# Why the average transfer fee misleads. The Log-Normal distribution

Source: https://www.footballdatascience.co.uk/learn/log-normal-distribution
Published: 2026-09-27

> 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.

## 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](/learn/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:

$$\ln(X) \sim \text{Normal}(\mu,\ \sigma^2)$$

<div class="plain" markdown="1">
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.
</div>

- 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:

<div class="bars" markdown="1">

| Fee | Share of deals |
|---|---|
| Under £0.5m | 12.4% |
| £0.5m to £1m | 15.8% |
| £1m to £2m | 21.8% |
| £2m to £5m | 27.7% |
| £5m to £10m | 13.3% |
| £10m to £20m | 6.2% |
| Over £20m | 2.8% |

</div>

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:

$$\text{mean} = e^{\mu + \sigma^2/2} = 2 \times e^{0.72} \approx £4.1\text{m}$$

<div class="plain" markdown="1">
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.
</div>

<figure class="rank-chart">
<div role="img" aria-label="A skewed curve of transfer fees: it peaks below £1m and has a long tail to the right. The median is £2m and the mean £4.1m; 73% of deals cost less than the mean.">

</div>
<figcaption>The example market's fees. The curve peaks below £1m, then trails off in a long tail of big deals. The tail drags the mean (£4.1m) well to the right of the median (£2m), so the shaded 73% of deals all cost less than "the average fee".</figcaption>
</figure>

### 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.

<details markdown="1">
<summary><span>Show the maths<small>Where the shape comes from, and the formulas used above. Optional.</small></span></summary>

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](/learn/normal-distribution), working on the logarithms.

</details>

## 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](https://www.statlect.com/probability-distributions/log-normal-distribution), StatLect. The full derivations.
- [Log-normal distribution](https://en.wikipedia.org/wiki/Log-normal_distribution), Wikipedia. The formal definition, its properties and where it turns up.