Measurement and Its Uncertainties

Using and Expressing Measurements 3.1 Using and Expressing Measurements Using and Expressing Measurements How do measurements relate to science?

Using and Expressing Measurements 3.1 Using and Expressing Measurements A measurement is a quantity that has both a number and a unit. It is important to make measurements and to decide whether a measurement is correct.

Using and Expressing Measurements 3.1 Using and Expressing Measurements In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power. The number of stars in a galaxy is an example of an estimate that should be expressed in scientific notation. Expressing very large numbers, such as the estimated number of stars in a galaxy, is easier if scientific notation is used.

Accuracy, Precision, and Error 3.1 Accuracy, Precision, and Error Accuracy, Precision, and Error How do you evaluate accuracy and precision?

Accuracy, Precision, and Error 3.1 Accuracy, Precision, and Error Accuracy and Precision Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured. Precision is a measure of how close a series of measurements are to one another.

Accuracy, Precision, and Error 3.1 Accuracy, Precision, and Error To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements.

Accuracy, Precision, and Error 3.1 Accuracy, Precision, and Error The distribution of darts illustrates the difference between accuracy and precision. a) Good accuracy and good precision: The darts are close to the bull’s-eye and to one another. b) Poor accuracy and good precision: The darts are far from the bull’s-eye but close to one another. c) Poor accuracy and poor precision: The darts are far from the bull’s-eye and from one another.

Accuracy, Precision, and Error 3.1 Accuracy, Precision, and Error Determining Error The accepted value is the correct value based on reliable references. The experimental value is the value measured in the lab. The difference between the experimental value and the accepted value is called the error.

Accuracy, Precision, and Error 3.1 Accuracy, Precision, and Error The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%. Expressing very large numbers, such as the estimated number of stars in a galaxy, is easier if scientific notation is used.

Accuracy, Precision, and Error 3.1 Accuracy, Precision, and Error

Accuracy, Precision, and Error 3.1 Accuracy, Precision, and Error Just because a measuring device works, you cannot assume it is accurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate. There is a difference between the person’s correct weight and the measured value. Calculating What is the percent error of a measured value of 114 lb if the person’s actual weight is 107 lb?

Significant Figures in Measurements 3.1 Significant Figures in Measurements Significant Figures in Measurements Why must measurements be reported to the correct number of significant figures?

Significant Figures in Measurements 3.1 Significant Figures in Measurements Suppose you estimate a weight that is between 2.4 lb and 2.5 lb to be 2.46 lb. The first two digits (2 and 4) are known. The last digit (6) is an estimate and involves some uncertainty. All three digits convey useful information, however, and are called significant figures. The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated.

Significant Figures in Measurements 3.1 Significant Figures in Measurements The calculated answers often depend on the number of significant figures in the values used in the calculation.

Significant Figures in Measurements 3.1 Significant Figures in Measurements Rules for determining significant figures: 1.) Every nonzero digit in a measurement is significant. examples: all these measurements have 3 significant figures in them: 24.7 meters 0.743 meter 714 meters

Significant Figures in Measurements 3.1 Significant Figures in Measurements 2.) Zeros appearing between nonzero digits are significant. examples: These have 4 in them: 7003 meters 40.79 meters 1.503 meters

Significant Figures in Measurements 3.1 Significant Figures in Measurements 3.) Leftmost zeros appearing in front of nonzero digits are not significant. examples: these have only 2: 0.0071 meter 0.42 meter 0. 0000099 meter

Significant Figures in Measurements 3.1 Significant Figures in Measurements 4.) Zeros at the end of a number and to the right of a decimal point are always significant. examples: these have 4 in them: 43.00 meters 1.010 meters 9.000 meters

Significant Figures in Measurements 3.1 Significant Figures in Measurements 5.) Zeros at the rightmost end of a measurement that lie to the left of an understood decimal point are not significant if they show as placeholders. examples: 300 meters = 1 7000 meters = 1 27210 meters = 4

Significant Figures in Measurements 3.1 Significant Figures in Measurements 6.) All digits in a scientific notation is significant. ex. 1.28 x 10 3 = 3 5. 1 x 10 4 = 2 6.000245 x 10 9 = 7

Significant Figures in Measurements Animation 2 See how the precision of a calculated result depends on the sensitivity of the measuring instruments.

for Conceptual Problem 3.1 Problem Solving 3.2 Solve Problem 2 with the help of an interactive guided tutorial.

Significant Figures in Calculations 3.1 Significant Figures in Calculations Rounding To round a number, you must first decide how many significant figures your answer should have. The answer depends on the given measurements and on the mathematical process used to arrive at the answer.

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for Sample Problem 3.1 Problem Solving 3.3 Solve Problem 3 with the help of an interactive guided tutorial.

Significant Figures in Calculations 3.1 Significant Figures in Calculations Addition and Subtraction The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places.

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for Sample Problem 3.2 Problem Solving 3.6 Solve Problem 6 with the help of an interactive guided tutorial.

Significant Figures in Calculations 3.1 Significant Figures in Calculations Multiplication and Division In calculations involving multiplication and division, you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures.

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for Sample Problem 3.3 Problem Solving 3.8 Solve Problem 8 with the help of an interactive guided tutorial.

Section Assessment 3.1.

3.1 Section Quiz 1. In which of the following expressions is the number on the left NOT equal to the number on the right? 0.00456  10–8 = 4.56  10–11 454  10–8 = 4.54  10–6 842.6  104 = 8.426  106 0.00452  106 = 4.52  109

3.1 Section Quiz 2. Which set of measurements of a 2.00-g standard is the most precise? 2.00 g, 2.01 g, 1.98 g 2.10 g, 2.00 g, 2.20 g 2.02 g, 2.03 g, 2.04 g 1.50 g, 2.00 g, 2.50 g

3.1 Section Quiz 3. A student reports the volume of a liquid as 0.0130 L. How many significant figures are in this measurement? 2 3 4 5

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