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As a final remark in this chapter, we will suggest here a few general rules for the rounding off of numerical data for their final presentation.

The number of digits to be kept in a numerical result is determined by the errors on that result. For example, suppose our result after measurement and analysis is calculated to be x = 17.615334 with an error sigma(x) = 0.0233. The error, of course, tells us that the result is uncertain at the level of the second decimal place, so that all following digits have absolutely no meaning. The result therefore should be rounded-off to correspond with the error.

Rounding off also applies to the calculated error. Only the first significant digit has any meaning, of course, but it is generally a good idea to keep two digits (but not more) in case the results are used in some other analysis. The extra digit then helps avoid a cumulative round-off error. In the example above, then, the error is rounded off to sigma = 0.0233 -> 0.023; the result, x, should thus be given to three decimal places.

A general method for rounding off numbers is to take all digits to be rejected and to place a decimal point in front. Then

  1. if the fraction thus formed is less than 0.5, the least significant digit is kept as is,

  2. if the fraction is greater than 0.5, the least significant digit is increased by 1,

  3. if the fraction is exactly 0.5, the number is increased if the least significant digit is odd and kept if it is even.

In the example above, three decimal places are to be kept. Placing a decimal point in front of the rejected digits then yields 0.334. Since this is less than 0.5, the rounded result is x = 17.615 ± 0.023.

One thing which should be avoided is rounding off in steps of one digit at a time. For example, consider the number 2.346 which is to be rounded-off to one decimal place. Using the method above, we find 2.346 -> 2.3. Rounding-off one digit at a time, however, yields 2.346 -> 2.35 -> 2.4!

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