A. Accuracy and Precision
Chemistry is an exact science; its development has been based on careful measurements of properties of matter and careful observations of changes in these properties. Measurements in chemistry must be both accurate and precise (Figure 2.4).

FIGURE 2.4 Accuracy and precision: Accurate measurements are close to the true value; precise measurements are close to one another.

An accurate measurement is one that is close to the actual value of the property being measured. The accuracy of a measurement depends on the calibration of the tool used to make the measurement. For example, if you are measuring the distance between two cities by driving between them, an accurate measurement requires that the odometer in your car read 1.00 km for each kilometer driven. If it reads only 0.95 km for each kilometer driven, the accuracy of your measurement will be reduced.

A precise measurement is one that can be reproduced. For example, the driving distance between Detroit and Chicago is 493 km. An accurate odometer - one that reads 1.00 km for every kilometer traveled - measures the distance between these two cities as 493 km. However, an odometer that reads 1.00 km for every 0.05 km traveled will measure the Detroit-Chicago distance as 519 km. Each time it is used on the trip, the inaccurate odometer records the same value of 519 km. The odometer reading is precise because it can be reproduced time after time. However, it is not accurate because the odometer itself is not properly calibrated. Note that accuracy requires precision, but precision does not guarantee accuracy.

The importance of obtaining measurements that are both accurate and precise is rarely greater than in a medical laboratory. Patients and doctors alike want to be certain that instruments give readings that are both precise and accurate. For this reason, instruments used in medical laboratories are calibrated each day (and often at the beginning of each shift) against samples of known concentrations. Periodically, accrediting agencies send the laboratories samples to analyze. The results obtained on these samples must be accurate within the range allowed. Precision is not enough; the determinations must also be accurate.

B. Exponential Notation
Measurements in chemistry often involve very large or very small numbers. An example of a very small number is the mass of a hydrogen atom, given by:

Mass of a hydrogen atom = 0.000 000 000 000 000 000 000 001 67 g

At the opposite extreme is the mass of the Earth:

Mass of the Earth = 5,976,000,000,000,000,000,000,000,000 g

Such numbers are hard to deal with, difficult to copy without making mistakes, and almost impossible to name. To simplify the writing and tabulating of such numbers, we use exponential notation (also called scientific notation). When exponential notation is used, the measurement is expressed as a number between 1 and 10 multiplied by a power of 10. Table 2.2, which lists the SI prefixes, shows the decimal equivalent of some powers of 10. When exponential notation is used to express a measurement greater than 1, the original number is expressed as a number between 1 and 10 multiplied by 10ⁿ, where n is the number of places the decimal point was moved to the left.

If the number to be expressed in exponential notation is less than 1, the original number is expressed as a number between 1 and 10 multiplied by 10 ^-n, where n equals the number of places the decimal point was moved to the right.

C. Uncertainty in Measurement; Significant Figures
Each time we make a measurement of length, volume, mass, or any other physical quantity, the measurement has some degree of uncertainty. Suppose you have a quantity of liquid whose volume you wish to measure. You are given three different containers in which you might make the measurement - a 50-mL beaker, a 50-mL graduated cylinder, and a 50-mL buret. Figure 2.5 shows these containers, each holding an identical volume of liquid.

FIGURE 2.5 Experimental uncertainty in measuring volume.

Look first at the 50-mL beaker [Figure 2.5(a)]. It has divisions or calibrations every 10 mL. You can see that the level of the liquid in the beaker is between the 20-mL and 30-mL marks. If you look more closely, you can see that the level of liquid is approximately midway between the two marks. You estimate that the volume is 25 mL; however, there is some uncertainty. The volume could be as little as 24 mL or it could be as much as 26 mL. If you record this volume as

In Figure 2.5(b), the same volume of liquid has been placed in a 50-mL graduated cylinder. Divisions on the cylinder are marked every 1 mL. You can read that the volume is between 25 mL and 26 mL and estimate that it is about 0.2 mL above the 25-mL mark. However, it could be as little as 25.1 mL or as much as 25.3 mL. Therefore, you should record the volume of the liquid as

To summarize, the uncertainty of any measurement is assumed to be ±1 in the last recorded digit. This uncertainty is rarely shown but is understood to be present. For example, if we write a measurement as 372, we understand that the uncertainty is ±1; if we write 0.017, we understand that the uncertainty is ±0.001.

