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# Stop confusing clients with weighted averages

Occasionally in client meetings, something happens that I like to call “Sudden Onset Confusion Syndrome”. This occurs when an explanation, which was being followed quite well, stops making sense. Nodding heads transform into confused expressions, and effective mental processing grinds to a halt.

This disorder has various triggers, but a particularly common pathogen is the weighted average. Symptoms of the disorder include asking, “What? That can’t be right?”, frowning slightly, and looking a bit cross.

But weighted average-induced Sudden Onset Confusion Syndrome can be prevented by using an easier, more straightforward statistic for spreading comp ratios: An aggregate.

Mean and median averages used in benchmarking comps are legitimate, but don’t account for different company sizes as each observation is weighted equally; the averages are informative but don’t capture what the industry is doing.

For instance, the table below lists eight companies of vastly different sizes. The average cash/sales is 31%—this seems a little low. But, an average weighted by sales, which gives larger firms greater influence, brings the group’s cash/sales to 61%, much closer to Apple’s numbers as its impact is now proportional to its size.

So weighted averages are very important because they account for this size difference in a sensible way. And often the right weighting factor is the denominator of the ratio you are trying to average.

But, calculating the average of cash/sales, weighted by sales feels weird, like sales will cancel out or something (which it kind of does—see algebraic explanation below.) This is usually where Sudden Onset Confusion Syndrome kicks in.

Fortunately, an aggregate provides the same answer and isn’t as cumbersome. It’s also robust; it doesn’t matter if some of the individual denominators are negative or zero, as long as the sum across all companies is positive.

The aggregate is calculated by summing the numerators, the denominators, and then dividing one by the other. Essentially a synthetic merger is conducted, combining all the cash and all sales of the whole comp group. It’s an easier and less counter-intuitive calculation, and will always give the same result as if you took the weighted average approach and weighted by the denominator. And this also works for any ratio, not just cash/sales.

And if you can’t believe mathematical assertions without getting down and dirty with the algebra (which you really shouldn’t because it’s full of mathy goodness), you can see that:

AA n_i in RRAAd_i in RR_(>0) sum_i(n_i/d_i*d_i/(sum_j d_j))=sum_i n_i/(sum_j d_j)=(sum_i n_i)/(sum_i d_i)

An aggregate is the preferred methodology of Pellucid Analytics and hopefully, it will vaccinate you and your clients against Sudden Onset Confusion Disorder the next time you’re considering companies of vastly different sizes. Are there other financial calculations you’d like fixed? Let’s talk. Email me at jamie@pellucid.com.

Learn how Pellucid Analytics' data analysis can help you deliver sharper insights to your clients. Visit www.pellucid.com.

Author

### Jamie Ballingall

Chief Scientist & Co-Founder at Pellucid Analytics. Former Wall St. strategist & quant and Columbia University adjunct professor. Solving complicated technical and mathematical problems.