What an average doesn't tell you

Recently, I had a hunch that the average age of males was lower than the average age of females. My logic was that women tend to live longer than men and that slightly more male babies are born than female babies. I wondered if my reasoning was correct so I downloaded the demographic statistics for NSW from the Australian Bureau of Statistics and I found the following:

• There were 4.3% more males than females under 31 years of age
• There were equal numbers of males and females aged 31 years of age
• There were 6.3% more females than males over 31 years of age
• 41.2% of the population were less than 31 years old while 57.4% were more than 31 years old

When I crunched the figures I found that the average male is 1 year 8 months younger than the average female. So my hunch was correct. But the question is: what does this tell us about any given person?

The answer is: nothing!

Averages are about populations not about individuals. So if you were to see a statistic such as "Males are on average 1 year 8 months younger than females", you aren't being given an answer so much as a provocation to ask "Why would this be the case? What does it mean?"

When you are surprised at an average then it may help you to surface your assumptions about whatever it was that was being measured.

For instance, if I were to ask you what the average height for a human being was, the chances are that you would say something like 5'2" or thereabouts. So it would surprise you if I were to tell you that the true figure is under 5'. I will explain this claim in a moment, but first would it surprise you to learn that short people often have lower literacy levels than taller people? Yes?

Well interestingly the reason for both of these statistics is the same: children are human beings too!

Most children are under 5' tall and most childen have lower literacy levels than adults and these two facts result in the given statistics. So if you were surprised at the claims I made, it would be because you assumed that we weren't counting children in the averages. Even if we weren't aware of it, we implicitly assumed that we were talking about adult humans.

This can be true of almost any average. The question we always need to ask is: what is the true population that is being averaged? What assumptions are we making about this population? Are there groups we are excluding which we should be including? Does it make sense to use a single average when you are dealing with a 'mixed' population (e.g. adults vs children, males vs females). Does comparing separately calculated averages lead us to ask new questions about the reasons for any differences? Are the reasons obvious or do we need to dig deeper?

The other question is whether we are comparing the right averages.

In Australia, a serious issue is that average Aboriginal life expectancy is significantly lower than that of other Australians. However is this the right comparison? If we were to segment the two populations by socio-economic level would we find that the issue isn't race but poverty i.e. that people in lower socio-economic groups have much lower life expectancies than those in higher socio-economic groups? In other words, the underlying reason may not be racial but economic. Whatever the answer may be, it determines what sort of strategy you use to approach the issue: do we directly target health services to Aboriginal people or do we adopt a broader strategy of reducing poverty across the board?

I don't know what the answer is, but I raise this issue as just one example where the populations you choose for comparing averages can make a significant difference to what strategy you use to investigate the causes of any differences and the consequential strategies you adopt to deal with such causes.

Averages are one of the simplest statistics to calculate and this is one of the reasons that they are so frequently used. However they may conceal differences and assumptions that need to be surfaced if you truly want to understand what is actually going on.

Compared to what?

The judgements that we make can at times depend on the subtlety of the measures we use.

Consider an insurance company where the employees have some latitude to waive the excess on a policy if the person making the claim has mitigating circumstances. Each person may make decisions on a variety of different policies (home. contents. car, boat, accident, personal liability etc.)

Now suppose you wanted to judge whether any of your employees were being overly generous in waiving excesses.

One measure you might use is to simply look at the proportion of claims that a person processed during some time period for which they waived the excess. You might compare this to the total proportion waived over all claims processed by all employees and target those employees who deviated too much from this average. However this is a very crude measure since it fails to take into account the types of claims each individual is processing.

A more refined measure might be to take the average proportion waived by all employees for each individual claim type and then calculate what each individuals waive rate would have been had they waived at these rates for the claims they actually processed. This is a fairer measure since it takes into account that the waive rate might differ between different kinds of claims. And again, you might target those employees for whom the deviation between actual and expected waive rates was too great. However, if any of your employees is just processing a single type of claim then they may be making the largest contribution to the waive rate for that claim type so you would effectively be comparing that person against themselves with little chance of a major deviation.

Both of these methods compare what an individual is doing against what everyone else is doing so it is a comparison against a norm of behavior rather than against any objective measure of what should be happening. For all you know, most employees might be being less generous than you might like, so the person who appears to be being more generous might actually be doing what you want. However, the attraction of these methods is that they are simple to implement.

The more important question however is what is the intended outcome of this policy and whether what your employees are doing is achieving this outcome. Every time an employee waives the excess it is a cost to your company, so presumably you want an outcome that will compensate for this cost. Otherwise, you would just be throwing money out the window. You would need to establish some metrics against which to measure the success of your policy (e.g. customer loyalty as measured by policy cancellations, taking out additional policies etc).

In the case of increased customer loyalty, you might gain an increase in policies due to word of mouth or conversely be able to reduce your advertising costs. Whatever measure you use, you should be able to measure the costs of the policy to your company versus the benefits gained and this in turn might give you a clearer indication as to how generous your employees should be. This would take a lot more gathering and analysis of data but would allow a more targeted approach. You might find that waiving the excess for one type of claim might result in no benefit at all. Or that placing conditions (e.g. 10 years without a claim) around a policy type in order to realise any benefits.

The major differences between this approach and the approach based on averages are:

• In this approach the policy has a clear purpose and you are more interested in whether the total effect of employees actions is achieving that purpose. In the 'average based' approach, you are interested only in how employees compare with each other (whether or not this benefits the business)
• This approach provides information that enables you to evaluate and refine the policy itself. The 'average based' approach allows you to change the behavior of individuals but such change might actually adversely affect the business.

In other words, one approach is looking at te big picture of the businesses actions and its customers reactions, whereas the other is treating the business as if the customers don't matter at all. It seems like a no brainer as to which approach to use but all too often managers employ the easier option rather than investing time and effort in analysis and refinement or more importantly on thinking through what a policy is intended to achieve.