# What Happens When Bill Gates Walks Into a Bar

The mathematician John Allen Paulos in his book Beyond Numeracy writes: “The fourth-grader notes that half the adults in the world are men and half are women and concludes therefrom that the average adult has one breast and one testicle.”

This is a rather extreme example of how the concept of mean or average is misused. An average of X numbers is obtained by adding those numbers and dividing it by X.

Here is another example of a situation where the concept of average is misused.  As Charles Wheelan writes in Naked Statistics—Stripping the Dread from the Data: “Imagine that ten guys are sitting on bar stools in a middle-class drinking establishment…each of these guys earns \$35,000 a year, which makes the mean annual income for the group \$35,000.”

The software billionaire, Bill Gates, walks into this bar. As Wheelan writes: “Let’s assume for the sake of the example that Bill Gates has an annual income of \$1 billion. When Bill sits down on the eleventh bar stool, the mean annual income for the bar patrons rises to about \$91 million. Obviously none of the original ten drinkers is any richer. If I were to describe the patrons of this bar as having an average annual income of \$91 million, the statement would be both statistically correct and grossly misleading.”

The point being that the average or the mean of a given set of numbers can be very misleading. One thing that clearly comes out of this example is that the majority of the numbers that constitute an average can be lower than the average.

As was clear in this example, ten out of 11 men in the bar had a lower income than the average income of \$91 million. Here is another interesting example. As Robert Matthews writes in Chancing It—The Laws of Chance and How They Can Work for You: “The world’s men provide an excellent example – in the shape of their penises. Or, to be more precise, size: according research, the average length is 13.24 centimetres, but the median value is 13.00 centimetres.”

And what is median value? As Paulos writes: “The median of a set of numbers is the middle number in the set.” Let’s go back to the bar example for a moment. Let’s say the eleven individuals in the bar are made to sit in the ascending order of their income. The individual setting on the sixth stool will represent the median income of the group.

Now let’s get back to the penis example. The average length of a man’s penis is 13.24 centimetres. But the median value is 13 centimetres. What does this mean? As Matthews writes: “First, it shows that the global distribution of penis sizes is skewed towards smaller values, and second that most men really do have below-average-sized penises.”

This becomes very important when we are discussing issues like per capita income of a country or the average income earned by a citizen of a country.
The economic health of a nation is also judged by the rise in its per capita income. But should that always be the case? Take the Indian case. A survey carried out by Gallup in December 2013, put India’s median income at \$616. Data from the World Bank shows that India’s per capita income during the same year was \$1455.

Hence, the median income was around 58% lower than the average income or the per capita income. And that is not a good sign at all. The difference is obviously because the rich (Bill Gates in the example) make substantially more than the poor and drive up the average income. Data from World Bank shows that the top 10% of India’s population makes 30% of the total income.

The point being that economic growth as measured by growth in per capita income is not always the correct way of going about things. Is this growth really trickling down? And that can only become clear if the median income is going up. The tragedy is that no regular data is available on this front.

(Vivek Kaul is the author of the Easy Money trilogy. He can be reached at [email protected])

The column originally appeared in the Bangalore Mirror on April 13, 2016