Imagine that, for whatever reason, you do not wish to engage a lawyer. Perhaps they are too expensive; perhaps lawyers do not have your trust. But you are numerate, and approach the endeavour with the seriousness of a bookie.

The context is you believe you have a strong employment claim (whether the claim is yours or simply one that your organisation anticipates paying for). Now, while normally people would reach into their pockets and commission an expert assessment from a specialist employment barrister, you say you did not wish to do so — *how else might you know the right level of award to settle the claim and avoid tribunal?*

*Making inferences from statistics*

One obvious way would be to look at the published statistics for levels of award for your type of claim. You would also want to look at the statistics over a number of years (to average out any outliers from unusual years). Then you would consider not only the average award, but above all the distribution of awards so that you could with some confidence estimate *the likely range of values* the award might take.

To do this, you would have regard to the shape of the distribution, and would calculate confidence intervals — what would be the upper and lower bounds of the award that you could be 90% confident any award handed down by the tribunal would fall within?

*But.. statistics are not available in an immediately useful format*

The truth is you would need to dig around quite a bit in order to get any meaningful data on this. There *are* repositories of tribunal statistics (see the Footnote 1), published each quarter — but despite being publicly available, they are not at all designed for public consumption — there is no convenient way to obtain, for example, lists of the amounts of each award (footnote 2) which are a basic prerequisite for constructing a distribution (footnote 3).

Solicitors’ firms regularly use their websites to publish updates listing the average awards for each type of claim and the upper thresholds — because those summary statistics are published by the Ministry of Justice. But there is a limit to the usefulness of knowing the outlier award, and to the unfamiliar it is actually quite a misleading figure to headline with: how valuable is it to the majority of claimants to learn that the highest claim in a quarter was worth £1/4m? It is in fact about as helpful as knowing the size of the National Lottery jackpot — good to know, but unlikely to be relevant to your life. It is equally a stressful statistic for employers — and again, not commonly the level of award for which they would be found liable. Nor is the *mean* award a good measure, since it is a moment that is unduly influenced by outlier amounts.

*Claimants will hate this skewness*

Coming back to modelling the likely bounds of any award, one can see from the plot of the cumulative distribution function above on the right that in 2020 **90% of the unfair dismissal awards fall below the value of £30**k. And what type of distribution actually fits the data? In the Cullen and Frey graph below, we can see that the blue dot represents the empirically observed data (the string of values representing 2020 unfair dismissal awards). Reading off the x-axis’s ‘Square of Skewness’ we can see our data is highly *skewed* (that is, highly asymmetrical if one looks at the distribution of the data on both the left and right sides of the distribution’s centre point). It is asymmetric with the skew to the left of the centre point — in other words, skewed in a way claimants will hate.

*Employers will hate this kurtosis*

Looking at the y-axis, we can see the blue dot representing the observations has a high positive score for kurtosis. Kurtosis tells us where the risk exists: is risk evenly spread through the distribution, or does it suddenly hit all at once, concentrated in the tail events?

Low (or negative) kurtosis means most observations fall within a predictable range, with little tail risk. High kurtosis, by contrast, indicates the presence of extreme surprises in the tails of the distribution. Our data has very high kurtosis. So: not good for those responding to claims. In terms of fitting the data to a known type of distribution, one can see that the unfair dismissal data appears quite close to a gamma distribution.

*Conclusions: hope for the best; prepare for the worst*

From this cursory analysis, one can clearly see why there is appetite to pay into insurance schemes covering employment claims: while most claims for unfair dismissal resolve for a relatively low sum of money (whether through settlement or by tribunal order), there is significant tail risk that should make any HR professional think twice before declining conciliation — we are not dealing with normal distribution but a distribution that is characterised by skewness, kurtosis, and a fat tail.

The gamma distribution is commonly used in finance as an alternative to the normal distribution — to model the returns on stock indices as well as various other areas of finance such as options pricing (Footnote 4). From our consideration of the 2020 unfair dismissal data, the gamma distribution is a solid candidate for describing the likelihood of that data (see the plots below fitting a gamma curve to the data).

Of course, if one did engage a lawyer, these non-specific insights can be used in conjunction with expert advice on the particular facts of your case to determine the most likely outcome with more precision than a general statistical approach can afford.

- https://www.gov.uk/government/statistics/tribunal-statistics-quarterly-april-to-june-2020
- However, there are indications of the numbers of awards falling within pre-defined bands. It is an odd way of presenting data to the public, but is enough to use bootstrapping to generate useful statistics — which is the technique I had to use to generate the density histogram above.
- I am grateful for the help of the Ministry of Justice in promptly answering my Freedom of Information request and indicating the extent of the statistics (and confirming to me the limits as to what is publicly available).
- See “Modeling and Risk Analysis Using Parametric Distributions with an Application in Equity-Linked Securities” by Sun-Yong Choi and Ji-Hun Yoon published in
*Mathematical Problems in Engineering*(2020, Special Issue) https://doi.org/10.1155/2020/9763065.