Uncertainty in measurements is indicated by the number of significant figures used. Significant figures (or significant digits) are all those figures measured plus one that is estimated. Using our volume measurements taken from Figure 2.5, we count the significant figures as follows:

1. Zero as a significant figure
A zero that serves only to locate the decimal point is not significant; zeros that are not needed to locate the decimal point are significant, for they report a measurement. If the above measurements were given in terms of liters, they would be 0.025 L, 0.0252 L, and 0.02528 L. The number of significant figures in each measurement is the same as before; the zeros have been added only to show the location of the decimal point.

Suppose you had reported the volume of liquid in a buret as 30.50 mL, or 0.03050 L. Are any of these zeros significant? The zeros to the left of the 3 are not significant, for their purpose is to locate the decimal point. The zero between the 3 and the 5 is significant because it shows that the measured volume in that place is 0. The zero after the 5 is also significant. It does not locate the decimal point; rather, it reports a measurement.

The use of exponential notation clarifies the significant figures. Any zero that disappears when a number is expressed exponentially is not significant. For example, the mass of a hydrogen atom has been given as

0.000 000 000 000 000 000 000 001 67 g

In exponential notation this number becomes

1.67 X 10^-24g

Because the zeros in the number have disappeared, we know that they merely showed the location of the decimal point and the magnitude of the number; they were not significant. Similarly, the mass of the Earth expressed exponentially is

5.976 X 10²⁷g

The zeros shown in the original expression of the measurement (Section 2.2B) have disappeared; they were not significant. Table 2.6 gives further examples.

A problem arises when a zero shows both a measurement and the location of the decimal point. The problem is solved by putting a decimal point after such a zero. Thus 250. means that the zero reflects a measurement; 250 means that the zero shows only the magnitude of the number. Similarly, 480,000 means the same as 4.8 X 10⁵, but 480,000. means 4.80000 X 10⁵.

2. Rounding off
When a calculation is performed, the number of significant figures in the numerical answer is determined by the precision of the measurements used in the calculation. It is often necessary to round off the calculated result to the proper number of significant figures. The rules for rounding off are:

1. If the digit following the last one to be kept is less than 5, all unwanted digits are discarded. For example, to report the quantity 36.723 mL using four significant figures, write 36.72 mL; to report it using three significant figures, write 36.7 mL.
2. If the digit following the last one to be kept is 5 or greater, the digit to be kept is increased by 1. Thus, to report 38.785 mL using four significant figures, write 38.79 mL; for three significant figures, write 38.8 mL; for two significant figures, write 39 mL.
   

3. The use of significant figures in calculations
Most often the measurements we make are not final answers in themselves. Rather, they are used in further calculations involving addition, subtraction, multiplication, or division. These calculations cannot improve the accuracy of the measurements but must record them.

Multiplication and division.
In calculations involving multiplication and division, the answer should contain the same number of significant figures as the measurement in the calculation that contains the fewest significant figures.

Addition and subtraction.
In calculations involving addition or subtraction, the answer can show only as many decimal places as are common to all the measurements used in the calculation. Note that the location of the decimal point in each of the measurements, rather than the number of significant figures, determines the number of significant figures in the answer.

4. Significant figures and electronic calculators
The use of electronic calculators has greatly increased the importance of understanding significant figures and of observing the rules governing their use. A calculator with an eight-digit display capability may display eight digits in the answer to a calculation, regardless of the number of digits entered. For example, dividing 5.0 by 1.67 on a calculator may give the following answer:

The correct answer, 3.0, has only two significant figures, as in the least accurate number (5.0) in the problem. All other digits displayed by the calculator are insignificant.