Measuring Average
Mathematicians use three different measures to describe the overarching concept of average: mean, median, and mode.
Mean
Mean is the calculation of average that we’re most used to and probably use fairly often in our day-to-day lives. To calculate it, first compute the sum of all the values in a group (ages, heights, incomes, etc.) by adding them together and then divide that total by the number of values in the group.
Example: Five friends have gathered for lunch and are discussing their yearly incomes, which are as follows:
Lee: $20k
Brian: $20k
Tracy: $50k
Warren: $80k
Stu: $200k
What’s their average (mean) income?
Adding all the incomes together, we get $370k. We then divide that number by the total number of friends, which is 5, to get an average of $74k.
I know what you’re thinking - $74k is only really close to Warren’s income - it’s miles away from everyone else’s. In fact, that observation leads us to the fundamental flaw in the concept of average (at least when talking about the mathematical mean - we’ll come onto other measures of average later): Averages can be virtually meaningless.
I think it’s worth reading that statement a few times as this is the number one stumbling block and difficulty most have in getting their heads around this concept.
Of course, a mathematician would argue that this is untrue, and he’d be right: a mathematical mean no doubt has a mathematical meaning - it is the sum of all the members divided by the number of members - no more, no less. But from a conceptual viewpoint, that doesn’t help us very much.
When I say “The average income of the group of friends is $74k”, does it have any meaning beyond expressing the result of a mathematical calculation?
Consider these questions:
Does any member of the group actually earn $74k?
Do some members of the group earn significantly less than $74k?
Do some members of the group earn significantly more than $74k?
By now it should be clear that the number $74k has little real meaning beyond the purely mathematical - no member of the group actually earns that amount, and most members earn significantly more or significantly less.
For $74k to represent anything in the real world, the friends would have to redistribute their incomes - with those earning above average giving part of their incomes to those earning below average, so that each friend finally ended up with $74k.
That’s actually quite a good way to conceptualise the mean: it’s what everyone would be left with if there was a fair redistribution.
Averages Ignore Variability
It sounds obvious: you can’t capture variability within a group with an average, but it’s so easy to forget that an average not only ignores variability, but really paints right over it - almost fooling us into imagining the group as a cohesive unit, where the members have similar traits that are neatly represented by a single number.
The truth is, that is very rarely the case.
Example: Consider the following excerpt from a report which appeared in the UK newspaper The Guardian in July 2015:
The price of the average property in the UK burst through the £200,000 barrier in June for the first time since records began, following an unexpected 1.7% surge in monthly house prices.
As a reader, casually skimming over this article, we might be inclined to conclude that in June 2015, a lot of houses in the UK were breaching the £200,000 price tag.
But was that the case?
Well, probably not. Like most places, house prices in the UK vary wildly by location.
According to Nationwide, a leading UK mortgage lender, the average house price in London in June 2015 was £429,711 whilst the average price in the north was £125,189.
In other words, houses in London sailed past the £200,000 price tag long ago, whilst prices in the North still have a long way to rise before breaching that level. Only East Anglia, with an average price of £198,000, seems to be well represented by the average.
So, this ‘average’ £200,000 house that we picture when reading the article is really an illusion.
Distribution: The Key to Interpreting Averages
In the examples above, the calculated or quoted average was rendered useless because the values that made up the group were too spread out. In the example of the five friends, the income ranged from $20,000 to $200,000. In the house price example, the range was £125,000 to £420,000. In both cases, the wide distribution of values produced an average that was misleading.
The good news is that where values are not spread out over such a large range and in such an uneven manner, the average might just be more useful as both a number and concept.
Example: Imagine a class of 30 10-year olds. If we say that the average height is 55 inches, there is a fairly good chance that most members of the class will be close to 55 inches tall, give or take a few inches. Of course there will be a few outliers - particularly tall or short children. But the average height of 55 inches is quite a descriptive and useful metric - this stems from the fact that the heights are fairly tightly clustered around the 55-inch mark.
Take a look at the graph above, which represents the distribution of heights I just described.
Notice how most children in the class are tightly clustered around the average (the vertical dashed line). You might have come across this ‘shape’ before - it’s called a normal distribution.
Don’t worry if you’ve never heard that term before - just think of it as a distribution of values clustered around the average, with fewer and fewer outliers as the numbers get bigger and smaller.
This type of distribution tends to occur quite regularly in many natural situations, from children’s heights, to people’s IQs.
The more tightly the values are clustered around the average, the more we can interpret the average as representative of, well, ‘averageness’.
