Why the average transfer fee misleads. The Log-Normal distribution
Transfer fees, wages and market values can't go below zero, cluster low and have a few enormous outliers. The Log-Normal distribution describes them, and shows why the average fee is a poor guide to a typical one.
Intermediate Part 12 of Statistics Through Football
Contents
The football question
A club's fans hear that the average transfer fee in their league is £4 million. Is that what a typical signing costs?
Almost certainly not. Most deals are far smaller, and a handful of huge ones drag the average up. The Normal distribution can't describe that shape. The Log-Normal can.
The concept
Some football numbers behave differently from sprint speeds or season points:
- They can't go below zero. There's no such thing as a negative transfer fee.
- Most are fairly small. Plenty of deals are six-figure sums.
- A few are enormous. The occasional £50m or £100m signing sits far out on its own.
That gives a lopsided shape with a long tail to the right. Transfer fees, wages, market values and social media followings all look like this.
The trick is to take the logarithm of each value. On the log scale, the long tail is pulled in and the shape becomes a symmetric bell curve:
$$\ln(X) \sim \text{Normal}(\mu,\ \sigma^2)$$
In plain football
- X is a transfer fee. ln(X) is its natural logarithm, which turns multiplying into adding: every doubling of the fee adds the same amount.
- The formula says: the fees themselves are lopsided, but their logarithms follow a normal bell curve.
- So a £1m, £2m and £4m deal are evenly spaced on the log scale, just as 1, 2 and 3 are on an ordinary one.
- Normal: balanced around the mean.
- Log-Normal: positive, right-skewed, with a long upper tail.
A football example
Take an example market, for illustration: the median fee is £2m, and fees typically sit about three times above or below that. (On the log scale that's μ = ln 2 and σ = 1.2.) Here's how the deals spread out:
Half the deals are under £2m, and the middle half run from about £0.9m to £4.5m. But one deal in a hundred is over £32m.
The average is dragged upwards
The mean of a Log-Normal isn't the median. It's pulled up by the tail:
$$\text{mean} = e^{\mu + \sigma^2/2} = 2 \times e^{0.72} \approx £4.1\text{m}$$
In plain football
- The median is £2m: half of deals cost less, half cost more.
- The average is £4.1m, more than double, because a few huge fees inflate it.
- 73% of deals cost less than the average. "The average fee" describes almost nobody.
A few deals hold most of the money
In this example market, the top 10% of deals account for 47% of all the money spent, and the top 1% alone for 13%. That's the long tail at work: the few big transfers are where most of the spending is.
The same idea, all over football
- Transfer fees: most modest, a few eye-watering.
- Wages: a squad of steady earners and one or two stars on several times more.
- Market values: the same pattern across a whole league's players.
- Social media followings: most players have thousands, a handful have hundreds of millions.
Whenever you see a football number that can't go negative and has a few giants, think Log-Normal, and reach for the median before the mean.
Show the mathsWhere the shape comes from, and the formulas used above. Optional.
If \(\ln X \sim \text{Normal}(\mu, \sigma^2)\), then X has density
$$\begin{aligned} f(x) &= \frac{1}{x\,\sigma\sqrt{2\pi}}\, e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}} \\ &\text{for } x > 0 \end{aligned}$$
Its median is \(e^{\mu}\), its mean \(e^{\mu + \sigma^2/2}\), and its most likely value \(e^{\mu - \sigma^2}\): the mode sits below the median, and the median below the mean.
Probabilities come from the Normal: \(P(X > x) = 1 - \Phi\!\left(\frac{\ln x - \mu}{\sigma}\right)\). For £20m, that's \(1 - \Phi(1.92) \approx 2.8\%\).
The share of all money held by the top fraction of deals above the q-th quantile is \(1 - \Phi(z_q - \sigma)\). For the top 10%, \(z_q = 1.28\), giving \(1 - \Phi(0.08) \approx 47\%\).
Why it appears: a fee is roughly a product of many factors: the player's ability, age, contract length, the buying club's wealth, how many clubs want him. Products become sums on the log scale, and sums of many independent pieces tend towards a Normal. That's the central limit theorem from the Normal distribution, working on the logarithms.
Why it matters
Skewed numbers are everywhere in football's money, and averages hide what's going on. Use the median to describe a typical deal or wage. And when modelling these numbers, work on the log scale: a model fitted to raw fees gets dominated by the few giant transfers, while one fitted to log-fees treats a jump from £1m to £2m the same as one from £10m to £20m.
Limitations
- The example market is made up to show the shape. Real leagues have their own median and spread.
- Free transfers and loans don't fit. A fee of zero has no logarithm, so they have to be handled separately.
- Real tails can be heavier still. The biggest record-breaking fees can sit even further out than a Log-Normal predicts.
Try it yourself
List your club's incoming transfer fees from the last few seasons. Work out the mean and the median: how far apart are they? Then take the logarithm of each fee and see whether the values look more evenly spread.
Further reading
- Log-normal distribution: properties, proofs, exercises, StatLect. The full derivations.
- Log-normal distribution, Wikipedia. The formal definition, its properties and where it turns up.