Ramblings, findings, opinion and the occasional fact.

#5: The confusion between Mean, Median, and Average

#5: The mean average

Sometime in the 60s we heard the news that now every third citizen of the world was Chinese. I remember turning around in the classroom to count the people behind me, but when I pointed my finger at them, doing the one-two-threes, they all looked just as they had done before our teacher told us that fact. I was very disappointed and haven’t believed any statistics since then. At the time I hadn’t heard the famous dictum There are three kinds of lies: lies, damned lies, and statistics”, which was – probably wrongly – attributed to Disraeli by Mark Twain.

The first thing I had to learn is that the term average” may be very familiar, but not correct. When we add the points (perhaps of a test) and then divide that sum by the total number of marks we got, we might call that the average mark. Statistically, however, this is the Mean! I still have no idea why that makes a difference, but then I never wanted to become an MBA, let alone a statistician (I can’t even pronounced that word without having three attempts at it, on average). We are all so familiar with that operation that I don’t need to spend valuable words and figures here on an example. But I have to introduce another term before I come to the practical part of this piece. This would be the Median, the middle value. If you have a list of numbers, say 9, 2, 48, 12, 15, you line them: 2, 9, 12, 15, 45, smallest to largest. As this is an odd amount of numbers, the Median is 12, the number in the middle. In an even line of numbers, you add the middle numbers and divide that sum by 2 (i.e. the mean of those two numbers).

Enter a rich man

Now for a practical example. Say you’re on a train with 49 other passengers when George Soros enters, one of the richest men on earth. How does that affect the average income of the people on that train? By a few percent? Let’s look at that example closely: Let’s presume that each of those 50 people who happen to be on the same train have an income of $67,521, which happens to be mean household income (after tax) in the USA for 2020. Add a presumed income for the additional passenger of some $250 million (I have no source for that) and the mean income for everybody on the train shoots up to $6350 and 54 cents. The expression average” makes no sense anymore at all.

Let’s take the same 49 people and put them on a bus. Then enters the fattest bloke in Britain (The Sun)” who weighs in at 980 pounds, or 444.5 kg for us metric people. How much does our average weight go up? A few percent only? Yes, if the average was 75kg without the man from Ipswich, his added bulk brings it to a mere 80kg. 

Never cross a river that has an average depth of one meter”, 

says Nassim Taleb, an essayist who runs a website www​.fooled​byran​dom​ness​.com and who offers similar examples to the ones above. That river can be only a few centimeters shallow over long stretches but turn into a fast water, ten meters deep in its middle. The average disguises the distribution and you would drown.

The world is becoming too complex to be explained in simple terms like averages of test marks. How many visits does an average website have? There is no such thing; a few gigantic sites like Google or Facebook attract the bulk of all traffic while billions of the rest share whatever little is left. Extremes dominate the distribution and the concept of average makes no sense anymore.

What is the average bonus of a banker? When between 5 or 8% of all men in the US have admitted to being homosexual, there should be at least 75 of them in the National Football League with its 1500 or more players. How many people live in an average town? What is an average war? Do you measure the length of the conflict or the amount of victims? Is there such a thing as average weather? The average UV radiation on a normal summer’s day is perfectly harmless. If you, however, spend all your summer in a dark office in front of a screen and then fly to a Greek island to lie in the sun for a week, you’ll have a problem although your average exposure to UV light has been less than that of somebody who spends a lot of his time outside. 

Can you measure the average success of a marketing campaign? And is there such a thing as an average marketing campaign? How many fires are there on average in your part of the country? All it needs is one to burn your house to the ground, but luckily, the insurance company will also look at the average likelihood when they work out your premium. (Houses do burn down, I had it happen to me, against all odds. And I wasn’t insured because I thought such an event highly unlikely.) Would you advise your kids to become actors because a few stars make millions per movie? Millions of actors, on the other hand, wait tables. The average income would look quite good, the chances to get there do not.

Average means below best

As the examples demonstrate, we should be very careful when it comes to using the word average” or mean” to explain anything. As long as average is used in the colloquial sense of below best”, we know what to make of it. But in cases where a few extremes dominate, the average means nothing.