0.1 A guide to significant figures  (Page 2/2)

In contrast, multiplying 257 times 0.20 tells you that the answer must be between 50.1 and 52.7 (257 x 0.195 and 257 x 0.205, respectively), and this allows you to express your answer (51) using two significant figures. Note that these last two calculations ignored the fact that 257 may range from 256.5 to 257.5, and including this uncertainty would only increase the possible range of the final answer by 0.1. Since 257 already has three significant figures, the uncertainty associated with 257 is trivial in comparison to the uncertainty associated with 0.20.

Even with all these guidelines, some situations are still ambiguous. For example, if we multiply 3.5 times 2.7 we get 9.45, which should be rounded to 9.5 as we have two significant figures. If we then multiply 3.8 times 2.7, we get 10.26. Some might claim that this should be rounded to 10., as we only have two significant figures. However, it may not always make sense that by adding one digit to the left we have to round off one more digit on the right. Again we can rigorously evaluate these situations by applying the full range of values to each number, and seeing the resulting range of values. Using this as our acid test, the “true” answer for 3.8 times 2.7 should be between the minimum value of 9.94 (3.75 times 2.65) and the maximum value of 10.59 (3.85 times 2.75). Thus our median answer 10.3 is approximately plus or minus 0.3. Deleting the one place to the right of the decimal (i.e., rounding to 10.) is losing information, while the value of 10.3 implies more precision than we actually have (i.e., this would imply that the true answer is known to fall between 10.25 and 10.35). In these cases I would be inclined to keep the “extra” digit and use 10.3, but in the text and subsequent calculations I would note that this value is not as precise as implied by the presence of three significant figures, but it is more precise than two significant figures. In many cases some other factor will limit the precision of the final answer, and in these cases I would delete the extra digit as soon as it ceases to have any material effect on the result.

• Changes in units can "create" or "remove" significant figures, but it is often difficult to decide exactly how many digits are significant. For example, 38 mm of precipitation is 0.125 ft (actually between 0.123 and 0.126). In this case the third digit is a bit suspect, but omitting it loses some precision that was inherent in the original datum of 38 mm. In such cases the decision as to whether a third digit should be included depends on: (1) the use to which the data will be put; and (2) the quality of the original data. If we're using the 38 mm of precipitation as input to a runoff prediction model, the model is likely to have more uncertainty than the precipitation estimate, and your output (estimated runoff) should have no more than two significant figures. If the 38 mm of precipitation was snowfall caught in a heated tipping bucket gage, the measurement itself could easily be off by several millimeters, so two significant figures would be appropriate. These conversion issues are another reason why your data and results should be reported in the original units whenever possible.

• The bottom line is that there are some important rules, but in some situations there may be some leeway in terms of the exact number of significant figures. As a consultant or a computer modeler one tends to add on extra "significant" figures in order to impress the client with the precision of your work (e.g., specifying that the 100-year flood is 43,560 cfs, or the hydraulic conductivity is 2.34 x 10-3 cm/sec). People with field experience tend to delete digits because they recognize the problems associated with obtaining accurate and representative data, while modelers tend to report things with unrealistic precision. When in doubt it is generally best to err slightly on the side of having one too many significant figures rather than one too few, as it is never a good policy to lose information. Conversely, if I see more than one too many significant figures, I immediately assume that you don’t appreciate the limitations of your data, and that calls into question your basic understanding of hydrology. In this course unjustifiable gains or losses in the number of significant figures will adversely affect your grade on each homework and exam. After studying this handout please feel free to ask questions to clarify specific points, but note that the presence or absence of the last significant figure may be ambiguous in some cases.