Error Estimation

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Error Estimation (?????)
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Definition of error

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

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Significant Figures ( ???? )

• 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.)

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• 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

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Significant Figures ( ???? ) 2.   Addition and
Subtraction

• Round?off the first column from the left and drop
  all the digits to its right

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2.   Addition and Subtraction

• Eg2 The 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?

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• (a) A student has two 100 note in his pocket.
  After he has spent 3, he left ?
• Exactly 200!!!!!!!
• answer (200 3) 197
• (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

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Multiplication and Division

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

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• Eg4
• 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
• Since the lowest no. of s.f. is 2, the no. of
  s.f. in the final result should also be 2.

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remarks

• It is better to carry extra two significant
  figures along the intermediate steps and the
  final answer is then rounded off appropriately.
• Dont copy all the digits displayed by the
  calculator.

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(C) Sources of Errors

• 1. Instrumental limitations ( ????? )
• 2. Systemic errors ( ???? )
• 3. Random errors ( ???? )

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Instrumental limitations ( ????? )

• All measuring instruments have their limitations.
• These errors cannot be reduced by taking repeated
  measurements.
• Example Meter rule having mm scale has a
  limitation of 0.5 mm.

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Systemic errors ( ???? )

• cause all measurement to be shifted
  systematically in one direction ? either larger
  or smaller than it should be.
• These errors cannot be reduced by taking
  repeated measurements.

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Examples of systemic errors

• Parallax ( ?? ) in reading scale when viewing
  the scale always from one side.
• A zero error ( ???? ) on any scale.
• A calibration error ( ???? ).
• A background count ( ???? ) in a radioactivity
  experiment.
• A biased stray magnetic field, electric field (
  ?/??????? ).
• An error in meter rules due to thermal expansion.

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Random errors ( ???? )

• They result from unknown and unpredicted
  variations in experiments.
• The effect of the random errors can be reduced by
  
• (I) improving experimental techniques and
• (II) repeating the measurement a number of times
  i.e. becoming statistically insignificant.

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Examples of random errors ( ???? )

• l    Parallax in reading scale when viewing the
  scale in different directions.
• Unpredicted fluctuation ( ??????? ) in air
  temperature or line voltage.
• Unbiased estimates ( ?????? ) of measurement
  readings by the observer.
• Non?uniformity of diameter of a wire.

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Treatment of errors ( ????? )
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• Instrumental limitations
• The scale error is usually taken as half of the
  smallest division on the scale.
• answer (20 ? 0.5)?C

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• Eg6
• Timing 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
•  

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Systematic error

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

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Random errors

• ?n-1
• Sample standard deviation ( ?????? ) of the data
  gives the measure of the random errors.

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Estimation of errors
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• Ex7
• 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).
  

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Combing errors
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• (a) Sum and Difference
• Z A B or Z A ? B where A and B are
  independent
• The errors are always added

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• 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

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• 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

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Product and Quotient

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

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Power

• Z k ? An
• where k and n are non?zero constants with error
  free

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E.g. 9

• E ½ mv2
• (?E)/E (?m)/m 2(?v)/v

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Other combinations

• If 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 ymax y , y - ymin