Chapter 1(5)Measurement and Error

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Chapter 1(5)Measurement and Error

  1. 1. Measurement and Error
  2. 2. Some terminology <ul><li>True value – standard or reference of known value or a theoretical value </li></ul><ul><li>Accuracy – closeness to the true value </li></ul><ul><li>Precision – reproducibility or agreement with each other for multiple trials </li></ul>
  3. 3. Types of Errors <ul><li>Determinate (or Systematic) </li></ul><ul><ul><li>Sometimes called bias due to error in one direction- high or low </li></ul></ul><ul><ul><li>Known cause </li></ul></ul><ul><ul><ul><li>Operator </li></ul></ul></ul><ul><ul><ul><li>Calibration of glassware, sensor, or instrument </li></ul></ul></ul><ul><ul><li>When determined can be corrected </li></ul></ul><ul><ul><li>May be of a constant or proportional nature </li></ul></ul>
  4. 4. Types of Errors continued <ul><li>Indeterminate (or Random) </li></ul><ul><ul><li>Cannot be determined (no control over) </li></ul></ul><ul><ul><li>Random nature causes both high and low values which will average out </li></ul></ul><ul><ul><li>Multiple trials help to minimize </li></ul></ul>
  5. 5. Accuracy and Precision The center of the target is the true value. Can’t hit the broad side of a barn! Gun barrel must be bent! Great shooting! Target shooters comments Neither accurate nor precise Precise only Both accurate and precise Nature of accuracy and precision
  6. 6. Standard Deviation of the… <ul><li>Population </li></ul><ul><li>Actual variation in the population </li></ul><ul><li>Sample – part of population </li></ul><ul><li>Estimates </li></ul><ul><li>the variation </li></ul><ul><li>in the population </li></ul><ul><li>May not be representative sample </li></ul>
  7. 7. <ul><li>THE SAMPLE STANDARD DEVIATION allows for more variation in the sample compared to the population, since sample is only part of population. Dividing by n-1 increases the estimate of the population variation. This attempts to eliminate the possibility of bias . </li></ul>Population Sample
  8. 8. Measurements- The Uncertainty <ul><li>Example: </li></ul><ul><li>True value of thickness of a book is 5cm . </li></ul><ul><li>Student A uses meter ruler and measures the thickness to be 4.9cm with an uncertainty of 0.1cm . </li></ul><ul><li>Student B , with Vernier caliper, found it to be 4.85cm with an uncertainty of 0.01cm . </li></ul><ul><li>We may say, </li></ul><ul><li>Student A has more accurate value, but less precise . </li></ul><ul><li>Student B got a more precise value, but less accurate (due to the faulty caliper. Un-calibrate !) </li></ul><ul><li>However, after sending the caliper to be calibrated, student B performs the measurement again and found the thickness is 4.98cm . So, now he has more accurate and more precise value. </li></ul><ul><li>Note: We always report a measurement in a way that would includes the uncertainty carried by the instrument. </li></ul><ul><li> For instance: </li></ul>
  9. 9. Combining uncertainties + and - <ul><li>Adding or subtracting quantities then sum all individual absolute uncertainties </li></ul><ul><li>eg 2.1 ± 0.1 + 2.0 ± 0.2 = 4.1 ± 0.3 </li></ul><ul><li>eg 2.1 ± 0.1 - 2.0 ± 0.2 = 0.1 ± 0.3 </li></ul><ul><li>this method overestimates the final uncertainty </li></ul>
  10. 10. Combining uncertainties x and / <ul><li>When Dividing or multiplying quantities, then sum all of the individual relative uncertainties </li></ul><ul><li>eg (2.5 ± 0.1) x (5.0 ± 0.1) </li></ul><ul><ul><li>= (2.5 ± 4%) x (5.1 ± 2%) =12.5 ± 6% (or 0.75 or 0.7) </li></ul></ul><ul><li>eg (21 ± 6%) / (5.0 ± 4%) </li></ul><ul><ul><li>= 4.12 ± 10% or 4.2 ± 0.42 or 4.2 ± 0.4 </li></ul></ul><ul><li>also overestimates final uncertainty </li></ul>
