Allowing for measurement error – a principled approach to peak bagging

Recent debate on the latest revision of Munro's Tables has overshadowed the fact that past revisions have tended to be driven by resurveying rather than by subjective judgment on what is deserving of being designated a Munro or Munro Top. Older readers may remember the consternation when the OS remapped heights in metres rather than feet. Why can't the OS give a decimal place, some asked? - then we would know whether a peak mapped as 914 metres was over or under the magic 914.4 (the metric equivalent of 3000ft), wouldn't we?

The reason why the OS declines to quote decimal places is that the measurements aren't that accurate. How accurate are they, then? When I contacted the OS, their Geodetic Advisor referred me to Chapter 11 of J B Harley's Ordnance Survey Maps - a descriptive manual. Harley gives a maximum error of ±2ft for triangulation stations and ±3.3m for spot heights from aerial plotting. So a hill mapped as 913m, say, has a distinct possibility of being over 914.4m in reality. Take several such hills and the probability of at least one of them being over 914.4m becomes quite large. In fact, if someone has climbed only the 284 Munros in the SMC list, the probability that they have climbed all significant Scottish 3000ft hills is only 21%.

A lot of problems with lists can be resolved by changing the issue from "what hills are currently considered to be over 3000ft?", to "what hills should I climb to give a reasonable degree of assurance that I have climbed all hills over 3000ft?" The difference is that the second approach takes into account measurement error (including rounding error) while the first does not. Producing such a list requires an elementary statistical calculation, requiring only the assumption that the errors are normally distributed and that the quoted accuracy is to be interpreted as ±3 standard deviations, as elsewhere in Harley's book. The calculation for Munros is shown below.

The first step is to compile a list of "near misses". Thanks to the efforts of Alan Dawson and others, details of 'subs' for most published lists are readily obtainable. In the case of Munros, the obvious candidates are Corbetts of 911m and above. (The probability of a hill mapped at 910m being above 914.4m is too remote to be worth considering). The only non-Corbetts meriting consideration are Creag na h-Eighe (913m) and Meall Gaineimh on Ben Avon (914m), both of which have c 69m of reascent. Although several Munros have less reascent than this, neither of these two peaks exerts sufficient individuality to make a strong case for inclusion - Creag an Dail Mhor to the south of Ben Avon (972m and a Top) has a similar amount of reascent to Meall Gaineimh, for example.

The probability of any single hill mapped at x being above 914.4m is given by F((x-914.4)/s) where s is the standard deviation of the measurement error and F the cumulative normal distribution function. For example, Beinn Dearg in Torridon is mapped at 914m - here taken as 2998ft from the old imperial map. The standard deviation for a spot height is 1.1m, so the above formula gives a probability of 0.29 that Beinn Dearg is over 914.4m (on a scale of 0 to 1, where 1 equals certainty). Similarly, the probabilities of Sgurr a'Choire-bheithe in Knoydart (913m) and Beinn Bhreac in Atholl (912m) being above 914.4m are 0.10 and 0.015 respectively, and, for The Fara, 0.001. We can exclude the possibility of Leathad an Taobhain being over 914.4m because, despite being mapped at 912m, this is a triangulation point and the error much lower^[1]. Foinaven presents a slight problem: the recent mapping at 914m would give a Munro probability of 36%, but information given by the OS to the SMC in 1990 quotes a range of error of 913.8m to 915.2m, which implies the measurement was rounded down from 914.5m (a 914.5 measurement may be rounded up or down).

Assuming that the errors on different hills are independent of each other, the probability that all the aforementioned hills are beneath the critical 914.4m is given by [1-P₁][1-P₂][1-P₃] [1-P₄][1-P₅], where P_j is the probability that hill j exceeds 914.4m. This comes to 0.21, ie the 21% probability mentioned above. If you climb Foinaven, the probability that none of the remaining hills is over 914.4m is [1-P₂][1-P₃][1-P₄][1-P₅], or 63%. Adding Beinn Dearg increases this to 88%, and climbing Sgurr a'Choire-bheithe as well gives you 98.5%. Throw in Beinn Bhreac and the probability reaches 99.9%, enough to satisfy most baggers and to fulfil the more general usage of "probability". If you prefer to take Foinaven as 914m, then the 21% probability becomes 40%, ie still odds-on that you haven't climbed them all; the other probabilities are unchanged.

Similar calculations can be made for other lists. For example, neglecting sub-Murdos will give you only a 22% probability of having climbed all subsidiary peaks likely to merit the status of Munro Top - using the Murdo criterion of >30m of reascent. To get a probability of 99.6%, you need to climb Carn na Caim S Top in Drumochter (914m), Sgurr na Ruaidhe E Top in Strathfarrar (913m), Meall Buidhe SE Top in Glen Lyon (912m and a Munro Top until 1974), Beinn a'Chlachair E Top (977m but 29m of reascent), The Chancellor on the Aonach Eagach (924m, 28m of reascent), and Bidean a'Ghlas Thuill N Top (919m, 26m of reascent), Creag na h-Eighe (not the top in the 1891 Tables but a very awkward peak at the end of the spur NE of Carn Eige) and Meall Gaineimh on Ben Avon, mentioned above.

For Corbetts, sticking to the published list gives a 51% probability of having climbed those hills meeting Corbett's criterion. "Near misses" include Grahams Beinn Talaidh, Sgurr a'Chorainn, and Cnoc Coinnich, all mapped at 761m, and Shee of Ardtalnaig, Beinn a'Chapuill, and Carn an Tionail, mapped at 759m. To these add Glas Mheall Mor (830m), Marg na Craige (834m), Beinn Gharbh (896m), and Kirriereoch Hill (786m), each with nominally 149m of reascent and an individual 1.4% probability of the actual drop being at least 152.4m (500ft). Climbing all these gives a "completion" probability of 99.98%. Miss out the three 759m hills and it's still 99%.

Finally, there is a quite different source of error which I have not attempted to evaluate: the possibility that the measurement is not at the highest point. Spot heights are obtained by positioning a cursor on a computer screen and reading off the co-ordinates; there is no guarantee that the operator has spotted the highest point (or lowest in the case of a bealach), particularly if the hill is not very "peaky". This accounts for some of the differences between 1:50000 and 1:25000 maps, as they are produced by different teams.

All this will doubtless seem pedantic to many - but isn't peak bagging pedantic by its very nature? If your objective is to climb Munros, or Corbetts, or some other named list, then fine; but if it is to climb all Scottish 3000ft hills, all English 2000s, or whatever, then a "probabilistic" approach will at least give you confidence of having achieved your goal.