $ 30 
   wk3   $100      $ 95 
   wk4   $ 60      $ 60 
   wk5   $ 80      $ 75 
       -----------------
   sum   $325      $305

And average sales are down:

       Month 1   Month 2
         $ 65      $ 61 

But median is identical to the month prior:

       Month 1   Month 2
         $ 60      $ 60 

Why? Median is still just the middle number, and doesn't allude to those
overall lower sales.

3. In month 2, sales don't change:

       Month 1   Month 2
   wk1   $ 50      $ 45 
   wk2   $ 35      $ 35 
   wk3   $100      $100 
   wk4   $ 60      $ 65 
   wk5   $ 80      $ 80 
       -----------------
   sum   $325      $325 

And average doesn't change:

       Month 1   Month 2
         $ 65      $ 65 

But median goes up:

       Month 1   Month 2
         $ 60      $ 65 

Is the higher median here an indicator of success? Probably not.

4. Finally, in month 2 sales don't change:

       Month 1   Month 2
   wk1   $ 45      $ 50 
   wk2   $ 35      $ 35 
   wk3   $100      $100 
   wk4   $ 65      $ 60 
   wk5   $ 80      $ 80 
       -----------------
   sum   $325      $325 
  
And average doesn't change:

       Month 1   Month 2
         $ 65      $ 65 

But median goes down:

       Month 1   Month 2
         $ 65      $ 60 

Yet the median going down doesn't suggest trouble, because sales overall
weren't really down.

Summary:
Above, I showed examples where median doesn't indicate when something
really is happening and I also showed that medians can change when nothing
is happening.

Measures aren't perfect, but we can use them to gain insight. Ultimately,
a metric should measure what's important to leaders or be connected to a
behavior that a company wants to change. Median is a useful measure of
"central tendency" and it can be useful to look at. Often it's
used when outliers can skew the average, e.g., median home prices in an
area, or median salary.

Looking at sum, average, and median simultaneously is certainly more
insightful than looking at median by itself. This could be one solution to
the problem. If outliers are a problem, it may be worth looking at a
"truncated mean" as another solution.

I think perhaps a more insightful measure is a moving average. Looking at
a 4-week moving average, for example, might be a good indicator of
momentum and show a big-picture trend. Watching this change over time
certainly does have meaning that management would care about. I find
trends over time usually provide the best insight.

Does this help? Please write back if you need anything else.  :-)

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


Date: 05/19/2010 at 10:48:31
From: Nathan
Subject: Using median as an analytical tool

Thanks for your answer. This is pretty much what I was thinking without
being able to put it into the right words.

Your mention of a "truncated mean" is intriguing. Is there a specific
formula for that, or would I just essentially take the data, lop off the
top and bottom 5-10% or so, and then calculate a mean? That sounds like a
better way to measure what I think my boss is trying to get at.

Given that the end goal is to see how much value we are adding to the
average order (of course not letting outliers skew the data), are there
any other measures or any other advice you might have to meet that end?

Thanks again for your insight on this.



Date: 05/19/2010 at 13:46:47
From: Doctor Wilko
Subject: Re: Using median as an analytical tool

Hi,

Thanks for writing back to Ask Dr. Math!

I'll put the burden on you to do some looking into how the truncated mean
is used in practice; but yes, essentially it works as you described: lop
off extremes and get to the heart of the average.

But that's also what moving averages do: they try to dampen the noise and
get to the underlying signal to see the trend.

Researching "business metrics" or "dashboards" should get you in the right
direction for how to show your metrics.

I can't say I've seen medians used a whole lot as the main metric for a
measure. But I did personally consider medians recently for something when
the number of samples got really small and the averages were being skewed
by outliers. (In the end, I didn't feel like medians were doing any
justice to my sample of 2 or 3 data points, either. So we wound up
reporting the average and just explaining that it was "made up of three
values and the individual values were blah, blah, and blah.")

The point of metrics is to provide insight to leaders and help them make
decisions. So as long as we were keeping that in mind, we could talk
around the imperfection of the specific metric and get to the heart of
what mattered.

Does this help?

Please write back if you need anything else.  :-)

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



Date: 05/20/2010 at 10:10:36
From: Nathan
Subject: Thank you (Using median as an analytical tool)

Thank you so much for your help on this question.  I'm going to look into
other metrics to use in addition to the median to try to help show where
the change in median may be coming from, rather than just making
assumptions based off of the median alone. I hope this will help us see
what we want regarding our change in average order size and total sales.

Thanks again!
Associated Topics:
High School Statistics

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