Are you above average? Why mean, mode & median matter for MoneySavers (especially for Premium Bonds)

This all started with my Friday Facebook joke (1) just over a week ago,

I just intervened in a lovers fight. I went for a coffee after Radio 2, as I had a meeting later. There’s a couple next to me, she was quite pretty and if I’m honest he was a bit plain. She then blasts out, ‘you’re the most average man I’ve ever met, average car, average money, average job, average looks, you’re even average in bed.’ I couldn’t help it, I just said, ‘that’s not nice, in fact its mean’!"

Quite commonly the responses are far funnier than the jokes (not difficult), but what I noted here was not everyone knew what the ‘mean’ was, or even that there are different types of averages. That’s worrying as companies often deliberately use different averages to help their spin.

How well do you know your averages?

So this morning to follow it up, I asked the following question on my Twitter and Facebook pages:

If a lottery allowed a million people to buy £1 tickets, and gave one £1million prize only, how much would people get back on average?"

Most people answered ‘£1’, many said nothing, and a few nerds (meant affectionately) correctly answered, ‘depends on which average you use’ (a few also confused it with the National Lottery and talked re prize funds – but this was a purely made up non-profit lottery for mathematical ease).

Now before I explain where EXACTLY this type of logic is used in a sales pitch, it’s worth running through the different types of averages.

• The ‘mean’ average is £1. When we say average this is the one people most commonly think of. If you had a range say (5,4, 2, 1,1,1,14) to find the mean you’d simply add them up (28) and divide by the number of them (7) so the mean average is 4.

  Using the lottery example the prize fund is a total £1,000,000 and there are a million entrants so the mean return is £1. Therefore accurately, though very disingenuously (interesting one for the Advertising Standards), you could argue:

  on average everyone gets their money back!"

• The ‘median’ average is nothing. This average is what the person in the middle would win.

  Taking again the number set (5,4, 2, 1,1,1,14), to work out the median you’d put them in order (1,1,1,2,4,5,14) and count up half way (there are seven numbers so the half way point is the fourth) so here the median average is 2.

  Using the lottery question, as one person wins the jackpot and everyone else wins nothing, if you lined all the people up in order of winnings and took the half way person they’d have won nowt. I would think this is probably the most fair average to use in this case.

• The ‘modal’ average is nothing. The modal average is the result which occurs most frequently.

  So taking the same number set (5,4, 2, 1,1,1,14) here the modal average is 1 as it appears most often.

  Using the lottery question, the most common result was to win nowt, so that’s the modal average.

  While this gives the same answer as the median in this case, I wouldn’t use it for a lottery in general as it would be easy to jerry-rig with the prize distribution.

Premium bonds are mean

I deliberately used the lottery question, because it’s actually quite close to how premium bonds are marketed (see the Premium Bonds: are they worth it? guide).

The key figure given by NS&I who run the bonds is,  "Annual Prize fund interest rate" which is currently 1.5% – in effect this is the MEAN AVERAGE prize that you will win.

So put £100 in over a year and on mean average you’d expect to win £1.50. Yet that isn’t possible as the smallest prize is £25. In fact, nineteen in 20 people will win nowt and just one wins £25 or more.

And this works all the way up the scale, just like in my lottery question, for every big prize winner lots of people have to win substantially less than the average.   

I’ve grabbed the figures from the Premium Bond Probability Calculator to show the difference, which is especialy important at lower amounts.

	Winnings over one year
Amount Saved	Mean Average (based on interest rate)	Median Average	Modal Average
£100	£1.50	£0 (95% of people win nothing)	£0 (95% of people win noting)
£1,000	£15	£0 (60% of people win nothing)	£0 (60% of people win nothing)
£3,000	£45	£25(78% of people win £25 or more, 45% win £50 or more)	Not avail

There’s no average average

Take a quick look at this UK income page on Wikipedia, calculated for the tax year 2004-05 for people aged 35-39. While the mean income is £26,800 the median income is £20,100. In other words some very rich people are pushing the mean average up but in reality a ‘typical’ person was earning quite a bit less than the mean average – so which is the correct one to use?

I suppose the answer is that it depends on which point you’re trying to prove and it takes us to the over-used cliché ‘lies, damn lies and statistics’. Having said that, in common parlance if someone says the average they mean the mean.

A quick final thought to finish, take a look at the latest site poll – how much should petrol cost? It has a rather strange distribution, as many have selected the psychological £1 barrier, so which average would you use to give the result? I’m tempted that this time its modal…

Comment and Discuss

(1) Every Friday on my Facebook page (see Martin Lewis Facebook) I post a joke, which always starts off reading like a normal post; on average it’s met with riotous laughter and much acclaim from worldwide comedy officianados groans.