Uncertainties in Measurement

1. 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 ?

2. Expressing Uncertainty We will express the results of measurements in this laboratory as (measured value  uncertainty) units For example g  Dg = (9.803  0.008) m/s2

3. 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.

4. 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.

5. Accuracy and Precision Accuracy 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.

6. 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 ?

7. _____ accuracy _____ precision _____ accuracy _____ precision _____ accuracy _____ precision Accuracy and Precision Answer with GOOD or POOR ...

8. 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

9. 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 102 g 1.230 x 102 g 1.2 x 10-3 m 1.203 x 10-4 s 1.230 x 10-4 s 1 x 103 cm 1.000 x 103 cm 150

10. 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.

11. 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/s2 to g  Dg = (9.803  0.008) m/s2

12. 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 mm * 5.5 m 21 C° 34.5015 N·m 0.00097142857 mm/C° ________ N·m _________ mm/C°

13. 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

14. Percent Discrepancy quantifies the __________ of a measurement. Comparing Experimental and Accepted Values E ± DE = An experimental value and its uncertainty. A = An accepted (published) value. Percent Uncertainty quantifies the __________ of a measurement.

15. Comparing Two Experimental Values E1 and E2 = Two different experimental values.

16. Average (Mean) Value Let x1, x2,… xN represent a set of N measurements of a quantity x. The average or mean value of this set of measurements is given by

17. Mean = 19.6 Frequency Distribution (N=10)

18. Mean = 19.89 Frequency Distribution (N=100)

19. Mean = 19.884 Frequency Distribution (N=1,000)

20. Mean = 19.9879 Frequency Distribution (N=10,000)

21. Expressing Uncertainty The Standard Deviation of a set of N measurements of x is given by: The Standard Deviation of the Mean (or Standard Error of the Mean) of a set of N measurements of x is given by:

22. Expressing Uncertainty 19.6  0.9 19.9  0.2 19.88  0.08 19.99  0.03

23. Combining Uncertainties:Propagation of Uncertainty Let A ± DA and B ± DB represent two measured quantities. The uncertainty in the sum S = A + B is DS = DA + DB The uncertainty in the difference D = A - B is ALSO DD = DA + DB

24. Combining Uncertainties:Propagation of Uncertainty Let A ± DA and B ± DB represent two measured quantities. The uncertainty in the product P = A * B is The uncertainty in the quotient Q = A / B is ALSO