Significant Figures or Digits

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Significant Figures or Digits

Imagine that a nurse takes a child's temperature. However, the mercury thermometer being used is marked off in intervals of one degree. There is a 98° F mark and a 99° F mark, but nothing in between. The nurse announces that the child's temperature is 99° F. How does someone interpret this information? The answer becomes clear when one has an understanding of significant digits.

The Importance of Precision

One possibility, of course, is that the mercury did lie exactly on the 99°F mark. What are the other possibilities? If the mercury were slightly above or below the 99°F mark (so that his actual temperature was, say, 99.1°F or 98.8°F), the nurse would probably still have recorded it as 99°F. However, if his actual temperature was 98.1°F, the nurse should have recorded it as 98°F. In fact, the temperature would have been recorded as 99° F only if the mercury lay within half a degree of 99°F—in other words, if the actual temperature was between 98.5°F and 99.5°F. (This interval includes its left endpoint, 98.5°F, but not its right endpoint, because 99.5°F would be rounded up to 100°F. However, any number just below 99.5° F, such as 99.49999°F, would be rounded down to 99°F.) One can conclude from this analysis that, in the measurement of 99°F, only the first 9 is guaranteed to be accurate—the temperature is definitely 90-something degrees. The second 9 represents a rounded value.

Now suppose that, once again, the child's temperature is being taken, but this time, a more standard thermometer is being used, one that is marked off in intervals of one tenth of a degree. Once again, the nurse announces that his temperature is 99° F. This announcement provides more information than did the previous measurement. This time, if his temperature were actually 98.6° F, it would have been recorded as such, not as 99° F. Following the reasoning of the previous paragraph, one can conclude that his actual temperature lies at most halfway toward the next marker on the thermometer—that is, within half of a tenth of a degree of 99°F. So the possible range of values this time is 98.95° F to 99.05°F—a much narrower range than in the previous case.

When marking down a patient's temperature in a medical chart, it may not be necessary to make it clear whether a measurement is precise to one half of a degree or one twentieth of a degree. But in the world of scientific experiments, such precision can be vital. It is clear that simply recording a measurement of 99°F does not, by itself, give any information as to the level of precision of this measurement. Thus, a method of recording data that will incorporate this information is needed. This method is the use of significant digits (also called significant figures).

In the second of the two scenarios described earlier, the nurse is able to round off the measurement to the nearest tenth of a degree, not merely to the nearest degree. This fact can be communicated by including the tenths digit in the measurement—that is, by recording the measurement as 99.0°F rather than 99°F. The notation 99.0°F indicates that the first two digits are accurate and the third is a rounded value, whereas the notation 99°F indicates that the first digit is accurate and the second is a rounded value. The number 99.0°F is said to have three significant digits and the number 99°F to have two.

What Is Significant?

When one looks at a measurement, how can she tell how many significant digits it has? Any number between 1 and 9 is a significant digit. When it comes to zeros, the situation is a little more complicated. To see why, suppose that a certain length is recorded to be 90.30 meters long. This measurement has four significant digits—the zero on the end (after the three) indicates that the nine, first zero, and three are accurate and the last zero is an estimate. Now suppose that she needs to convert this measurement to kilometers. The measurement now reads 0.09030 kilometers. She now has five digits in the measurement, but she has not suddenly gained precision simply by changing units. Any zero that appears to the left of the first nonzero digit (the 9, in this case) cannot be considered significant.

Now suppose that she wants to convert that same measurement to millimeters, so that it reads 90,300 millimeters. Once again, she has not gained any precision, so the rightmost zero is not significant. However, the rule to be deduced from this situation is less clear. How should she consider zeros that lie to the right of the last nonzero digit? The zero after the three in 90.30 is significant because it gives information about precision—it is not a "necessary" zero, in the sense that mathematically 90.30 could be written 90.3. However, the two zeros after the three in 90300 are mathematically necessary as place-holders; 90300 is certainly not the same number as 9030 or 903. So you cannot conclude that they are there to convey information about precision. They might be—in the example, the first one is and the second one is not—but it is not guaranteed. The following rules summarize the previous discussion.

Any nonzero digit is significant.
Any zero that lies between non-zero digits is significant.
Any zero that lies to the left of the first non-zero digit is not significant.
If the number contains a decimal point, then any zero to the right of the last non-zero digit is significant.
If the number does not contain a decimal point, then any zero to the right of the last non-zero digit is not significant.

Here are some examples:

0.030700 — five significant digits (all but the first two zeros);

400.00 — five significant digits;

400 — one significant digit (the four); and

5030 — three significant digits.

Consider the last example. One has seen earlier that the zero in the ones column cannot be considered to be significant, because it may be that the measurement was rounded to the nearest multiple of ten. If, however, someone is measuring with an instrument that is marked in intervals of one unit, then recording the measurement as 5030 does not convey the full level of precision of the measurement. One cannot record the measurement as 5030.0, because that indicates a jump from three to five significant digits, which is too many. In such a circumstance, avoid the difficulty by recording the measurement using scientific notation, i.e. writing 5030 as 5.030 × 10³. Now the final zero lies to the right of the decimal point, so by rule four, it is a significant digit.

Calculations Involving Significant Figures

Often, when doing a scientific experiment, it is necessary not only to record data but to do computations with the information. The main principle involved is that one can never gain significant digits when computing with data; a result can only be as precise as the information used to get it.

In the first example, two pieces of data, 10.30 and 705, are to be added. Mathematically, the sum is 715.30. However, the number 705 does not have any significant digits to the right of the decimal point. Thus the three and the zero in the sum cannot be assumed to have any degree of accuracy, and so the sum is simply written 715. If the first piece of data were 10.65 instead of 10.30, then the actual sum would be 715.65. In this case, the sum would be rounded up and recorded as 716. The rule for numbers to be added (or subtracted) with significant digits is that the sum/difference cannot contain a digit that lies in or to the right of any column containing an insignificant digit.

In a second example, two pieces of data, 10.30 and 705, are to be multiplied. Mathematically, the product is 7261.5. Which of these digits can be considered significant? Here the rule is that when multiplying or dividing, the product/quotient can have only as many significant digits as the piece of data containing the fewest significant digits. So in this case, the product may only have three significant digits and would be recorded as 7260.

In the final example, the diameter of a circle is measured to be 10.30 centimeters. To compute the circumference of this circle, this quantity must be multiplied by . Since is a known quantity, not a measurement, it is considered to have infinitely many significant digits. Thus the result can have as many significant digits as 10.30—that is, four. (To do this computation, take an approximation of to as many decimal places as necessary to guarantee that the first four digits of the product will be accurate.) A similar principle applies if someone wishes to do any strictly mathematical computation with data: for instance, if one wants to compute the radius of the circle whose diameter she has measured, multiply by 0.5. The answer, however, may still have four significant digits, because 0.5 is not a measurement and hence is considered to have infinitely many significant digits.

see also Accuracy and Precision; Estimation; Rounding.

Naomi Klarreich