1.11:
Uncertainty in Measurement: Significant Figures
All of the numbers in a scientific measurement are certain, except for the last digit. The certainty of measurement depends on two factors: the number of digits in the measurement and the precision of the instrument used.
In a measured quantity, all of the digits, including the last uncertain digit, are called significant figures and can be determined using specific rules.
Any non-zero digits and all captive zeros, which lie between two non-zero digits, are significant. For example, 28 has two significant figures, while 26.25 has four, and 208 has three.
Leading zeros are never significant, as they just locate the decimal point. For example, 0.00208 has three significant figures. Such quantities can be expressed using exponential notations. Thus, 0.00208 can be written as 2.08 × 10−3.
Trailing zeros are only significant in decimal formatted numbers. 2200 has two trailing zeros and two significant figures, whereas 2200.0 and 2200.1 both have 5 significant figures.
For quantities without decimal points, the significance of trailing zeros becomes ambiguous. Thus, 2200 can be written as 2.2 × 103 with two significant figures or 2.20 × 103 with three significant figures.
Significant figures help achieve certainty in mathematical operations, too. In addition or subtraction, the result should be rounded off to have the same number of decimal places as the measurement with the fewest decimal places.
Rounding down should be performed when the last digit is below 5, and rounding up carried out when it is 5 or above. Other rounding methods are sometimes used when the last digit is 5. For instance, the sum of 2.052 and 1.2 is rounded off as 3.3.
However, while multiplying or dividing, the result should be rounded to have the same number of significant figures as the measurement with the fewest significant figures. Thus, the product of 2.052 and 1.2 is rounded off as 2.5.
Scientists often repeat experiments to achieve precision in their measurements. Standard deviation is the statistical expression of such precision and measures the dispersion from the expected value. If precision is high, the standard deviation is small, and vice versa.
For example, two groups measured the thickness of a book in centimeters. They found the same average: 10.6 cm. However, the measurements by the first group are more precise, and thus have a lower standard deviation. The second group has more spread out measurements and a higher standard deviation.
1.11:
Uncertainty in Measurement: Significant Figures
All the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits. Note that zero may be a measured value; for example, if a scale that shows weight to the nearest pound reads “140,” then the 1 (hundreds), 4 (tens), and 0 (ones) are all significant (measured) values.
A measurement result is properly reported when its significant digits accurately represent the certainty of the measurement process. Below are a set of rules to determine the number of significant figures in a measurement:
- All nonzero digits are significant. Starting with the first nonzero digit on the left, count this digit and all remaining digits to the right. This is the number of significant figures in the measurement. For example, 843 has three significant digits, 843.12 has 5 significant digits.
- Captive zeros, which are zeros between two nonzero digits, are significant. For example, 808.101 has two captive zeros and 6 significant figures.
- Leading zeros are zeros to the left of the first nonzero digit. Leading digits are never significant; they merely represent the position of the decimal point. For example, the leading zeros in 0.008081 are not significant. This number can be expressed using exponential notation as 8.081 × 10−3, then the number 8.081 contains all the significant figures, and 10−3 locates the decimal point.
- The significance of trailing zeros, which are zeros at the end of a number, depends on their position. Trailing zeros before (but after a nonzero digit) and after the decimal point are significant. However, for numbers that do not have decimal points, trailing zeros may or may not be significant. This ambiguity can be resolved with the use of exponential notation. For example, the measurement 1300 can be written as 1.3 × 103 (two significant figures), 1.30 × 103 (three significant figures, if the tens place was measured), or 1.300 × 103 (four significant figures, if the ones place was also measured).
Significant Figures in Calculations
Uncertainty in measurements can be avoided by reporting the results of calculation with the correct number of significant figures. This can be determined by the following the rules for rounding numbers:
- When adding or subtracting numbers, round the result to the same number of decimal places as the number with the least number of decimal places.
- When multiplying or dividing numbers, round the result to the same number of digits as the number with the least number of significant figures.
- If the digit to be dropped (the one immediately to the right of the digit to be retained) is less than 5, “round down” and leave the retained digit unchanged.
- If the digit to be dropped (the one immediately to the right of the digit to be retained) is 5 or greater, “round up” and increase the retained digit by 1. Alternative rounding methods may also be used if the dropped digit is 5. The retained digit is rounded up or down, whichever yields an even value.
An important note is that rounding of significant figures should preferably be done at the end of a multistep calculation to avoid the accumulation of errors at each step due to rounding. Thus, significant figures and rounding, facilitate correct representation of the certainty of the measured values reported.
This text is adapted from Openstax, Chemistry 2e, Section 1.5: Measurement Uncertainty, Accuracy, and Precision.