The mathematics of flies in ointment

By Andrew Clifford

You can use simple mathematics to highlight areas that demand management attention and to stop bad news being hidden.

Since ancient times we have recognised that, when assessing a situation, a few bad things can have a disproportionate effect. Take, for example, this passage from the Bible, from which we get the phrase "fly in the ointment" (Ecclesiastes 10:1, RSV):

Dead flies make the perfumer's ointment give off an evil odor; so a little folly outweighs wisdom and honor.

In business, the measures we use for goodness and badness do not show this. At Metrici, we help organisations assess things: projects, IT systems, people, maturity, risk, and so on. It is useful to have summary measures, and we tend to use normal averages (arithmetic means) based on a weighting and scoring scheme. This is easy to understand, but it can hide bad news among good. For example, Project A has an excellent sponsor relationship, effective project management, adequate resourcing, but is seriously over budget. Project B has a reasonable sponsor relationship, project management and resourcing, and is within budget. The averaging effect means that Project A and Project B have the same score, hiding from management the problems with Project A's budget.

We try to find ways of highlighting areas that require management attention. Sometimes we show a second figure next to the score to indicate the number and severity of recommendations. Sometimes we have ad hoc rules, like "the average of the worst three". Sometimes we add commentary.

It would be good to have a single number that represents where management attention is required. People are not good at making decisions based on lots of numbers, and if they are considering multiple items (like a portfolio of projects), additional information soon gets lost. One score that reliably measures goodness or badness would be better.

The power mean is a variation of the arithmetic mean that allows you to emphasise one or other end of a range. Each individual factor is raised to a power before an average is taken, and then the average is raised to the inverse of the power to give the final result. For example, you might take the square root of each factor, average them, and then take the square of the result. You can use the power mean so that a situation with a mixture of good and bad comes out worse than a consistently mediocre situation. In a typical scoring scheme which goes from 0 (bad) to 100 (good), using a power less than 1 (such as 0.2) means that measures at the bad end of the scale are emphasised and not lost.

To be confident in the numbers you give them, people need to understand how they were calculated. Although they are more complicated than simple arithmetic means, power means are easier to understand than presenting multiple numbers and commentary.

We have found the power mean a useful addition to our tool box of analysis techniques. If you are involved in assessing or judging business situations, it is worth understanding the power mean.

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