no. of sig. digits
Suppose that an object is found to have a weight of 3.98 ± 0.05 g. This would place its true weight somewhere in the range of 3.93 g to 4.03 g. In judging how to round this number, you count the number of digits in 3.98 that are known exactly, and you find none! Since the 4 is the leftmost digit whose value is uncertain, this would imply that the result should be rounded to one significant figure and reported simply as 4 g. An alternative would be to bend the rule and round off to two significant digits, yielding 4.0 g. How can you decide what to do?
In a case such as this, you should look at the implied uncertainties in the two values, and compare them with the uncertainty associated with the original measurement.
rounded value |
implied max |
implied min |
absolute uncertainty |
relative uncertainty |
3.98 g | 3.985 g | 3.975 g | ±.005 g or 0.01 g | 1 in 400, or 0.25% |
4 g | 4.5 g | 3.5 g | ±.5 g or 1 g | 1 in 4, 25% |
4.0 g | 4.05 g | 3.95 g | ±.05 g or 0.1 g | 1 in 40, 2.5% |
Clearly, rounding off to two digits is the only reasonable course in this example.
The same kind of thing could happen if the original measurement was 9.98 ± 0.05 g. Again, the true value is believed to be in the range of 10.03 g to 9.93 g. The fact that no digit is certain here is an artifact of decimal notation. The absolute uncertainty in the observed value is 0.1 g, so the value itself is known to about 1 part in 100, or 1%. Rounding this value to three digits yields 10.0 g with an implied uncertainty of ±.05 g, or 1 part in 100, consistent with the uncertainty in the observed value.
Observed values should be rounded off to the number of digits that most accurately conveys the uncertainty in the measurement.
In science, we frequently need to carry out calculations on measured values. For example, you might use your pocket calculator to work out the area of a rectangle:
rounded value |
precision |
1.58 | 1 part in 158, or 0.6% |
1.6 | 1 part in 16, or 6 % |
Comment: Your calculator is of course correct as far as the pure numbers go, but you would be wrong to write down 1.57676 cm2 as the answer. Two possible options for rounding off the calculator answer are shown below:
It is clear that neither option is entirely satisfactory; rounding to 3 significant digits leaves the answer too precisely specified, whereas following the rule and rounding to 2 digits has the effect of throwing away some precision. In this case, it could be argued that rounding to three digits is justified because the implied relative uncertainty in the answer, 0.6%, is more consistent with those of the two factors.
The above example is intended to point out that the rounding-off rules, although convenient to apply, do not always yield the most desirable result.
Other examples of rounding calculated values based on measurements are given below.
calculator result |
rounded |
remarks |
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1.6 |
Rounding to two significant figures yields an implied uncertainty of 1/16 or 6%, three times greater than that in the least-preciseely known factor. This is a good illustration of how rounding can lead to the loss of information. |
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1.9E6 |
The "3.1" factor is specified to 1 part in 31, or 3%. In the answer 1.9, the value is expressed to 1 part in 19, or 5%. These precisions are comparable, so the rounding-off rule has given us a reasonable result. |
A certain book has a thickness of 117 mm; find the height of a stack of 24 identical books:
|
2810 mm |
The 24 and the 1 are exact, so the only uncertain value is the thickness of each book, given to 3 significant digits. The trailing zero in the answer is only a placeholder. |
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10.4 |
In addition or subtraction, look for the term having the smallest number of decimal places, and round off the answer to the same number of places. |
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23 cm |
[see below] |
The last of the examples shown above represents the very common operation of converting one unit into another. There is a certain amount of ambiguity here; if we take "9 in" to mean a distance in the range 8.5 to 9.5 in, then the uncertainty is ±0.5 in, which is 1 part in 18, or about ± 6%.
The relative uncertainty in the answer must be the same, since all the values are multiplied by the same factor, 2.54 cm/in. In this case we are justified in writing the answer to two significant digits, yielding an uncertainty of about ±4 cm; if we had used the answer "20 cm" (one significant digit), its implied uncertainty would be ±5 cm, or ±25%.
The excellent Significant figures tutorial by David Dice allows you to test your understanding as you go along.
Make sure you thoroughly understand the following essential ideas which have been presented above. It is especially imortant that you know the precise meanings of all the highlighted terms in the context of this topic.
You can download a pdf document suitable for viewing or printing that contains
all five sections of the "Matter-and-measure" unit.
Page last modified: 12.08.2010
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