  11. 11. Measurements- The Significant figures, What is ? <ul><li>In general, </li></ul><ul><li>The number of Significant figures of a numerical quantity is the number of reliably known digits it contains. </li></ul><ul><li>For measured quantity , it is defined as all of the digits that can be read directly from the instrument used in making the measurement plus one uncertain digit that is obtained by estimating the fraction of the smallest division of the instrument’s scale. </li></ul><ul><li>Note: Exact quantities are considered as having unlimited number of significant figures. We need to be concerned with significant figures only when dealing with measurements that have required some estimation. </li></ul><ul><li>For example, </li></ul><ul><li>Reading of the thickness of a book is </li></ul><ul><li>5.0cm or 50mm from meter ruler (with 2 sf) </li></ul><ul><li>5.00cm or 50.0 mm from vernier caliper. (with 3 sf) </li></ul>
  12. 12. Significant Figures <ul><li>The rules of significant figures: </li></ul><ul><li>1. Any figures that is non-zero, are considered as a significant figure. </li></ul><ul><li>2. Zeros at the beginning of a number are not significant </li></ul><ul><li>Example: 0.254 ----------------- 3 s.f </li></ul><ul><li>3. Zeros within a number are significant. </li></ul><ul><li>Example: 104.6 m ---------------- 4 s.f </li></ul><ul><li>4. Zeros at the end of a number after the decimal point are significant. </li></ul><ul><li>Example: 27050.0 ------------------- 6 s.f </li></ul>
  13. 13. Significant Figures … <ul><li>5. Zeros at the end of a whole number without a decimal point may or may not be significant. </li></ul><ul><li>It depends on how that particular number was obtained, using what kind of instrument, and the uncertainty involved. </li></ul><ul><li>Example: 500m ------------------- could be 1 or 3 sf. </li></ul><ul><li>Convert the unit: </li></ul><ul><li>500m = 0.5km (would you say it has 1 sf ? ) </li></ul><ul><li>500m = 50 000cm (would you say it has 1 or 5 sf ? ) </li></ul><ul><li>How to solve this problem ? </li></ul>
  14. 14. Significant figures – Addition and Subtraction processes <ul><li>The rule: </li></ul><ul><li>The final result of an addition and/or subtraction should have the same number of significant figures as the quantity with the least number of decimal places used in the calculation. </li></ul><ul><li>Example: </li></ul><ul><li>23.1 + 45 + 0.68 + 100 = 169 </li></ul><ul><li>Example: </li></ul><ul><li>23.5 + 0.567 + 0.85 = </li></ul>24.9
  15. 15. Significant figures – Multiplication and division processes <ul><li>The rule: </li></ul><ul><li>The final result of an multiplication and/or division should have the same number of significant figures as the quantity with the least number of significant figures used in the calculation. </li></ul><ul><li>Example: </li></ul><ul><li>0.586 x 3.4 = 1.9924 </li></ul><ul><li> = 2.0 </li></ul><ul><li>Example: </li></ul><ul><li>13.90 / 0.580 = 23.9655 </li></ul>= 24.0
  16. 16. Estimating the slope <ul><li>1. Simple conservative method </li></ul><ul><li>Don’t forget the uncertainty on the slope </li></ul>
  17. 17. Simple conservative method <ul><ul><li>Plot error bars on the graph </li></ul></ul><ul><ul><li>Draw maximum (m max ) and minimum (m min ) slopes. </li></ul></ul><ul><ul><li>m = ( m max + m min )/2 </li></ul></ul><ul><ul><li> m = ( m max - m min )/2 </li></ul></ul><ul><ul><li>C = ( C mak -C min )/2 </li></ul></ul>

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