I was reading a very interesting article on behavioural psychology today and its effect with relation to Peak oil, climate change etc. I really liked the article, it does a pretty good job of explaining skepticism towards change, it is a must read for certain. I could not help but notice a common enough error with statistics.

To illustrate a point about human optimism particularly with regards to our own abilities, the author quotes research showing how people think that they are better drivers than average (88%) or that professors think that they are better than average (94%), which is obviously impossible as "By definition, almost half of those surveyed are 'overly optimistic'" or in other words 38% of drivers surveyed and 94% of professors must be wrong. Interestingly, it is possible for the surveyed people to be right, although very unlikely and here is why.

To illustrate a point about human optimism particularly with regards to our own abilities, the author quotes research showing how people think that they are better drivers than average (88%) or that professors think that they are better than average (94%), which is obviously impossible as "By definition, almost half of those surveyed are 'overly optimistic'" or in other words 38% of drivers surveyed and 94% of professors must be wrong. Interestingly, it is possible for the surveyed people to be right, although very unlikely and here is why.

The arithmetic mean or average of a distribution is simply the sum of all values of the distribution divided by the number of items in the distribution. Say that we quantify driver ability on a scale of 0 to 1000, where 1000 is the driving skills of Michael Schummacher coupled with the law abiding ethos of a 50 year old German protestant pastor and 0 is the law abiding ethos of an Italian teenager coupled with the driving skills of a 100 year old arthritic, half blind grandmother.

We take a random selection of 10 people and measure their driving skills. These are their results:

We take a random selection of 10 people and measure their driving skills. These are their results:

- 501
- 500
- 504
- 502
- 503
- 508
- 516
- 490
- 491
- 532

The average of the above distribution is 504.7 and thus we have 80% above average. what is going on here, I hear you ask?

Nothing much, just good old fashioned confusion, here is another distribution, this one is the annual salaries of ten friends who have met for a dinner party.

Nothing much, just good old fashioned confusion, here is another distribution, this one is the annual salaries of ten friends who have met for a dinner party.

- £50100
- £50000
- £50400
- £50200
- £50300
- £50800
- £51600
- £49000
- £49100
- £532000

On this distribution 90% are

*below*average, the average is close to £95k.

The author confused arithmetic mean with the median, which is described in Wikipedia as: a median is described as the numeric value separating the higher half of a sample, a population, or a probability distribution, from the lower half.

As I have shown it is possible to have almost everybody above or below average it is impossible to have more than 50% of the drivers above the median, by definition.

This post was just a little bit of facetiousness, his point still stands even if he conveyed it poorly. People think that they are above average

**and**above the median.

Incidentally, the median for the first distribution is: 503 and for the second: £50300.