Chemistry: 1.4 - Uncertainty in Measurments

Although precision and accuracy are frequently used interchangeably in everyday life, they have different meanings in the scientific context. What is the difference between precision and accuracy?

Accuracy refers to the agreement of a particular value with the true value. Precision refers to the degree of agreement among several measurements of the same quantity. Precision reflects the reproducibility of a given type of measurement. The difference between precision and accuracy is illustrated by the results of three different dart throws shown in Fig. 1.8.

figure 1.8

Dart Board A: neither accurate nor precise -darts not near center and scattered all over board Dart Board B: precise but not accurate -darts not near center but are close together Dart Board C: accurate and precise -all darts on (or almost on) bullseye

What is precision often used as in quantitative work?

In quantitative work, precision is often used as an indication of accuracy

systematic error

an error that always occurs in the same direction (also called a determinate error)

random error

an error that has an equal probability of being high or low (also called an indeterminate error) -occurs in estimating the value of the last digit of a measurement.

What are two terms often used to describe the reliability of measurements

precision and accuracy

accuracy

the agreement of a particular value with the true value

significant figures

the certain digits and the first uncertain digit of a measurement

uncertainty (in measurement)

the characteristic that any measurement involves estimates and cannot be exactly reproduced.

precision

the degree of agreement among several measurements of the same quantity; the reproducibility of a measurement

uncertain digit

the digit to that must be estimated and therefore varies

What types of measurement errors are illustrated in Fig. 1.8?

(a) indicates large random errors (poor technique) (b) indicates small random errors but a large systematic error (c) indicates small random errors and no systematic error

For example, consider the measurement of the volume of a liquid using a buret. The meniscus (curved upper surface of a liquid tube) of the liquid occurs at about 20.15 mL. This means that about 20.15 mL of liquid has been delivered from the buret (if the initial position of the liquid meniscus was 0.00 mL). We must estimate the last number of the volume reading by interpolating (adding/inserting/filling in) between the 0.1-mL marks. Since the last number is estimated, its value may be different if another person makes the same measurement. If five different people read the same volume, the results might be as follows: Person / Results of Measurement 1 / 20.15 mL 2 / 20.14 mL 3 / 20.16 mL 4 / 20.17 mL 5 / 20.16 mL These results show that the first three numbers (20.1) remain the same regardless of who makes the measurement; these are called certain digits. However, the digit to the right of the 1 must be estimated and therefore varies; it is called an uncertain digit. We customarily report a measurement by recording all the certain digits plus the first uncertain digit. In our example it would not make any sense to try to record the volume to thousandths of a milliliter because the value for hundredths of a milliliter must be estimated when using the buret.

It is very important to realize that a measurement always has some degree of uncertainty. The uncertainty of a measurement depends on the precision of the measuring device. For example, using a bathroom scale, you might estimate the mass of a grape- fruit to be approximately 1.5 lb. Weighing the same grapefruit on a highly precise balance might produce a result of 1.476 lb. In the first case, the uncertainty occurs in the tenths of a pound place; in the second case, the uncertainty occurs in the thousandths of a pound place. Suppose we weigh two similar grapefruits on the two de- vices and obtain the following results: # / Bathroom Scale / Balance Grapefruit 1 / 1.5 lb / 1.476 lb Grapefruit 2 / 1.5 lb / 1.518 lb Do the two grapefruits have the same mass? The answer depends on which set of results you consider. Thus a conclusion based on a series of measurements depends on the certainty of those measurements. For this reason, it is important to indicate the uncertainty in any measurement. This is done by always recording the certain digits and the first uncertain digit (the estimated number). These numbers are called the significant figures of a measurement. The convention of significant figures automatically indicates something about the uncertainty in a measurement. The uncertainty in the last number (the estimated number) is usually assumed to be ± 1 unless otherwise indicated. For example, the measurement 1.86 kg can be taken to mean 1.86 ± 0.01 kg.

How is the number associated with a measurement obtained?

The number associated with a measurement is obtained using some measuring device.

certain digit

The numbers that remain the same regardless of who makes the measurement