Take this to the extreme and imagine every child in the above class is exactly 55 inches tall, and you’ll see that the average height of 55 inches is now extremely representative and useful - in fact it’s the height of every child in the class!
So averages don’t have to be that useless after all. Just remember to think about the distribution.
Alternative Measures of Average: Median and Mode
Let’s go back to the example of the group of friends and their income:
Lee: $20,000
Brian: $20,000
Tracy: $50,000
Warren: $80,000
Stu: $200,000
To make things easier, I’ve arranged them in ascending order, from Lee’s $20k to Stu’s $200k.
No prizes for working out the middle value, or the median, it’s Tracy’s $50k.
You might want to consider whether you feel that $50k is a more useful representation of average than the mean of $74k that we came up with in the previous section. What do you think?
In defense of the median - at least it corresponds to someone’s income, rather than the mean, which doesn’t correspond to anyone’s. And that’s really the median’s ‘strength’ - if you will - it’s just the value of the middle guy (or girl) in our group.
But is $50k really any better a representation of average than $74k?
Probably not - it’s still miles away from everyone else’s income - so the median still suffers from variability, just as the mean does.
If we were to take our class of 10-year olds and line them up in height order (like they do when preparing for the class photo), the median height would be that of the 15th or 16th pupil, which, to be honest, would probably be almost exactly 55 inches - in other words, approximately the same as the mean.
In a tightly clustered distribution, the median will be very close to the mean, both of which will be a good representation of average.
So, when distributions of values are normal(ish) and tightly clustered around the average, both mean and median are useful. But when distributions are a bit more ‘all over the place’, then both suffer from the same problem, and in fact, can be quite different from each other.
In our friends’ incomes example, there is a $24k difference between mean and median - that’s quite a lot. However, if Tracy’s income had been $30k instead of $50k, then things would look quite different. She’d still be the middle member of the group, so the median would now be $30k, but the mean would have only moved down to $70k - which is not that much of a difference from $74k (perhaps the mean isn’t so useless after all!).
So you can see just how dependent on one member the median really is - it just seems a bit arbitrary, doesn’t it?
Again, it all comes back to the distribution - if we’re dealing with a large, fairly homogenous group, there shouldn’t be any problem - the median should be close to the mean and fairly representative of average. On the other hand, if we’re dealing with a small group with a somewhat funny distribution, like our group of friends, the median can be really dependent on just one member, which really affects how useful it is and implies we should approach it with caution.
The mode is just the most common value in the group, and of course suffers from the exact same issues as the mean and median - its usefulness is determined largely by the distribution of values.
Looking back at our group of friends, the mode would be $20k. How does that compare to our mean of $74k and our median of $50k?
Truthfully, it’s a world away - and we’d have to conclude that it’s the least useful metric.
Why? Because our small group meant that just two members are determining the mode for the whole group, which is leading to an extremely skewed value.
Just as in the case of the median, the result is somewhat arbitrary, but just happens to be different from the median of $50k, which was determined by just one person!
What a mess! In the end, the mean of $74k seems to do the job of ‘averaging’ quite well, even though it’s not that representative of any one particular person’s income. Perhaps you can now see why the mean is the most commonly used of the mathematical averages, with its power to simply brush over the whole group and come up with a single number that describes everyone, without getting thrown off by any one member.
Just remember how to interpret that number!
What Does it Mean to be Average?
That’s enough maths for today. But we’re still left with the question of just what it means to be ‘average’.
Consider the following statement:
I’m an average tennis player.
Does this mean that I have considered the distribution of tennis playing abilities of every single person in the world? Or just those that play tennis regularly? Am I referring to mean, median or mode? What number am I using to represent ability?
The same doubts apply to pretty much any similar statement that casually refers to ‘average’:
Is he good looking? Not really - about average.
I’ll leave you to decide what it means to be an average tennis player, or ‘averagely’ good looking. You should now have the tools in place to make that judgement.
Use the Concept
A good way to conceptualise average is to think of it as the way everyone would end up (money, height, tennis ability, good looks) if those who had more gave to those who had less until everyone had the same amount.
Think twice before drawing any conclusions from any averages you see quoted - they can be almost meaningless.
If you absolutely must apply a global value to a group in order to encapsulate some numerical property (like income, price or height), think carefully about the distribution of the values in that group. Are they likely to be tightly clustered around the average? Is the average a useful, or representative metric?
Be extra careful interpreting the median or mode of small groups with odd distributions of values - they may be very skewed by the particular value of the middle member or a few common values which are not representative of any other values.