Читать книгу U Can: Chemistry I For Dummies - Hren Chris - Страница 9

Whenever you make measurements, you must consider two factors, accuracy and precision. Accuracy is how well the measurement agrees with the accepted or true value. Precision is how well a set of measurements agree with each other. In chemistry, measurements should be reproducible; that is, they must have a high degree of precision. Most of the time chemists make several measurements and average them. The closer these measurements are to each other, the more confidence chemists have in their measurements. Of course, you also want the measurements to be accurate, very close to the correct answer. However, many times you don’t know beforehand anything about the correct answer; therefore, you have to rely on precision as your guide.

Suppose you ask four lab students to make three measurements of the length of the same object. Their data follows:

The accepted length of the object is 27.55 cm. Which of these students deserves the higher lab grade? Both students 1 and 3 have values close to the accepted value, if you just consider their average values. (The average, found by summing the individual measurements and dividing by the number of measurements, is normally considered to be more useful than any individual value.) Both students 1 and 3 have made accurate determinations of the length of the object. The average values determined by students 2 and 4 are not very close to the accepted value, so their values are not considered to be accurate.

However, if you examine the individual determinations for students 1 and 3, you notice a great deal of variation in the measurements of student 1. The measurements don’t agree with each other very well; their precision is low even though the accuracy is good. The measurements by student 3 agree well with each other; both precision and accuracy are good. Student 3 deserves a higher grade than student 1.

Neither student 2 nor student 4 has average values close to the accepted value; neither determination is very accurate. However, student 4 has values that agree closely with each other; the precision is good. This student probably had a consistent error in his or her measuring technique. Student 2 had neither good accuracy nor precision. The accuracy and precision of the four students is summarized below.

Usually, measurements with a high degree of precision are also somewhat accurate. Because the scientists or students don’t know the accepted value beforehand, they strive for high precision and hope that the accuracy will also be high. This was not the case for student 4.

So remember, accuracy and precision are not the same thing:

✔ Accuracy: Accuracy describes how closely a measurement approaches an actual, true value.

✔ Precision: Precision, which we discuss more in the next section, describes how close repeated measurements are to one another, regardless of how close those measurements are to the actual value. The bigger the difference between the largest and smallest values of a repeated measurement, the less precision you have.

The two most common measurements related to accuracy are error and percent error:

✔ Error: Error measures accuracy, the difference between a measured value and the actual value:

✔

Percent error: Percent error compares error to the size of the thing being measured:

Being off by 1 meter isn’t such a big deal when measuring the altitude of a mountain, but it’s a shameful amount of error when measuring the height of an individual mountain climber.

Examples

Q. A police officer uses a radar gun to clock a passing Ferrari at 131 miles per hour (mph). The Ferrari was really speeding at 127 mph. Calculate the error in the officer’s measurement.

A. –4 mph. First, determine which value is the actual value and which is the measured value:

✔ Actual value = 127 mph

✔ Measured value = 131 mph

Then calculate the error by subtracting the measured value from the actual value:

Q. Calculate the percent error in the officer’s measurement of the Ferrari’s speed.

A. 3.15 %. First, divide the error’s absolute value (the size, as a positive number) by the actual value:

Next, multiply the result by 100 to obtain the percent error:

Practice Questions

1. Two people, Reginald and Dagmar, measure their weight in the morning by using typical bathroom scales, instruments that are famously unreliable. The scale reports that Reginald weighs 237 pounds, though he actually weighs 256 pounds. Dagmar’s scale reports her weight as 117 pounds, though she really weighs 129 pounds. Whose measurement incurred the greater error? Who incurred a greater percent error?

2. Two jewelers were asked to measure the mass of a gold nugget. The true mass of the nugget is 0.856 grams (g). Each jeweler took three measurements. The average of the three measurements was reported as the “official” measurement with the following results:

✔ Jeweler A: 0.863 g, 0.869 g, 0.859 g

✔ Jeweler B: 0.875 g, 0.834 g, 0.858 g

Which jeweler’s official measurement was more accurate? Which jeweler’s measurements were more precise? In each case, what was the error and percent error in the official measurement?

Practice Answers

1. Reginald’s measurement incurred the greater magnitude of error, and Dagmar’s measurement incurred the greater percent error. Reginald’s scale reported with an error of , and Dagmar’s scale reported with an error of . Comparing the magnitudes of error, you see that 19 pounds is greater than 12 pounds. However, Reginald’s measurement had a percent error of , while Dagmar’s measurement had a percent error of .

2. Jeweler A’s official average measurement was 0.864 grams, and Jeweler B’s official measurement was 0.856 grams. You determine these averages by adding up each jeweler’s measurements and then dividing by the total number of measurements, in this case three. Based on these averages, Jeweler B’s official measurement is more accurate because it’s closer to the actual value of 0.856 grams.

However, Jeweler A’s measurements were more precise because the differences between A’s measurements were much smaller than the differences between B’s measurements. Despite the fact that Jeweler B’s average measurement was closer to the actual value, the range of his measurements (that is, the difference between the largest and the smallest measurements) was 0.041 grams (). The range of Jeweler A’s measurements was 0.010 grams ().

This example shows how low-precision measurements can yield highly accurate results through averaging of repeated measurements. In the case of Jeweler A, the error in the official measurement was . The corresponding percent error was . In the case of Jeweler B, the error in the official measurement was . Accordingly, the percent error was 0 %.