'Are you above average? Why mean, mode & median matter...' blog discussion

Former_MSE_Helen

This is the discussion to link on the back of Martin's blog. Please read the blog first, as this discussion follows it.

Read Martin's "Are you above average? Why mean, mode & median matter for MoneySavers (especially for Premium Bonds)" Blog.

Please click 'post reply' to discuss below.

BackOnTrack

If 50% of the lottery entrants each won £2 the mean would be £1, the median would be £1 but what would the mode be?

JimmyTheWig

BackOnTrack wrote: »

If 50% of the lottery entrants each won £2 the mean would be £1, the median would be £1 but what would the mode be?

There would be two modes - £2 and zero.

JimmyTheWig

Median average saving £3,000: £25(78% of people win £25 or more, 45% win £50 or more)
Modal average: Not avail

Add in the statistic (from your own tool) that 22% win £75 or more and you can establish that the modal average is £25. [22% win zero, 32% win £25, 24% win £50 and 22% win other amounts.]

jamesd

The mode would be whatever was paid out most often. Random numbers wouldn't produce an equal distribution even if there was an even number of participants. With a perfect distribution an odd number of participants would have a single clear value for the mode.

happyinflorida

When I read the question I thought, well he's given the answer and it's obvious - it said there was one £1million pound prize - so that would be my answer but when ML gave the answer it wasn't in there?!!!

Oh well, I guess I'm wrong again!!

thelawnet

"on average everyone gets their money back!"

Ah, your regularly scheduled rant about Premium Bonds.

The fact is that for gambling it is entirely appropriate to use the mean for payout. (And trust me when I say I'm well-qualified to talk about this.)

Let's consider something with shorter odds.

For example, a lot of people like to bet on favourites at the bookies.

Let's say for sake of example that you can get odds of 1-4 (4 to 1 on), for Manchester United to win the FA Cup. So if you bet 100 pounds, you get your 100 pounds stake back, plus 25 pounds winnings.

Is that a good bet? It depends entirely on how likely Manchester United are to win. For example, let's say Manchester United will win 2 times in 3.

According to both the 'mode' and 'median' average, the 'average' gambler is a winner from this bet: 2 times in 3, you will win money. But only the mean tells us whether it's a good bet or not:

Win: 2/3 * £25 = £50/3
Lose: 1/3 * -£100 = -£100/3
Mean return: -£50/3 = -£16.67

The mean payback is only 83.33% - a poor bet, despite the fact that most people will win.

And if we go to the other side, let's say that we can get 3-1 against another team winning the Cup.

Since there's only a 1 in 3 chance of that happening, the median and modal gamblers are both losers, but:

Win: 1/3 * £300 = £100
Lose: 2/3 * -£100 = -£66.67
Mean return: £33.33

So even though you will lose most of the time, the bet is an outstanding one, and if you were able to repeatedly make such a bet you would become, in the long run, infinitely rich.

By far and away the most important measure of the value of a given gamble is the mean return. If it is absurdly low, as with the lottery, it's a poor bet, if it's up towards 100% it's ok, and if it's above 100% it's prima facie a good bet.

Clearly a gambling operator that pays back 100% consistently is not going to stay in business very long, so for them the '"on average everyone gets their money back!" claim is a huge one and certainly not one to sneer at as you seem to be doing - in fact for the operator, the mean payback percentage is the only thing that matters, because it determines their profitability, because they have exposure to ALL the players, so the reality is they do give back all the money.

The real issue here is not whether the mean is an appropriate measure of the value of the bet - it certainly is, and it's the first thing you should look at, but also whether there are any other metrics to look at. The answer to that is yes - but the mode and median are utterly inappropriate ones to use. More apt would be the standard deviation (higher numbers mean more risk, which is generally a bad thing, although we have to consider that people seek a certain amount of risk, so it is not always bad if the risk is controlled), the probability distribution (which can be used to derive the standard deviation - here we are looking at the chance of winning each prize), as well as percentiles for the return (for example, if I bought 10 lottery tickets per week for 10 years, I would spend 5,200 pounds, and if I did this, what would be (say) the 5th percentile (representing a 1 in 20 loss) return).

The real issue of course is that with premium bonds 'On average everyone gets their money back', but of course they don't because the money is given back at a later date, by which time it has less buying power. So while normally, given that there is a positive return, you would evaluate the utility of the bet using the Kelly Criterion (which determines the amount of money you should wager on a given positive return gamble, according to the probability of winning (smaller probability means smaller bet), the fact is the real return of premium bonds is actually negative so it's a moot point - but then the money I've got sitting in my Lloyds current account is also losing value too.

