Sound as a Pound? The statistics of counterfeiting

Author: Tom King

Statistics have been important in the working of the Royal Mint for centuries, with Sir Isaac Newton introducing a form of statistical process control in the 18th century. Newton understood variation about an average value before there was a coherent theory of distribution. This was very important as the King didn’t want too much gold being used and merchants wanted their fair share. Checking the specifications on a production line is fairly straightforward to organise and it was very ceremonial in ‘the trial of the pyx’.

The Pound coin on the left is a forgery

but the one on the right is genuine but

badly worn. Image by Zephyris/Wikimedia.

Currency can only hold its value if it can be guaranteed to be accepted in exchange for goods and services. Therefore policing counterfeiting is very important. For notes there are effective systems in place with numerous security features in the design and scanners are used by many establishments to check for fakes. Note designs can also be changed fairly easily as the individual notes may last as little as one year in circulation.

Coins are more difficult as they don't allow for watermarks or other security features. However, as they are lower value they attract less interest from forgers. Indeed for small coins the metal may be worth more than the face value of the coinage (The Royal Mint has a video explaining how Pound coins are made). Counterfeiting was common in Newton’s day and many features of modern coinage are to prevent crude forging.

Nevertheless there is a significant problem with circulation of fake One Pound coins in the UK. The Royal Mint estimates that 3.1 % of coins in circulation are fakes, around 44 million coins. When they first started surveying in 2003 the figure was around 0.9% representing a rapid increase given that the coin was first issued in 1983. This proportion is also much greater than any other coins in circulation and has continued to increase despite public information and improving technology.

While it is interesting to speculate how this change may have come about: is it that Euro coins are harder to forge than those they replaced? There is also a statistical dimension to this issue which hasn't been covered in any of the press reports.

It is obvious that the 3% figure is not calculated by checking every Pound coin each year, partly as there are more than one billion but also this would allow all of the fakes to be withdrawn. Thus some kind of sampling is necessary to get an estimate which can be extrapolated to the whole population. The design of this sampling scheme is very important to make inferences about changes over time, e.g. to estimate the number of fakes being introduced.

The approach taken is to check 15 thousand Pound coins twice annually. These coins are sourced from cash centres in the form of around 30 bags in total i.e. £500 per bag. This represent a cluster sampling approach with all of the bags being taken from a similar point in the circulation (cash centres). These methods were reported in more detail in response to a Freedom of Information request in 2010 and neither the Mint nor the Treasury publish these reports routinely. This sampling approach is based on advice from the Office for National Statistics which changed to biannual samples and what back of an envelope calculations show is confidence of around ±0.1% (absolute).

Clustering makes sampling easier while reducing the information in an item because items in a cluster are more likely to be similar in source. Taking all items at the same point may over or underestimate their prevalence depending on how close this point is to the source. However, as counterfeit coins may have been in circulation for some time this may not be too important (coins have a life of around 40 years). There are some changes in the use of coinage which may affect the chance of coins going to one of these cash centres, such as the increase of use in debit cards for certain transactions and the ability of machines to detect and reject fakes.

Any inference asserting a change over time requires that the chance of a fake coin going to a cash centre is the same over time, even though it may not be the same as a real coin. Any inference about the true number of counterfeit coins in circulation requires that these two probabilities be equal, but there is no firm evidence that either is even plausible. What makes things worse is that none of the data presented give any indication of a confidence interval of the proportion estimated.

However, this is not to suggest that the changes in the rates of counterfeits are an artefact of the sampling. Past data on the fakes shows changes in the material being used, with a rapid growth in the number using materials similar to the true blanks. This does beg the question of how the sampled coins are analysed and this does raise a few concerns.

Public information indicates that the whole sample of 15,000 is examined visually by one person. Those identified as fake are then analysed by spectrometry, with a consistently null false positive rate. While this is impressive, it is concerning that no effort is made to assess for false negatives. Thus changes in the number found could relate to an improvement in the visual screening process. This is a particular concern for consecutive surveys showing a substantial change in the materials being used.

Overall, the most interesting question is about what use the data is put to. As mentioned before, notes do not persist in circulation but coins do, so any taken out may have been introduced years ago. However, there should still be enough information to estimate the production of counterfeits on a biannual basis but that would be best done using a diffusion model. The ideal would be to invisibly mark a sample of coins and track their progress in circulation but the expense might be prohibitive against a possible gain capped at only £44million. Without more data a graphical approach may be just as effective.

It is tempting to infer many things from the simple time series plot but this ignores other data about the number of counterfeit coins being removed from circulation, the uncertainty of the sampling, and the type of counterfeits being recorded. The graph increases steadily until one point (2009) which corresponds to a time when ten times as many fakes were removed from circulation. We could have a certain amount of confidence about the proportion of counterfeits at cash centres but change over time requires much more information about circulation.

This is a good example of the conflict expressed in publishing public data. The Royal Mint will see this as important management information to monitor where counterfeits are appearing and any characteristic of their production, which is why they collect more data such as region and material. However, the proportion of counterfeits in the circulating coins is an important indicator of the confidence the public can have in using the pound coin in financial exchange. In this case, only the overall proportion is important, but as part of a long term trend. Indeed confidence intervals might help – there is a great deal of uncertainty about all of these figures.

Time series of Pound coins surveys of cash centres 2002-2011. Sources: HM Treasury and The Royal Mint.