# How GPS usually exaggerates your performance

Could this be an early development version of the iTrundle? Photo by Norbert Kaiser via Wikimedia Commons.

Just a few days ago, I remarked here that distances measured using iPhones and fitness tracking systems seem to vary much more greatly than you would expect from the apparent precision of the GPS systems being used.

I am delighted now to be able to explain why this is so, and how most of those hard-won performances in Garmin Connect, Strava, and other tracking systems are actually inaccurate.

It all comes down to the statistical performance of how our devices handle the error in GPS positions, as detailed by Peter Ranacher and his colleagues in their paper here.

The way that your iPhone works out how far you travel is to periodically obtain your position using GPS, and then connect those locations with straight lines. It is a simple geometric calculation to work out the length of each straight line segment, and those lengths are then totalled to give the distance travelled. Distance travelled divided by time gives speed, and so on for all the variables which you proudly post after your workout.

So long as those GPS positions are accurate, your figures will be remarkably accurate too. Only using GPS, those positions are not, and any system which uses GPS in this way is prone to error when measuring distance. Worst case error can overestimate the distance by as much as 20%.

Inherent in GPS (as with any system to determine position) is measurement error, a spatial uncertainty. Normally such errors are random, and might be expected to result in random error in the distance estimated between two points. Thus if you measured a distance using GPS a hundred times, you would expect the average to be very close to the actual distance, and roughly half the measurements would be greater, and half smaller, than the actual distance. In the long run (!) these would effectively cancel one another out.

Unfortunately that expectation is wrong: when Ranacher and others analysed it mathematically (see their paper for details), it turns out that the distances measured by GPS are usually (sometimes always) greater than the actual distance. The higher the GPS measurement error, the greater this over-estimation will be. The authors measured this experimentally in someone walking, and in a car driven around an urban area over a total period of about six days, and found that in real measurements distances were far more likely to be overestimated.

Even worse, the amount of overestimation varies. Although there are corrections which could be made at the time, there is no easy way of back-correcting old measurements.

If most of the distances travelled in performance trackers are overestimated, most of those performances are also overestimated, but by variable amounts.

Rumours of the imminent launch of an iTrundle may not be so far-fetched after all.

Thanks to I Programmer for bringing this otherwise obscure paper to light.