Dagmar’s measurement incurred the greater percent error. Reginald’s scale reported with an error of 
2. Jeweler A’s official average measurement was 0.864 grams, and Jeweler B’s official measurement was 0.856 grams. You determine these averages by adding up each jeweler’s measurements and then dividing by the total number of measurements, in this case three. Based on these averages, Jeweler B’s official measurement is more accurate because it’s closer to the actual value of 0.856 grams.
However, Jeweler A’s measurements were more precise because the differences between A’s measurements were much smaller than the differences between B’s measurements. Despite the fact that Jeweler B’s average measurement was closer to the actual value, the range of his measurements (that is, the difference between the largest and the smallest measurements) was 0.041 grams (
This example shows how low-precision measurements can yield highly accurate results through averaging of repeated measurements. In the case of Jeweler A, the error in the official measurement was
Identifying Significant Figures
Significant figures (no, we’re not talking about supermodels) are the number of digits that you report in the final answer of the mathematical problem you’re calculating. If we told you that one student determined the density of an object to be 2.3 g/mL and another student figured the density of the same object to be 2.272589 g/mL, we bet that you’d believe that the second figure was the result of a more accurate experiment. You may be right, but then again, you may be wrong. You have no way of knowing whether the second student’s experiment was more accurate unless both students obeyed the significant figure convention.
If we ask you to count the number of automobiles that you and your family own, you can do it without any guesswork involved. Your answer may be 0, 1, 2, or 10, but you know exactly how many autos you have. Those numbers are what are called counted numbers. If we ask you how many inches are in a foot, your answer will be 12. That number is an exact number – it’s exact by definition. Another exact number is the number of centimeters per inch, 2.54. In both exact and counted numbers, you have no doubt what the answer is. When you work with these types of numbers, you don’t have to worry about significant figures.
Now suppose that we ask you and four of your friends to individually measure the length of an object as accurately as you possibly can with a meter stick. You then report the results of your measurements: 2.67 meters, 2.65 meters, 2.68 meters, 2.61 meters, and 2.63 meters. Which of you is right? You are all within experimental error. These measurements are measured numbers, and measured values always have some error associated with them. You determine the number of significant figures in your answer by your least reliable measured number.
Remember: When you report a measurement, you should include digits only if you’re really confident about their values. Including a lot of digits in a measurement means something – it means that you really know what you’re talking about – so we call the included digits significant figures. The more significant figures (sig figs) in a measurement, the more accurate that measurement must be. The last significant figure in a measurement is the only figure that includes any uncertainty, because it’s an estimate. Here are the rules for deciding what is and what isn’t a significant figure:
✔ Any nonzero digit is significant. So 6.42 contains three significant figures.
✔ Zeros sandwiched between nonzero digits are significant. So 3.07 contains three significant figures.
✔ Zeros on the left side of the first nonzero digit are not significant. So 0.0642 and 0.00307 each contain three significant figures.
✔ One or more final zeros (zeros that end the measurement) used after the decimal point are significant. So 1.760 has four significant figures, and 1.7600 has five significant figures. The number 0.0001200 has only four significant figures because the first zeros are not final.
✔ When a number has no decimal point, any zeros after the last nonzero digit may or may not be significant. So in a measurement reported as 1,370, you can’t be certain whether the 0 is a certain value or is merely a placeholder.
Be a good chemist. Report your measurements in scientific notation to avoid such annoying ambiguities. (See the earlier section “Using Exponential and Scientific Notation to Report Measurements” for details on scientific notation.)
✔ If a number is already written in scientific notation, then all the digits in the coefficient are significant. So the number
✔ Numbers from counting (for example, 1 kangaroo, 2 kangaroos, 3 kangaroos) or from defined quantities (say, 60 seconds per 1 minute) are understood to have an unlimited number of significant figures. In other words, these values are completely certain.
Remember: The number of significant figures you use in a reported measurement should be consistent with your certainty about that measurement. If you know your speedometer is routinely off by 5 miles per hour, then you have no business protesting to a policeman that you were going only 63.2 mph in a 60 mph zone.
Example
Q. How many significant figures are in the following three measurements?
a.
b.
c.
A. a) Five, b) three, and c) four significant figures. In the first measurement, all digits are nonzero, except for a 0 that’s sandwiched between nonzero digits, which counts as significant. The coefficient in the second measurement contains only nonzero digits, so all three digits are significant. The coefficient in the third measurement contains a 0, but that 0 is the final digit and to the right of the decimal point, so it’s significant.
Practice Questions
1. Identify the number of significant figures in each measurement:
a.
b. 0.000769 meters
c. 769.3 meters
2.
