1. Error Estimation ( 誤差之評定 )

2. Definition of error measure of the accuracy of the result.Error is a measure of the accuracy of the result. how the result closes to the true value.It indicates how the result closes to the true value.

3. Significant Figures ( 有效數字 ) To quote an error associated with the measured value, i.e. (measured value  error) unit. E.g.(9240  5) mg To express in scientific notation. E.g.9240 mg (3 s.f.) 9.240 x 10 3 mg (4 s.f.)

4. Eg1An object of mass is estimated to lie between 9.235 g and 9.245 g. Write the result in appropriate form. (9.24  0.005) g

5. Significant Figures ( 有效數字 ) Significant Figures ( 有效數字 ) 2. Addition and Subtraction Round  off the first column from the left and drop all the digits to its right

6. 2. Addition and Subtraction 1.36 cm, 16.72 cm, 5 cm, 0.89 cm and 9.3 cm.Eg2The length of 5 rods are 1.36 cm, 16.72 cm, 5 cm, 0.89 cm and 9.3 cm. What is the total length of the rods when placed in a straight end to end?

7. two $100 note(a)A student has two $100 note in his pocket. After he has spent $3, he left $? Exactly $200!!!!!!!Exactly $200!!!!!!! answer:$(200 – 3) = $197 estimated to be 200(b)No. of audience in a concert is estimated to be 200. If 3 men left, the estimated no. of audience becomes Figure “2” in 200 is a doubtful figure. –Answer:still 200

8. Multiplication and Division same number of significant figureslowest number of significant figures among the quantitiesThe final result has the same number of significant figures as the lowest number of significant figures among the quantities.

9. Eg4 1.204 kg(4 s.f.) 3.2 m s  1 (2 s.f.)A toy car of mass 1.204 kg (4 s.f.) moves on a horizontal ground with speed 3.2 m s  1. (2 s.f.) The kinetic energy of the car is lowest no. of s.f. is 2 2Since the lowest no. of s.f. is 2, the no. of s.f. in the final result should also be 2.

10. remarks carry extra two significant figures intermediate steps rounded offIt is better to carry extra two significant figures along the intermediate steps and the final answer is then rounded off appropriately. all the digitsDon’t copy all the digits displayed by the calculator.

11. (C)Sources of Errors 1.Instrumental limitations ( 儀器的限制 )1.Instrumental limitations ( 儀器的限制 ) 2.Systemic errors ( 系統誤差 )2.Systemic errors ( 系統誤差 ) 3.Random errors ( 隨機誤差 )3.Random errors ( 隨機誤差 )

12. Instrumental limitations ( 儀器的限制 ) All measuring instruments have their limitations. cannot repeated measurementsThese errors cannot be reduced by taking repeated measurements. Example: Meter rule having mm scale has a limitation of 0.5 mm.

13. Systemic errors ( 系統誤差 ) all measurement to be shifted systematically in one directioncause all measurement to be shifted systematically in one direction  either larger or smaller than it should be. cannot repeated measurements These errors cannot be reduced by taking repeated measurements.

14. Examples of systemic errors Parallax ( 視差 ) in reading scale when viewing the scale always from one side. A zero error ( 零點誤差 ) on any scale. A calibration error ( 校準誤差 ). A background count ( 本底輻射 ) in a radioactivity experiment. A biased stray magnetic field, electric field ( 雜 / 離散磁場、 電場 ). An error in meter rules due to thermal expansion.

15. Random errors ( 隨機誤差 ) unknown and unpredicted variationsThey result from unknown and unpredicted variations in experiments. can be reducedThe effect of the random errors can be reduced by (I)improving experimental techniques and repeating the measurement a number of times(II) repeating the measurement a number of times i.e. becoming statistically insignificant.

16. Examples of random errors ( 隨機誤差 ) Parallax in reading scale when viewing the scale in different directions. Unpredicted fluctuation ( 不可預期的波動 ) in air temperature or line voltage. Unbiased estimates ( 無偏私的估計 ) of measurement readings by the observer. Non  uniformity of diameter of a wire.

18. Treatment of errors ( 誤差的處理 )

19. Instrumental limitations half of the smallest division on the scaleThe scale error is usually taken as half of the smallest division on the scale. answer:(20  0.5)  C

20. Eg6 10.12 sreaction time0.1 sTiming Mr. Yip in running 100 m by a digital stop watch gives a reading of 10.12 s. If the reaction time of the stop watch controller is 0.1 s, the appropriate way of expressing the time will be (10.1  0.1)s

21. Systematic error There is no general rule for the estimation of these errors.

22. Random errors  n-1 : Sample standard deviation ( 樣本標準偏差 ) Sample standard deviation ( 樣本標準偏差 ) of the data gives the measure of the random errors.

23. Estimation of errors

24. Ex7 error in single measurement  ) 1 nm mean gives 589.1 = 589 nm ( x ) 2 nm (  n-1 )The error in single measurement of wavelength of sodium light is (  ) 1 nm. Taking more measurement will reduce the random errors. For instance, the results are 587, 589, 588, 591, 587, 588, 590, 592, 590 and 589. Then the mean gives 589.1 = 589 nm ( x ). The sample standard deviation is 2 nm (  n-1 ). correct expression: 589  2 nm( x   n-1 ) correct expression: 589  2 nm. ( x   n-1 ).

25. Combing errors

26. (a)Sum and Difference Z = A + B or Z = A  B where A and B are independent The errors are always added

27. Eg8 If B = (15  2) and A = (76  3), then Z = A  B =?solution: Z = A – B = 76 – 15 = 61  Z = 2 + 3 = 5  Z = 61  5

28. Eg9 (1.0  0.05)cm & (3.2  0.05)cm Length of wire = 3.2 –1.0 = 2.2 cm Max. error = 0.05 + 0.05 = 0.1 cm Therefore, length = (2.2  0.1)cm cm 01234

29. Product and Quotient Z = A  B or A  B where A and B are independent

30. Power Z = k  A n where k and n are non  zero constants with error free

31. E.g. 9 E = ½ mv 2 (  E)/E = (  m)/m + 2(  v)/v

32. Other combinations sine, log find the maximum and minimum possible valuesIf special function such as sine, log are involved, it will be easier to find the maximum and minimum possible values in order to find the errors. Error = max [y max – y, y - y min ]Error = max [y max – y, y - y min ]