Part B: Significant Figures = Precision

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Amanda Richardson
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1 Part A: Accuracy vs. Precision The terms precision and accuracy are often used in discussing measured values. Precision is a measure of how closely individual measurements agree with one another or is an indication of how much error can be expected in a measurement. Accuracy refers to how closely a measurement agrees with the correct, or true, value. The analogy of darts stuck in a dartboard pictured illustrates the difference between these two concepts. In lab we can always discuss precision, but we cannot discuss accuracy unless the true value is known. Part B: Significant Figures = Precision 1. Exact Numbers These numbers have no uncertainty and are taken to be exact. Examples are one dozen, 12, 1 liter = 1000 ml, etc. We will also assume all conversion factors are exact. For example, 1 inch = 2.54 cm exactly. As a rule, numeric definitions, conversion factors, and whole numbers used for counting are taken to be exact. 2. Inexact Numbers All digits of a measured quantity, including the uncertain one, are called significant figures. A measured mass reported as 2.2 g has two significant figures, whereas one reported as g has five significant figures. The greater the number of significant figures, the greater is the certainty implied for the measurement and therefore the greater is the precision of the measurement. When multiple measurements are made of a quantity, the results can be averaged, and the number of significant figures estimated by using statistical methods. These measurements or averages have an uncertainty and therefore limited precision. All experimental measurements and averages are inexact numbers and are treated accordingly. Rules for Counting Significant Figures in an Inexact Number 1. All nonzero digits are significant. 2. All leading zeros are not significant. 3. All trailing zeros are taken to be significant. 4. If the number is expressed in scientific notation, all digits in the decimal portion of the number are significant. Practice: How many significant digits are in the following numbers? (a) (b) (c) 1.30x Significant Figures and the ERROR DIGIT Rules for inexact numbers. Every measurement made in a laboratory comes with some uncertainty. The uncertain digit in any measurement is always the rightmost digit, the last digit written. This is called the ERROR DIGIT. Unless stated otherwise, the precision (uncertainty) in a number is taken to be ± one unit in the error digit. For example, a reported value of 95.4 ml indicates a precision of ± 0.1 ml (95.4 ± 0.1 ml). If the precision in the value is not ±1 in the error digit then the recorder has a responsibility to Chem. 1A Daley i Revised 4/8/08

2 report the actual precision. For example, if the precision is ± 0.5 ml then the recorder should report 95.4 ± 0.5 ml. The ± error in a measurement is often called the absolute error. In lab it will become your responsibility to determine the absolute error of a measurement and record the measured value to the proper number of significant digits. Sometimes precision is recorded as a percentage value instead as an absolute value. For example, 50 ml ± 4% instead of 50 ml ± 2 ml where 2 ml is 4% of 50 ml. Practice: Give the absolute error (±x) in the following measured values: (a) ml ± (b) g ± (c) 1.30x10 3 atm ± Significant Digits in Calculated Numbers When we do a calculation involving inexact numbers the result of the calculation will have some error. It is your responsibility to determine the error digit in the resultant calculation. To do this we apply the rules below to EACH STEP in the calculation. 1. A Multiplication or Division Step: The result of the calculation contains the same number of significant digits as there are in the measurement with the fewest significant digits. Find the error digit by counting significant digits from left to right in the result. Underline this digit. 2. An Addition or Subtraction Step: The answer has an absolute error equal to the greatest absolute error found in the numbers added and/or subtracted. Locate and underline the digit in the calculation result corresponding to this absolute error. 3. Rules for Rounding the ERROR DIGIT: (this is how your calculator will do round off) (a) If the digit to be removed after the error digit is 5 or greater then round up the error digit up. (b) If the digit to be removed after the error digit is 4 or less then round down. (c) When a calculation involves two or more steps and you write down answers for intermediate steps, retain at least one additional digit past the number of significant figures for the intermediate answers. This procedure ensures that small errors from rounding at each step do not combine to affect the final result. When using a calculator, you may enter the numbers one after another, rounding only the final answer. Accumulated rounding-off errors may account for small differences among results you obtain and the correct value Practice: Round each of the following to 3 significant digits. (a) (b) (c) (d) 3136 Practice: Round the following calculations to the proper number of significant digits using the rules above. (a) 6.19 x 2.8 (b) 3.18/1.702 (c) (4.10 x )/1.56x10 4 (d) (e) (f) 5.19x Chem. 1A Daley ii Revised 4/8/08

3 4. Logarithms and Antilogarithms: When taking the logarithm of a number, the mantissa of the logarithm (numbers after the decimal point) should contain the same number of digits as the number of significant digits in the measured number. # sig. digits in number = # of decimals in logarithm log (3.000) = log (3.0) = 0.48 log (3) = 0.5 log (2.78x10 6 ) = log (5.890x10 4 ) = When taking an antilogarithm, retain the same number of significant digits in the result as there are decimals in the mantissa: # of decimals in logarithm = # sig. digits in number = 7.8x = = = 2.18x10 5 Express the following to the proper number of significant digits. (a) log(6.19) (b) log(67.09/2.3x10 7 ) (c) (d) Complete the problems on the next page concerning the metric system and significant figures. Chem. 1A Daley iii Revised 4/8/08

5 Name 1. What decimal power do the following abbreviations represent: (a) d (b) c (c) f (d) μ (e) M (f) k (g) n (h) m (i) p 2. Use appropriate metric prefixes to write the following measurements without the use of exponents: (a) L (b) s (c) m (d) m 3 (e) kg 3. Indicate which of the following are exact numbers: (a) the mass of a paper clip (c) the number of inches in a mile (e) the number of microseconds in a week (f) g (b) the surface area of a dime (d) the number of ounces in a pound (f) the number of pages in this worksheet. 4. What is the number of significant figures in each of the following measured quantities? (a) 358 kg (c) cm (b) s (d) L 5. Round each of the following numbers to four significant figures, and express the result in standard exponential notation: (a) (b) 656,980 (c) (d) (e) Carry out the following operations, and express the answer with the appropriate number of significant figures: (a) (6104.5/2.3) (b) [( ) - ( )] (c) ( ,000.0) + ( ) (d) 863 [ ( )]. Chem. 1A Daley 1 Revised 4/8/08

Chapter 2: Measurement and Problem Solving

Chapter 2: Measurement and Problem Solving Determine which digits in a number are significant. Round numbers to the correct number of significant figures. Determine the correct number of significant figures