Uncertainty in Measurement and Significant Figures

1. Uncertainty in Measurement and Significant Figures Chapter 5, Section 2

2. Uncertainty in Measurement • Every measurement device has its limitations • You can only estimate between points, but not beyond • Example: a bathroom scale doesn’t give your weight to the thousandth of a pound

3. Uncertainty in Measurement • A beaker is never an accurate measurement device. • For accurate liquid measurement, use a graduated cylinder • The liquid in a graduated cylinder may for a curve, called a meniscus • When reading a graduated cylinder, always read the bottom of the meniscus

4. Significant Figures • Significant figures are numbers recorded in a measurement. This includes certain digits and the first uncertain digit in the measurement. • You will often see significant figures abbreviated as “sig figs” • Sig figs determine the maximum amount of numbers you can use in an answer

5. Rules for Counting Significant Figures • Nonzero integers always count as significant figures. Ex: 1457 is all no zero integers, so all count as sig figs • Zeros fall into three groups: • Leading zeros are zeros that preceded nonzero digits. They NEVER count as sig figs • Captive zeros are zeros that fall between nonzero digits. They ALWAYS count as sig figs • Trailing zeros are zeros at the right end of the number. They are only significant when written with a decimal point. • Exact numbers, like numbers obtained by counting, will never limit the number of sig figs in a calculation

6. Sig Fig Practice • How many significant figures do the following numbers have? • 0.0108 g of vitamin C • 3 sig figs; the leading zeros don’t count, but the captive zero does • 480 cars • 3 sig figs; this is an exact number • 5.030 x 103ft • 4 sig figs; both zeros are significant • 0.00100 m • 3 sig figs; the leading zeros don’t count, but the trailing zeros do

7. Rounding Off • When you perform a calculation on your calculator, the number displayed is usually greater than the number of sig figs • You must “round off” your calculations so your answer equals the correct number of sig figs allowed

8. Rules for Rounding Off • If the digits to be removed… • Is less than 5, the preceding digit stays the same • Is equal to or greater than 5, the preceding digit is increased by 1. • In a series of calculations, carry the extra digits to the end and then round off. Do not round off at each step.

9. Determining Sig Figs in Calculations • For multiplication and division, significant figures are counted. • Use the smallest number of sig figs in your answer • For addition and subtraction, the decimal places are counted • Use the smallest number of decimal places in your answer

10. Determining Sig Figs in Calculations • Multiplication Example: • 4.56 x 1.4 = 6.384 • 4.56 = 3 sig figs; 1.4 = 2 sig figs • Answer needs 2 sig figs • Rounding off answer = 6.4 • Division Example • 8.315/298 = 0.0279027 • 8.315 = 4 sig figs; 298 = 3 sig figs • Answers needs 3 sig figs • Rounding off answer = 2.79 x 10-2

11. Determining Sig Figs in Calculations • Addition Example • 12.11 + 18.0 + 1.013 = 31.123 • 12.11 = 2 decimals; 18.0 = 1 decimal; 1.013 = 3 decimals • Answer must have only 1 decimal • Rounding off Answer = 31.1 • Subtraction Example • 0.6875 – 0.1 = 0.5875 • 0.6875 = 4 decimals; 0.1 = 1 decimal • Answer must have only 1 decimal • Rounding off Answer = 0.6