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Measurement And Error

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Measurement And Error

1. 1. Measurement and Error
2. 2. Error in Measurement <ul><li>Types of Error </li></ul><ul><ul><ul><li>Systematic – one that always produces an error of the same sign; positive is a reading too high and negative error is a reading too low </li></ul></ul></ul><ul><ul><ul><li>Random – occur as variations that are due to a large number of factors each of which adds to its own contribution of the total error. These errors are a matter of chance </li></ul></ul></ul>
3. 3. Types of Systematic Error <ul><li>Instrumental Error – caused by faulty, inaccurate apparatus </li></ul><ul><li>Personal Error – caused by some peculiarity or bias of the observer </li></ul><ul><li>External Error – caused by external conditions (wind, temperature, humidity) </li></ul>
4. 4. Random Error <ul><li>Random errors are subject to the laws of chance. Taking a large number of observations may lessen their effect. When al errors are random, the value having the highest probability of being correct is the arithmetic mean or average. </li></ul>
5. 5. Propagation of Error <ul><li>Scientific measurements will always contain some degree of uncertainty. This uncertainty will depend on: </li></ul><ul><li>1. The instrument(s) used to make measurements </li></ul>
6. 6. Propagation of Error <ul><li>2. The object being measured </li></ul><ul><li>3. The proximity to the object being measured </li></ul>
7. 7. Variance <ul><li>The uncertainty of a measurement is indicated showing the possible variance with a plus and minus factor. </li></ul><ul><li>Example: You measure the length of an object five times and record the following measurements </li></ul><ul><li>53.33 cm, 53.36 cm, 53.32 cm, 53.34 cm, & 53.38 cm </li></ul><ul><li>The average is 53.35 cm; this should be written as </li></ul><ul><li>53.35 ± .03 cm </li></ul>
8. 8. Errors in Addition and Subtraction <ul><li>Example: 13.02  .04 cm </li></ul><ul><li>23.04  .03 cm </li></ul><ul><li>14.36  .03 cm </li></ul><ul><li>26.89  .04 cm </li></ul><ul><li>  77.31  .14 cm </li></ul><ul><li>The variance of the result is equal to the sum of all the individual variances </li></ul>
9. 9. Errors in Multiplication and Division <ul><li>Example: 13.2  .2 cm x 23.5  .3 cm </li></ul><ul><li>   </li></ul><ul><li>Maximum and Minimum: </li></ul><ul><li>  Maximum 13.4 cm x 23.7 cm = 319 cm 2 </li></ul><ul><li>  Minimum 13.0 cm x 23.2 cm = 302 cm 2 </li></ul><ul><li>  </li></ul><ul><li>  Average = 310. cm 2 </li></ul><ul><li>Answer 310.  9 cm 2 </li></ul><ul><li>The variance MUST be large enough to include both </li></ul><ul><li>maximum and minimum </li></ul>
10. 10. Accuracy <ul><li>The closeness of a measurement to the accepted value for a specific physical quantity. Accuracy is indicated mathematically by a number referred to as error. </li></ul><ul><li>Absolute Error (E A ) = (Average of observed values) – (Accepted Value) </li></ul><ul><li>  </li></ul><ul><li>  Relative Error (E R ) = X 100% </li></ul>
11. 11. Precision <ul><li>The agreement of several measures that have been made in the same way. Precision is indicated mathematically by a number referred to as deviation. </li></ul><ul><li>Absolute Deviation (D A ) = (Each observed value) – (Average of all values) </li></ul><ul><li>Relative Deviation (D R ) = x 100% </li></ul>
12. 12. Example for Measuring Error and Deviation <ul><li>Measured Values: 893 cm/sec 2 936 cm/sec 2 </li></ul><ul><li>1048 cm/sec 2 </li></ul><ul><li> 915 cm/sec 2 </li></ul><ul><li> 933 cm/sec 2 </li></ul><ul><li>  Accepted Value: 981 cm/sec 2 </li></ul>
13. 13. Example for Measuring Error and Deviation <ul><li>Step 1: Calculate the Average </li></ul><ul><li> 893 cm/sec 2 </li></ul><ul><li> 936 cm/sec 2 </li></ul><ul><li>1048 cm/sec 2 </li></ul><ul><li> 915 cm/sec 2 </li></ul><ul><li> 933 cm/sec 2 </li></ul><ul><li>Average = 945 cm/sec 2 </li></ul>
14. 14. Example for Measuring Error and Deviation <ul><li>Step 2: Calculate Absolute and Relative Error </li></ul><ul><li>  </li></ul><ul><li>Absolute Error (E A ) = (Average of observed values) – (Accepted Value) </li></ul><ul><li>  E A = 945 cm/sec 2 – 981 cm/sec 2 = 36 cm/sec 2 </li></ul><ul><li>  </li></ul><ul><li>Relative Error (E R ) =   x 100 % </li></ul><ul><li>E R = x 100% = 3.7 % </li></ul>
15. 15. Example for Measuring Error and Deviation <ul><li>Step 3: Calculate Absolute and Relative Deviations </li></ul><ul><li>  Absolute Deviation (D A ) = (Each Observed Value) – (Average of All Values) </li></ul><ul><li>  D A = 893 cm/sec 2 – 945 cm/sec 2 = 52 cm/sec 2 </li></ul><ul><li>  D A = 936 cm/sec 2 – 945 cm/sec 2 = 9 cm/sec 2 </li></ul><ul><li>  D A = 1048 cm/sec 2 – 945 cm/sec 2 = 103 cm/sec 2 </li></ul><ul><li>D A = 915 cm/sec 2 – 945 cm/sec 2 = 30 cm/sec 2 </li></ul><ul><li>  D A = 933 cm/sec 2 – 945 cm/sec 2 = 12 cm/sec 2 </li></ul><ul><li>  </li></ul><ul><li>  Average Absolute Deviation: 206 cm/sec 2 / 5 = 41 cm/sec 2 </li></ul>
16. 16. Example for Measuring Error and Deviation <ul><li>Relative Deviation: </li></ul><ul><li>  </li></ul><ul><li>Relative Deviation (D R ) = X 100% </li></ul><ul><li>  </li></ul><ul><li>D R = x 100% = 4.3% </li></ul>
17. 17. Significant Figures <ul><li>Usually, you will estimate one digit beyond the smallest division on the measuring tool if the object you are measuring has a well defined edge. </li></ul><ul><li>When reading a measurement that someone else has made, you must determine if the digits he/she has written down are significant to the measurement. </li></ul>
18. 18. Significant Figures <ul><li>Those digits in an observed quantity (measurement) that are known with certainty plus the one digit that is uncertain or estimated. </li></ul><ul><li>The number of significant figures in a measurement depends on: </li></ul>
19. 19. 1. Smallest divisions on a measuring tool
20. 20. 2. The size of the object being measured
21. 21. 3. The difficulty in measuring a particular object