### 8. SOME GENERAL RULES FOR ROUNDING-OFF NUMBERS FOR FINAL
PRESENTATION

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 (*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 = 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

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

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

- 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!