Scientific Measurement Measurements and their Uncertainty Dr. Yager Chapter 3.1

Objectives  Convert measurement to scientific notation  Distinguish between accuracy, precision, and error of a measurement  Determine the number of significant figures in a measurement and in a calculated answer.

A measurement is a quantity that has both a number and a unit. Measurements are fundamental to the experimental sciences. It is important to be able to make measurements and to decide whether a measurement is correct.

 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.

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.

Key Concepts 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.

Determining Errors Determining Errors  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.

The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%. The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%.

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.

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 Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significant figures in the values used in the calculation. Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significant figures in the values used in the calculation.

Rule 1 Every nonzero digit in a reported measurement is assumed to be significant. 24.7, 0.743, 714 all have three significant figures

Rule 2 Zeros between nonzero digits are significant. Zeros between nonzero digits are significant. 7004, 40.79, all have four significant figures

Rule 3 Look for a decimal point. If there is no decimal, then you are done. Look for a decimal point. If there is no decimal, then you are done.Otherwise: Look for zeros at the end of the number after the decimal point - they are significant. Look for zeros at the end of the number after the decimal point - they are significant , 42.0, 9.00 all have three significant figures

Note: Left most zeros in front of nonzero digits are not significant. Left most zeros in front of nonzero digits are not significant , 0.42, all have two significant figures Zeros to the right of nonzero digits but left of implied decimal point are not significant. Zeros to the right of nonzero digits but left of implied decimal point are not significant. 300, 7000, 1,000,000 all have one significant figure

Scientific Notation Scientific notation is a means of expressing significant figures: Scientific notation is a means of expressing significant figures: 300 = 3 x = 3 x ,000.0 = x ,000.0 = x = 4.56 x = 4.56 x The first part of scientific notation holds all the significant figures. The first part of scientific notation holds all the significant figures.

Rule 4 There are two special situations with an unlimited (infinite) number of significant figures: There are two special situations with an unlimited (infinite) number of significant figures: A. Counting i.e. 23 people B. Exactly defined quantities i.e. 60 min = 1 hr

Significant Figures in Calculations In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements from which it was calculated.

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.

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.

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. 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. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements. The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements.

1.In which of the following expressions is the number on the left NOT equal to the number on the right? a)  10 –8 = 4.56  10 –11 b)454  10 –8 = 4.54  10 –6 c)842.6  10 4 =  10 6 d)  10 6 = 4.52  10 9

1.In which of the following expressions is the number on the left NOT equal to the number on the right? a)  10 –8 = 4.56  10 –11 b)454  10 –8 = 4.54  10 –6 c)842.6  10 4 =  10 6 d)  10 6 = 4.52  10 9

2.Which set of measurements of a 2.00 g standard is the most precise? a)2.00 g, 2.01 g, 1.98 g b)2.10 g, 2.00 g, 2.20 g c)2.02 g, 2.03 g, 2.04 g d)1.50 g, 2.00 g, 2.50 g

2.Which set of measurements of a 2.00 g standard is the most precise? a)2.00 g, 2.01 g, 1.98 g b)2.10 g, 2.00 g, 2.20 g c)2.02 g, 2.03 g, 2.04 g d)1.50 g, 2.00 g, 2.50 g

3. A student reports the volume of a liquid as L. How many significant figures are in this measurement? a)2 b)3 c)4 d)5

3. A student reports the volume of a liquid as L. How many significant figures are in this measurement? a)2 b)3 c)4 d)5