1.5 Uncertainty in Measurement

In scientific work we recognize two kinds of numbers: exact numbers (those whose values are known exactly) and inexact numbers (those whose values have some uncertainty). Exact numbers are those that have defined values or are integers that result from counting numbers of objects. For example, by definition, there are exactly 12 eggs in a dozen, exactly 1000 g in a kilogram, and exactly 2.54 cm in an inch. The number 1 in any conversion factor between units, as in 1 m = 100 cm, is also an exact number.

Numbers obtained by measurement are always inexact. There are always inherent limitations in the equipment used to measure quantities (equipment errors), and there are differences in how different people make the same measurement (human errors). Suppose that 10 students with 10 different balances are given the same dime to weigh. The 10 measurements will vary slightly. The balances might be calibrated slightly differently, and there might be differences in how each student reads the mass from the balance. Remember: Uncertainties always exist in measured quantities.

Precision and Accuracy

Two terms are usually used in discussing the uncertainties in measured values: precision and accuracy. Precision is a measure of how closely individual measurements agree with one another. Accuracy refers to how closely individual measurements agree with the correct, or "true" value. The analogy of darts stuck in a dartboard pictured in Figure 1.25 illustrates the difference between the two terms.

Figure 1.25 The distribution of darts on a target illustrates the distinction between accuracy and precision.

In general, the more precise a measurement, the more accurate it is. We gain confidence in the accuracy of a measurement if we obtain nearly the same value in many different experiments. Thus, in the laboratory you will often perform several different "trials" of the same experiment. It is possible, however, for a precise value to be inaccurate. If a very sensitive balance is poorly calibrated, for example, the masses measured will be inaccurate even if they are precise.

Significant Figures

Suppose you weigh a dime on a balance capable of measuring to the nearest 0.0001 g. You could report the mass as 2.2405 0.0001 g. The notation (read "plus or minus") is a useful way to express the uncertainty of a measurement. In much scientific work we drop the notation with the understanding that an uncertainty of at least one unit exists in the last digit of the measured quantity. That is, measured quantities are generally reported in such a way that only the last digit is uncertain. All digits, including the uncertain one, are called significant figures. The number 2.2405 has five significant figures. The number of significant figures indicates the exactness of a measurement.

SAMPLE EXERCISE 1.4

What is the difference between 4.0 g and 4.00 g?

SOLUTION Many people would say there is no difference, but a scientist would note the difference in the number of significant figures in the two measurements. The value 4.0 has two significant figures, while 4.00 has three. This implies that the second measurement is more precise. A mass of 4.0 g indicates that the mass is between 3.9 and 4.1 g; the mass is 4.0 0.1 g. A measurement of 4.00 g implies that the mass is between 3.99 and 4.01 g; the mass is 4.00 0.01 g.

PRACTICE EXERCISE

A balance has a precision of 0.001 g. A sample that weighs about 25 g is weighed on this balance. How many significant figures should be reported for this measurement? Answer: 5

The following guidelines apply to determining the number of significant figures in a measured quantity:

  1. Nonzero digits are always significant—457 cm (3 significant figures); 2.5 g (2 significant figures).
  2. Zeros between nonzero digits are always significant—1005 kg (4 significant figures); 1.03 cm (three significant figures).
  3. Zeros at the beginning of a number are never significant; they merely indicate the position of the decimal point—0.02 g (one significant figure); 0.0026 cm (two significant figures).
  4. Zeros that fall both at the end of a number and after the decimal point are always significant—0.0200 g (3 significant figures); 3.0 cm (2 significant figures).
  5. When a number ends in zeros but contains no decimal point, the zeros may or may not be significant—130 cm (two or three significant figures); 10,300 g (three, four, or five significant figures). We next describe how to remove this ambiguity.

The use of exponential notation (Appendix A) avoids the potential ambiguity of whether the zeros at the end of a number are significant (rule 5). For example, a mass of 10,300 g can be written in exponential notation showing three, four, or five significant figures:

PM01009

PM01010

PM01011

In these numbers all the zeros to the right of the decimal point are significant (rules 2 and 4). (All significant figures come before the exponent; the exponential term does not add to the number of significant figures.)

Exact numbers can be treated as if they have an infinite number of significant figures. This rule applies to many definitions between units. Thus, when we say, "There are 12 inches in 1 foot," the number 12 is exact and we need not worry about the number of significant figures in it.

