Uncertainties in Measurement Laboratory investigations involve taking measurements of physical quantities. All measurements will involve some degree of experimental uncertainty. QUESTIONS 1. How does one express the uncertainty in an experimental measurement? 2. How does one determine the uncertainty in an experimental measurement? 3. How does one compare an experimental measurement with an accepted (or published) value? 4. How does one determine the uncertainty in a quantity that is computed from uncertain measurements?
Expressing Uncertainty We will express the results of measurements in this laboratory as (measured value ± uncertainty) units For example g ± g = (9.803 ± 0.008) m/s 2
Types of Experimental Uncertainty Random, Indeterminate or Statistical Results from unknown and unpredictable variations that arise in all experimental situations. Repeated measurements will give slightly different values each time. You cannot determine the magnitude (size) or sign of random uncertainty from a single measurement. Random errors can be estimated by taking several measurements. Random errors can be reduced by refining experimental techniques.
Types of Experimental Uncertainty Systematic or Determinate Associated with particular measurement instruments or techniques. The same sign and nearly the same magnitude of the error is obtained on repeated measurements. Commonly caused by improperly calibrated or zeroed instrument or by experimenter bias.
Accuracy Accuracy and Precision Is a measure of how close an experimental result is to the true (or published or accepted) value. Precision Is a measure of the degree of closeness of repeated measurements.
Accuracy and Precision Consider the two measurements: A = (2.52 ± 0.02) cm B = (2.58 ± 0.05) cm Which is more precise? Which is more accurate?
Accuracy and Precision Answer with GOOD or POOR... accuracy precision accuracy precision accuracy precision
Implied Uncertainty The uncertainty in a measurement can sometimes be implied by the way the result is written. Suppose the mass of an object is measured using two different balances. Balance 1 Reading = 1.25 g Balance 2 Reading = 1.248 kg
Significant Figures In a measured quantity, all digits are significant except any zeros whose sole purpose is to show the location of the decimal place. 123 123.0 0.0012 0.0001203 0.001230 1000 1000. 150 g g m cm s cm cm 1.23 x 10 2 g 1.230 x 10 2 g 1.2 x 10-3 m 1.203 x 10-4 s 1.230 x 10-4 s 1 x 10 3 cm 1.000 x 10 3 cm 150
Rounding If the digit to the right of the position you wish to round to is < 5 then leave the digit alone. If the digit to the right of the position you wish to round to is >= 5 then round the digit up by one. For multiple arithmetic operations you should keep one or two extra significant digits until the final result is obtained and then round appropriately. Proper rounding of your final result will not introduce uncertainty into your answer. ROUNDING DURING CALCULATIONS IS NOT A VALID SOURCE OF ERROR.
Expressing Uncertainty When expressing a measurement and its associated uncertainty as (measured value ± uncertainty) units Round the uncertainty to one significant digit, then round the measurement to the same precision as the uncertainty. For example, round 9.802562 ± 0.007916 m/s 2 to g ± g = (9.803 ± 0.008) m/s 2
Significant Figures in Calculations Multiplication and Division When multiplying or dividing physical quantities, the number of significant digits in the final result is the same as the factor (or divisor ) with the fewest number of significant digits. 6.273 N 0.0204 µm * 5.5 m 21 C 34.5015 N m 0.00097142857 µm/c N m µm/c
Significant Figures in Calculations Addition and Subtraction When adding or subtracting physical quantities, the precision of the final result is the same as the precision of the least precise term. 132.45 cm 0.823 cm + 5.6 cm 138.873 cm --> cm
Comparing Experimental and Accepted Values E ± E = An experimental value and its uncertainty. A = An accepted (published) value. E A Percent Discrepancy = 100% A Percent Discrepancy quantifies the of a measurement. E Percent Uncertainty = 100% E Percent Uncertainty quantifies the of a measurement.
Comparing Two Experimental Values E 1 and E 2 = Two different experimental values. Percent Difference = E E 2 1 E + E 2 1 2 100%
Average (Mean) Value Let x 1, x 2, x N represent a set of N measurements of a quantity x. The average or mean value of this set of measurements is given by 1 N 1 x = xi = 1 2 + N i= 1 N ( x + x +... x ) N
Frequency Distribution (N=10) data 14 22 20 18 20 23 18 19 19 23 hist.x hist.d 10 0 11 0 12 0 13 0 14 1 15 0 16 0 17 0 18 2 19 2 20 2 21 0 22 1 23 2 24 0 25 0 26 0 27 0 28 0 29 0 30 0 Frequency 2.0 1.5 1.0 0.5 0.0 10 Mean = 19.6 15 20 Value 25 30
Frequency Distribution (N=100) Mean = 19.89 hist.x hist.d 10 0 11 0 12 0 13 0 14 0 15 3 16 5 17 8 18 11 19 15 20 15 21 19 22 13 23 6 24 3 25 2 26 0 27 0 28 0 29 0 30 0 Frequency 15 10 5 0 10 15 20 25 30 Value
Frequency Distribution (N=1,000) Mean = 19.75 hist.x hist.d 10 0 11 0 12 1 13 5 14 11 15 20 16 49 17 74 18 127 19 152 20 161 21 148 22 100 23 81 24 44 25 17 26 6 27 3 28 1 29 0 30 0 Frequency 160 140 120 100 80 60 40 20 0 10 15 20 25 30 Value
Frequency Distribution (N=10,000) Mean = 19.99 hist.x hist.d 10 0 11 1 12 7 13 32 14 96 15 236 16 466 17 781 18 1184 19 1435 20 1541 21 1451 22 1159 23 796 24 453 25 240 26 98 27 18 28 3 29 3 30 0 Frequency 1400 1200 1000 800 600 400 200 0 10 15 20 25 30 Value
Expressing Uncertainty = = N i i x x x N s 1 2 ) ( 1 1 The Standard Deviation of a set of N measurements of x is given by: N s SEM s x x = = The Standard Deviation of the Mean (or Standard Error of the Mean) of a set of N measurements of x is given by:
Expressing Uncertainty N Mean S.D. SEM Result 10 19.6 2.71 0.857 100 19.89 2.26 0.226 1,000 19.884 2.48 0.0784 10,000 19.9879 2.52 0.0252 19.6 ± 0.9 19.9 ± 0.2 19.88 ± 0.08 19.99 ± 0.03
Combining Uncertainties: Propagation of Uncertainty Let A ± A and B ± B represent two measured quantities. The uncertainty in the sum S = A + B is S = A + B The uncertainty in the difference D = A - B is ALSO D = A + B
Combining Uncertainties: Propagation of Uncertainty Let A ± A and B ± B represent two measured quantities. The uncertainty in the product P = A * B is The uncertainty in the quotient Q = A / B is ALSO + = B B A A P P + = B B A A Q Q