Premium Bonds are pretty harmless, you put some money in there and it slowly depreciates and you might win something occasionally. It's much better than playing the lottery, because the amount of money at risk with premium bonds is only the marginal value of the interest sacrificed elsewhere (which might not, in reality for most people, be much anyway), which, relative to the principal, is by definition affordable, whereas people are FAR more likely to spend money that they can't afford on slot machines/roulette machines/lottery tickets/scratchcards. With Premium Bonds you demonstrate that you have capital and can afford to buy a little risk by foregoing the interest on it, whereas with lottery tickets you are spending the money in your pocket, which might be needed to buy food and pay for your accommodation.

As for incomes, the median is usually the best figure to use - if Lakshmi Mittal leaves the country, it might drag the mean income down a pound or two, but it doesn't make people any poorer. There are other issues of course, tax, benefits, multi-income households, regional disparities, etc., so knowing that the median income is 20k still doesn't tell us that much, because 20k gross could be 15k net or it could be 30k, depending on your various benefits.

thelawnet

Just to add to that, the reality is that the 'mean' is by far the best measure to use. I've undertaken certain wagers where I've had a small chance of winning - and sure enough, most of the time, I haven't won. But.... I've done those wagers many times, and I've come out well-ahead.

On the same basis, I would most certainly invest in Premium Bonds if they had the same sort of chance of winning as now, but proportionally higher prizes, such that the mean return was better than 1.5% - I would so safe in the knowledge that although the majority of players do not end up richer, as part of a portfolio of risk, on average (and yes, this is an appropriate term to use) it will make me richer.

There are really two problems with Premium Bonds:

1: overallocation (i.e. don't put all your eggs in this basket)
2. poor return (mean return is too low)

That's it. 1.5% is a problem for me. The chance that I'm unlikely to win is not the main issue at all, and personally I think it's disingenous to imply that it is - if the Premium Bond prizes were multiplied by a factor of 6 (9% return), it would be a great investment, but you'd still be able to produce the same article showing that the 'modal and median players are losers', but it would be an article that sensible people would ignore as they rushed to invest.

You need to look at risk-adjusted return, not simply say 'most people lose'. Have a look at http://en.wikipedia.org/wiki/Kelly_criterion For every person alive, it would be appropriate to invest (and in this case 'invest' means 'forego a small amount of interest in return for a chance of winning') some fraction of their bankroll (i.e. their net worth) IF the bet is in your favour (where in this case 'in your favour' is defined as 'having a better payback than the risk-free rate of return').

For example, if you have a net worth of x and the mean return on premium bonds is 2% in excess of the risk-free rate of return, then it is correct to invest some fraction of £x in premium bonds (noting that there maybe other, even more attractive, forms of investment opportunity for your cash - although be careful evaluationg their risk profile, which might be less straightforward than the simple probability distribution offered by a RNG).

This (the high mean return) is all hypothetical of course, but please give it a rest on the Premium Bond rants, I don't see many people coming out of newsagents on dodgy estates having spent their last ten quid on Premium Bonds....

Nile

Mode?..............do you mean Ugly Betty?:D

I am above average..................A+ in fact:cool:...............okay, I'm talking about my blood group.:o

Mobeer

"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"

Actually the figures for "people" in general would be lower, because people with income tax liability below the personal allowance threshold are ignored. Or it might be higher, because non-domicile people are excluded, yet still exist?

Tall_Ian

Of course calculating average salaries or wages is even more complex than just means or medians. It also depends on what you are using it for. Politicians and others have a habit of quoting average salaries in order to make a point (usually that they are apparently paying people above average).

These days you have many more people in work in very different situations than in the past when it was usually just the "man of the household" who went out to work, people these days may also look for more than just financial returns in choosing what work to take. For example, in many households with children, one parent might choose to have a lower paid job in order to have the flexibility of being around when the children are not at school. They would probably (but obviously not always as there are always exceptions) earn less than the main bread-winner in their household, both in terms of hours worked but also wages per hour. Similarly some retired people choose to take jobs they will enjoy in retirement to supplement their income, but because they have a pension are happy to take a lower paid job that they will enjoy more.

Such people working to supplement the main household income or another income will bring down average wages, both mean and median as the average makes no distinction of whether wages are main or supplementary. This means that we should always be suspicious of politicians and others who quote average wages as a way of showing that a jobholder who is probably earning a main household income is paid more than average. The chances are that they are being dishonest and using the statistics misleadingly (they never compare it to average main incomes in a household for example, if such figures exist). The statistics are truthful, it is those who use them to mean something they don't who are (to quote Disraeli) liars!!