Measurements and Errors Notes

# Uncertainty in addition or subtraction e f where e f

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Uncertainty in addition or subtraction = e f = Where: e f = absolute uncertainty of calculated value % Relative error Relative Uncertainty % Relative Uncertainty = Relative Uncertainty x 100

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e i = absolute uncertainty for individual value n = number of experimental values Example 1: take three measurements of a value and add or subtract. value 1. = +2.93 ± 0.01 <--- e 1 value 2. = -1.21 ± 0.03 <--- e 2 value 3. = +6.71 ± 0.05 <---e 3 (Summed value) 8.43 ± ? <------- e f e f = [( 0.01 ) 2 + ( 0.03 ) 2 + ( 0.05 ) 2 ] 1/2 = 0.059 (Really only one significant figure) e 4 = 0.06 so, 8.43 ± 0.06 is answer Example 2: 1.76 (±0.03) e 1 + 1.89 (±0.02) e 2 0.59 (±0.02) e 3 3.06 (± e f ) e f = = 0.0412 The answer is 3.06 ± 0.04; keep same number of significant figures in answer If necessary, you can determine the relative uncertainty at the end of the calculation %Relative Uncertainty = x 100 = 1.31% 1%
Multiplication and Division For multiplication and division, use percent relative uncertainty % e f = Where: %e f = percent relative uncertainty for calculated value %e i = percent relative uncertainty for individual value n = number of experimental values 1. Convert to % relative uncertainty 2. Square root of sum of squares 3. Convert to absolute uncertainty Example 1: (10.83 ± 0.07) x (3.87 ± 0.03) = 8.23 ± e f (5.09 ± 0.02) 1. % Relative uncertainty (0.07/10.83) * 100 = 0.65% (0.03/3.87) * 100 = 0.78% (0.02/5.09) * 100 = 0.39% 2. [ Σ i x i 2 ] 1/2 = 1.1%

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3. 8.23 x 1.1% = 0.09 so answer is 8.23 ± 0.09 Significant figures must be consistent with uncertainties . Example 2 = 5.6(± ?) First convert absolute uncertainty to percent relative uncertainty = 5.6(± ?) %e f = = 4.0% 4% You should retain extra "insignificant figures" throughout the calculation and round at the end. When using a calculator, don't worry about significant figures until the final answer If necessary, you can determine the absolute uncertainty at the end of the calculation 4% x 5.6 = 0.2 5.6(±0.2) (absolute uncertainty) 5.6(±4%) (relative uncertainty)
• Fall '11
• Tarr
• pH, Observational error, absolute uncertainty, relative uncertainty, antilog

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