Requisitos Legales

Rounding introduces bias. As mentioned earlier in the chapter, bias is an error that remains constant in a series of observations or calculations. To understand this point, consider a mass of 1.629 g, measured on a balance that the manufacturer claims has an uncertainty of 0.005 g. The true mass, then, is probably between 1.624 and 1.634 g.

If we round the measurement to “1.63 g,” the next person to read it will infer that the true measure- ment is between 1.625 and 1.635 g. The new range is higher than the original by 0.001; this difference is a positive bias—a constant error—that will accompany this value into any subsequent calculations, affecting the outcomes accordingly.

Rounding also sacrifices information about precision. For example, the value “3.8” implies a range of 3.75 to 3.85, which represents an uncertainty of 0.05. However, rounding “3.8” up to “4” changes the implied range to 3.5–4.5, an uncertainty of 0.5. The new uncertainty is 10 times greater than the previ- ous uncertainty.

Keeping FiguRe SigniFicAnce in peRSpecTive

Illusory precision is something we want to avert. The rules of figure significance were developed to pre- vent misunderstandings about the level of uncertainty in the measurements we share with each other. They keep us from implying more precision in our measurements than exists in the devices with which we make those measurements. Even so, the rules only approximate the uncertainty in our numbers, and sometimes that approximation is unacceptable. What follows are three examples of problems that illustrate why this chapter earlier described the rules as “guidelines.” In each example, we must check the rules against solid reasoning in order to find a trustworthy solution.

exAmple 1

Consider the following addition of four measured masses (in “grams”). 100 + 23 + 40 + 31

The reasonable answer is 194 grams (“190” if rounded to the nearest 10, or even “200” if rounded to the nearest 100). However, applying the rule of figure significance for addition gives the preposterous total of 100. In this case, reasonableness trumps the rules.

exAmple 2

Consider the following 10 measured volumes (in “mL”). If we choose to report only the average of these volumes, and not the individual volumes themselves, what is the proper value?

89 95 102 93 97 100. 96 92 100. 95

The unrounded average is “95.9 mL.” The exception of repeated measurements, pre- sented earlier, would remind us to write the average of these values such that only the last digit is uncertain. Unfortunately, there is no unquestionable digit in either the ones, tens, or hundreds place. Therefore, we might round the average up to “100” in order to report only one significant figure, but this would be grossly misleading for the following two reasons.

1. As the reported average, “100” is too far from most of the values and from the

unrounded average. There is no justification for declaring the average to be so near the upper end of these measurements.

2. The value “100” has only one significant figure, implying that the average may be as

high as 149 (which becomes “100” when rounded to one significant figure). That range, however, would be silly for the values given. So, we might try to circumvent this absur- dity by making two digits significant, that is, by writing “100 (2 s.f.)” or “100.” But if we do, then the implied range for the average becomes 95-105, which is also indefensible because the highest measured volume is only 102. If we make all three digits significant by writing “100.” or “100 (3 s.f.),” then the implied range is 99.5-100.5, which is still too high and which carries more precision than exists in any of the measurements.

Like “100 (3 s.f.)” above, reporting the unrounded average of 95.9 would also imply too much precision. The best option, then, is to round the average to the nearest whole number, 96. Even though it implies that the uncertain “9” is certain, this value is a ratio- nal compromise between inflating the average (by choosing “100”) and overstating the precision (by choosing “95.9”).

Summary

rounding. one way to prevent this is to round to an even number whenever the final digit is “5,” whether doing so amounts to a rounding up or a rounding down.

5. A significant figure is one that either is known with certainty

or has been estimated.

6. In a properly reported measurement, any nonzero digit is

significant.

7. A measurement is typically reported such that only the last

digit is uncertain.

8. Figure significance is related to precision, or the agreement

among repeated measurements, rather than to accuracy, or the degree to which the value is correct.

9. A trailing zero, which is any zero that follows the last non-

zero digit in a number, is significant only if there is a decimal point somewhere in the number.

1. Rounding is about keeping a number consistent with the

level of our certainty in it.

2. Rounding whole numbers. (a) Identify the rounding digit.

(b) If the first digit to the right is less than 5, then do not change the rounding digit but do change to “0” all digits to the right of it. (c) If the first digit to the right is 5 or greater, then add 1 to the rounding digit and change to “0” all digits to the right of it.

3. Rounding decimal numbers. (a) Identify the rounding digit.

(b) If the first digit to the right is less than 5, then do not change the rounding digit but do drop all digits to the right of it. (c) If the first digit to the right is 5 or greater, then add 1 to the rounding digit and drop all digits to the right of it.

4. Under the standard rules for rounding, the average of a

set of numbers is higher after rounding than it is before

exAmple 3

Suppose we measure the volume of a liquid to be 6.97 mL in a cylinder that has a stated uncertainty of 0.05 mL. We properly record the volume as “6.97 mL { 0.05 mL,” realizing that the true volume falls between 6.92 and 7.02 mL. Rounding this number, therefore, presents a puzzle in that the digit in the ones place is questionable. Because only the last digit in our measurement should be uncertain, is it reasonable to round off to one decimal place, or must we stop at a whole number?

We have two options. First, we could simply round the volume to “7 mL.” It would be correct for the entire range from 6.92 to 7.02, although it has only one significant figure and, thus, less precision than the measurement itself. Second, we could round the reading to “7.0,” which has a precision closer to that of the original, but which assumes the true reading to be at least 6.95. We can settle the issue by comparing the implied relative uncer- tainties of the two rounded values to the relative uncertainty of the original measurement.

value (ml) uncertainty (ml)Absolute minimum (ml)implied maximum (ml)implied uncertaintyRelative

6.97 0.05 6.92 7.02 0.7%

7 0.5 6.5 7.5 7%

7. 0 0.05 6.95 7.05 0.7%

As the above table shows, rounding to one decimal place (two significant figures) gives the same relative uncertainty as there is in the original measurement. Thus, if we round off, our choice should be “7.0 mL.” generally, a raw measurement should be rounded to the number of digits most consistent with the measurement’s uncertainty.