SAMPLE EXERCISE 1.5

How many significant figures are in each of the following numbers (assume that each number is a measured quantity: (a) 4.003; (b) 6.023 1023; (c) 5000?

SOLUTION (a) Four; the zeros are significant figures. (b) Four; the exponential term does not add to the number of significant figures. (c) One, two, three, or four. In this case the ambiguity could have been avoided by using standard exponential notation. Thus 5 103 has only one significant figure; 5.00 103 has three.

PRACTICE EXERCISE

How many significant figures are in each of the following measurements: (a) 3.549 g; (b) 2.3 104 cm; (c) 0.00134 m3? Answers: (a) four; (b) two; (c) three

Significant Figures in Calculations

In carrying measured quantities through calculations, observe this point: The precision of the result is limited by the precision of the measurements. Thus, you can't get exact calculations using inexact data. To keep track of significant figures in calculations, we will make frequent use of two rules. The first involves multiplication and division, and the second involves addition and subtraction. In multiplication and division the result must be reported with the same number of significant figures as the measurement with the fewest significant figures. When the result contains more than the correct number of significant figures, it must be rounded off.

For example, the area of a rectangle whose edge lengths are 6.221 cm and 5.2 cm should be reported as 32 cm2:

PM01012

We round off to two significant figures because the least precise number—5.2 cm—has only two significant figures.

In rounding off numbers, look at the leftmost digit to be dropped:

  1. If the leftmost digit to be removed is less than 5, the preceding number is left unchanged. Thus, rounding 7.248 to two significant figures gives 7.2.
  2. If the leftmost digit to be removed is 5 or greater, the preceding number is increased by 1. Rounding 4.735 to three significant figures gives 4.74, and rounding 2.376 to two significant figures gives 2.4. (Your instructor may wish you to use a slight variation on the rule when the leftmost digit to be removed is exactly 5, with no following digits or only zeros. One common practice is to round up to the next higher number if that number will be even, and down to the next lower number otherwise. Thus, 4.7350 would be rounded to 4.74, and 4.7450 would also be rounded to 4.74.)

The guidelines used to determine the number of significant figures in addition and subtraction are different from those for multiplication and division. In addition and subtraction the result cannot have more digits to the right of the decimal point than any of the original numbers. In the following example the uncertain digits appear in color:

PM01013

PM01014

PM01015

PM01016

SAMPLE EXERCISE 1.6

A person's height is measured to be 67.50 in. What is this height in centimeters?

SOLUTION There are 2.54 cm in an inch; this is an exact number and can be treated as if it had an infinite number of significant figures. The precision of the answer is thus limited by the measurement in inches and should be reported to four significant figures (67.50 has four significant figures). The answer is

PM01018

PRACTICE EXERCISE

There are exactly 1609.344 m in a mile. How many meters are in a distance of 1.35 mi? Answer: 2.17 103 m

SAMPLE EXERCISE 1.7

A gas at 25°C exactly fills a container previously determined to have a volume of 1.05 103 cm3. The container plus gas are weighed and found to have a mass of 837.6 g. The container, when emptied of all gas, has a mass of 836.2 g. What is the density of the gas at 25°C?

SOLUTION The mass of the gas is just the difference in the two masses: (837.6 - 836.2) g = 1.4 g. Notice that 1.4 g has only two significant figures, even though the masses from which it is obtained have four.

From the definition of density we have

PM01019

PM01020

There are two significant figures in this quantity, corresponding to the smaller number of significant figures in the two numbers that form the ratio.

PRACTICE EXERCISE

To how many significant figures should the mass of the container be measured (with and without the gas) in Sample Exercise 1.7 in order for the density to be calculated to three significant figures? Answer: 5

It is important to have a feeling for significant figures when you use a calculator, because calculators ordinarily display more digits than are significant. For example, a typical calculator would give 1.3333333 10–3 as the answer to the calculation in Sample Exercise 1.7. This result must be rounded off because of the uncertainties in the measured quantities used in the calculation.

When a calculation involves two or more steps, retain at least one additional digit—past the number of significant figures—for intermediate answers. This procedure ensures that small errors from rounding at each step do not combine to affect the final result. In using a calculator, you may enter the numbers one after another, rounding only the final answer. Accumulated rounding-off errors may account for small differences between results you obtain and answers given in the text for numerical problems.