Error Estimation ( 誤差之評定 )

Definition of error

• Error is a measure of the accuracy of the measure of the accuracy of the result.result.

• It indicates how the result closes to the how the result closes to the true value.true value.

Significant Figures ( Significant Figures ( 有效數字 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 103 mg (4 s.f.)

Eg1 An 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

Significant Figures ( Significant Figures ( 有效數字有效數字 ))

2.   Addition and Subtraction• Roundoff the first column from the left and

drop all the digits to its right

2.   Addition and Subtraction

• Eg2 The length of 5 rods are 1.36 cm, 16.72 1.36 cm, 16.72 cm, 5 cm, 0.89 cm and 9.3 cm.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?

• (a) A student has two $100 notetwo $100 note in his pocket. After he has spent $3, he left $?

• Exactly $200!!!!!!!Exactly $200!!!!!!!

answer:answer: $(200 – 3) = $197$(200 – 3) = $197

• (b) No. of audience in a concert is estimated to estimated to be 200be 200. If 3 men left, the estimated no. of audience becomes

Figure Figure “2”“2” in in 220000 is a doubtful figure. is a doubtful figure.

– Answer:Answer: still 200still 200

Multiplication and DivisionMultiplication and Division

• The final result has the same number of same number of significant figuressignificant figures as the lowest number lowest number of significant figures among the of significant figures among the quantitiesquantities.

• Eg4

• A toy car of mass 1.204 kg1.204 kg (4 s.f.)(4 s.f.) moves on a horizontal ground with speed 3.2 m s3.2 m s11. (2 s.f.)(2 s.f.) The kinetic energy of the car is

• Since the lowest no. of s.f. is 2lowest no. of s.f. is 2, the no. of s.f. in the final result should also be 22.

remarks

• It is better to carry extra two significant figurescarry extra two significant figures along the intermediate stepsintermediate steps and the final answer is then rounded offrounded off appropriately.

• Don’t copy all the digitsall the digits displayed by the calculator.

(C)(C) Sources of ErrorsSources of Errors

• 1.1. Instrumental limitations ( Instrumental limitations ( 儀器的限儀器的限制 制 ))

• 2.2. Systemic errors ( Systemic errors ( 系統誤差 系統誤差 ) )

• 3.3. Random errors ( Random errors ( 隨機誤差 隨機誤差 ))

Instrumental limitations Instrumental limitations ( ( 儀器的限制 儀器的限制 ))

• All measuring instruments have their limitations.

• These errors cannot cannot be reduced by taking repeated measurementsrepeated measurements.

Example: Meter rule having mm scale has a limitation of 0.5 mm.

Systemic errors ( Systemic errors ( 系統誤差系統誤差 ))

• cause all measurement to be shifted all measurement to be shifted systematically in one directionsystematically in one direction either larger or smaller than it should be.

These errors cannotcannot be reduced by taking repeated measurementsrepeated measurements.

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.

Random errors ( 隨機誤差 )

• They result from unknown and unpredicted unknown and unpredicted variationsvariations in experiments.

• The effect of the random errors can be reducedcan be reduced by

• (I) improving experimental techniques and

• (II) repeating the measurement a number of repeating the measurement a number of timestimes i.e. becoming statistically insignificant.

Examples of random errors ( Examples of random errors ( 隨機誤差 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.

• Nonuniformity of diameter of a wire.

Random errors

Systematic errors

True value

Mean value of the above measurements

Treatment of errors ( 誤差的處理 )

Instrumental limitations

• The scale error is usually taken as halfhalf of th of the smallest division on the scalee smallest division on the scale.

answer:answer: (20 (20 0.5) 0.5)CC

0

20

10

30

• Eg6

• Timing Mr. Yip in running 100 m by a digital stop watch gives a reading of 10.12 s10.12 s. If the reaction timereaction time of the stop watch controller is 0.1 s0.1 s, the appropriate way of expressing the time will be

(10.1 (10.1 0.1)s 0.1)s•  

Systematic errorSystematic error

• There is no general rule for the estimation of these errors.

Random errors

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

Estimation of errors

%100%100 ) (error tage Percen(c)

) (error onal Fracti(b)

) (error Absolute (a)

x

X

X

Xx

X

X

X

xXX

百分誤差

部份誤差

絕對誤差

• Ex7

• The error in single measurementerror in single measurement of wavelength of sodium light is () ) 1 nm1 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 mean gives 589.1 = 589 nm ( x )589.1 = 589 nm ( x ). The sample standard deviation is 2 nm (2 nm (n-1n-1)).

correct expression:correct expression: 589 589 2 2 nm nm. ( x ( x n-1n-1)).

Combing errors

• (a) Sum and Difference

• Z = A + B or Z = A B where A and B are independent

• The errors are always added

BAZ

• Eg8

• If B = (15 2) and A = (76 3),

• then Z = A B =?

solution:solution:

• Z = A – B = 76 – 15 = 61Z = 2 + 3 = 5

Z = 61 5

• 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

0 1 2 3 4

Product and Quotient

• Z = A B or A B where A and B are independent

B

B

A

A

Z

Z

PowerPower

• Z = k An where k and n are nonzero constants with error free

A

An

Z

Z

E.g. 9

E = E = ½ mv½ mv22

((E)/E = (E)/E = (m)/m + 2(m)/m + 2(v)/vv)/v

Other combinations

• If special function such as sine, logsine, log are involved, it will be easier to find the maximum and minfind the maximum and minimum possible valuesimum possible values in order to find the errors.

• Error = max [yError = max [ymaxmax – y , y - y – y , y - yminmin]]