4.1. Sums and Differences

For future space tourists, hiking on Mars will require more preparation and planning than will climbing Everest. So let's start early. The picture below was taken by NASA's Mars rover Spirit on March 2, 2005.

View on Mars. Image credit: NASA/JPL/Cornell

The area on the right is known as the Columbia Hills, and Spirit is on the way to one of its summits. We assume that one of the favorite hikes would follow Spirit's route. Suppose we know that the length of the path Spirit traveled from the landing area to the summit of Columbia Hills is 4825 meters ± 5 meters. We start our hike from Spirit's landing area and make a short stop after 3260 meters ± 10 meters. How many more meters do we have to go before we reach the summit?

The calculation of the central value is straightforward

( 4825 − 3260 ) meters = 1565 meters.

What is the error in our result? Clearly, we cannot subtract errors, since it would make the resultant error negative. To do it right, we should consider two cases: the largest possible answer consistent with the errors and the smallest possible answer consistent with the errors. We obtain the largest possible answer when we take the upper limit for the rover distance (4825 + 5) meters = 4830 meters and the lower limit for our distance (3260 − 10) meters = 3250 meters. The maximum possible distance to go is then (4830 − 3250) meters = 1580 meters. Similarly, the lower limit for the rover distance is 4820 meters and the upper limit for our distance is 3270 meters. So the smallest possible distance to go is (4820 − 3270) meters = 1550 meters. Thus we have

maximum distance: 1580 meters
central value: 1565 meters
minimum distance: 1550 meters

We see that the error of 15 meters would include both extreme cases and everything in between. Our final answer is then 1565 meters ± 15 meters.¹

You've probably noticed that our final error (15 m) is the sum of the two errors of the original quantities (5 m and 10 m). It's not a coincidence, since the maximal and minimal distances are obtained when the two errors conspire to add. The above argument holds true for all differences and sums. Therefore, we arrive at the general rule for the error in the case of sums and differences: The absolute error of the result is the sum of the absolute errors of the original quantities. Remember, even if you subtract two quantities you still add their absolute errors.

¹ In this example and in the General Physics Laboratory, we use the maximum possible error estimate. For uncorrelated quantities (such as the two distances in this example), the maximum possible error overestimates the most probable error. This is so because it is very unlikely that a measurement of the second quantity would yield its maximum possible value when a measurement of the first quantity also yielded its maximum possible value.