Physics Biology Education

insider the following two statements. First, your friend tells you, "It pick you up at your place at 7:15." Second. the daily newspaper tells you, 'The ninon will rise tonight at 7:15."

Mile these two time values look the same, they are really very different. You would not be surprised if your friend showed up at 7:10 or 7:20. On the other hand. if the moon peeked over the horizon at 7:16, an astronomer might scion he looking for a job. The magnitudes of these two measurement statements are the same, but their uncertainties are very different.

No measurement is perfect. Consider the measurement of the height of a cylinder, shown in Figure 1-1. The centimeter scale, even if you paid a lot for it, is not perfect. Whoever uses it does not have infinitely perfect eye­sight. It might not he perfectly aligned with the cylinder. and the cylinder might not he perfectly uniform all the way around. These pitfalls in the process of measurement can he reduced. but they can never be completely dim inated. Furthermore, measurements are necessarily inexact because of the nature of the thing being measured. What is the diameter of a tennis balk It is so fuzzy that no sharp boundary exists between the ball and the space it is in. Ultimately, on the atomic scale, all surfaces are hwy.

In scientific work, every measured value must he accompanied by a state¬ment of its uncertainty. The height of the cylinder in Figure 1-1 would be given as 11.4 centimeters t0.I centimeter; the 10.1 centimeter is the limits of uncertainty in the measurement. nit means that the actual height of the cylinder is probably between 11.3 centimeters and 11.5 centimeters. These limits are not absolute but only probabilistic; even the uncertainty has its uncertainty.

If the limits of uncet tainty are t0.1 centimeter, is the measurement highly accurate? It depends. Finding the height of the cylinder within 0.1 centimeters is not difficult. However, in measuring the distance to the moon, an uncertainty of 0.1 centimeter would be an extraordinatv level of accuracy. It would require a lot of complex and expensive equipment. The accuracy of a measurement is its limits of uncertainty compared with the men¬surrment itself

Accuracy is the ratio of the limits of uncertainty to the measurement, usually expressed as a percent. For the measurement of the cylinder, the accuracy (percent uncertainty) is



if the pert tint uncertainty is known, it is simple to find the limits of uncertainty. Just multiply the magnitude of the measurement by its percent uncertainty. For example, what are the limits of uncertainty of a mass mea¬surement given as 232 grams ± 2%?