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# Significant Others: Sig Figs

Mike Isley
Product Developer

Science is all about measurement. Any reported data must always reflect the precision of the measuring instrument. For example, if you record the mass of a coin as 5 g in a lab report, there is no way of knowing the precision of the balance used for this mass. However, if you record the measurement as 5.00 g or 5.000 g, then we know the precision of the instrument. A measurement of 5.00 g expresses use of a balance sensitive to a hundredth of a gram and a measurement of 5.000 g expresses use of a balance sensitive to a thousandth of a gram. If a digit is part of a measurement, then it is a significant digit or a significant figure (sig fig).

### Determining the number of significant figures

How can you determine the number of sig figs in a given number? Just remember that any digit that is part of a measurement is significant. Here are a few simple rules:

• All digits other than zero are significant. Example: 12.567 cm (5 sig figs)
• Zeros between 2 other numbers are significant. Examples: 11.05 g, 10.02 g (4 sig figs)
• Leading zeros are not significant. They are only placeholders to locate the decimal and are not part of a measurement. Example: 0.0015 g (2 sig figs)
• Only trailing zeros in a number with a decimal point are significant. Example: 130.00 g (5 sig figs). The number 1,300 has trailing zeros, but no decimal point. The zeros are just placeholders (2 sig figs). If the number had been 1,300., the expressed decimal would make all 4 digits significant

### Operations and significant figures

When doing mathematical operations with a calculator, answers may be expressed with multiple decimal places. How do we know how many to keep in our answers? When adding, subtracting, multiplying, or dividing, the accuracy of an answer can be no more accurate than the least accurate measurement.

For these 2 operations, accuracy depends on decimal places. The answers to addition and subtraction for sets of data cannot contain any more decimal places than the least accurate measurement (the one with the fewest decimal places). In the example below, the least accurate measurement is the mass of water. Therefore, the answer is rounded to 1 decimal place.

135.0 g H2O
+  1.686 g NaCl
136.7 g solution
• Multiplication and division
For these 2 operations, accuracy is dependent upon the number of sig figs. The answer to a calculation cannot contain any more sig figs than the least accurate measurement (the one with the fewest sig figs). See the examples for density and volume calculations below.

Density = 60.48 g  =  27 g/cm3        Volume = 2.2 cm × 1.15 cm × 3.586 cm = 9.1 cm3
2.2 cm3

### Scientific notation

One way to remove ambiguity regarding zeros is to express all measurements in scientific notation. All digits in scientific notation are significant. For example, if a measurement were written as 1,500 grams, the only digits that are significant would be the 1 and 5. The zeros are trailing zeros in the absence of a decimal point and therefore could be the result of rounding or they could be part of an exact number. If the measurement is exactly 1,500 grams, then in scientific notation it would be expressed as 1.500 × 103 g. The trailing zeros would be significant due to the presence of the decimal point.

### Rounding to the proper number of significant digits

Rounding to a given number of sig figs for a calculation is not a problem if the digit past the one you want to keep is greater than 5 (round the preceding digit up) or less than 5 (keep preceding digit as is). For example, find the volume with the following measurements: 2.0 cm × 3.20 cm × 5.500 cm = 35.2 cm3. The answer has to be rounded to 2 sig figs. Since the third digit is less than 5, the answer becomes 35. Had the answer been 35.6, it would have rounded up to 36.

What is the protocol if the digit past the one to keep is a 5? Mathematicians would say that if the digit is 5 or greater, round up the previous digit. Most scientists would say to look at the previous digit to keep and note if it is even or odd. If it is even (0 is also considered even), keep the number as is, and if it is odd, round up by 1. This would not inflate the final answer for a large set of data containing the number 5 past the one to keep, because the even/odd digits to keep would be statistically around 50% each for large sets of data. Thus, there is no inflation of the final answer.

Example: Round each measurement to 3 sig figs and add. The first column of numbers is rounded by even or odd digits for the third sig fig preceding the 5 and the second column is by “traditional” rounding. Notice how the rounding up of all numbers in the second column results in an inflated final answer as compared to the unrounded answer of 468.5.

123.5 = 124                             123.5 = 124
234.5 = 234                             234.5 = 235
+110.5 = 110                           +110.5 = 111
468.5    468                             468.5    470

### Student competition for learning significant figures

For a competitive activity to help students grasp the concept of sig figs, see our Carolina Tips® article “Racing for Significant Figures,” written by Shana McAlexander.