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PhysicsNTHUMFTai-戴明鳳
Part I of Fundamental Measurements:
Uncertainties and Error Propagation
http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart2.html
Vern Lindberg, Copyright July 1, 2000
PhysicsNTHUMFTai-戴明鳳
Contents-1 of Uncertainties and Error Propagation
1. Systematic vs Random Error2. Determining Random Errors (a) Instrument Limit of Error, least count (b) Estimation (c) Average Deviation (d) Conflicts (e) Standard Error in the Mean 3. What does uncertainty tell me? Range of possible values 4. Relative and Absolute error
PhysicsNTHUMFTai-戴明鳳
Contents-2 of Uncertainties and Error Propagation
5. Propagation of errors (a) add/subtract (b) multiply/divide (c) powers (d) mixtures of +-*/(e) other functions 6. Rounding answers properly 7. Significant figures 8. Problems to try9. Glossary of terms (all terms that are bold face
and underlined)
PhysicsNTHUMFTai-戴明鳳
Simple Content
1. Systematic and random errors.
2. Determining random errors.
3. What is the range of possible values?
4. Relative and Absolute Errors
5. Propagation of Errors, Basic Rules
PhysicsNTHUMFTai-戴明鳳
1. Systematic & Random Errors No measurement made is ever exact. The accuracy (correctness) and precision
(number of significant figures) of a measurement are always limited:
1. by the degree of refinement of the apparatus used,
2. by the skill of the observer, and
3. by the basic physics in the experiment.
PhysicsNTHUMFTai-戴明鳳
In doing experiments we are trying 1. to establish the best values for certain quantities, or2. to validate a theory. We must also give a range of possible true
values based on our limited number of measurements.
Why should repeated measurements of a single quantity give different values?
Mistakes on the part of the experimenter are possible, but we do not include these in our discussion.
A careful researcher should not make mistakes! (Or at least she or he should recognize them and correct the mistakes.)
PhysicsNTHUMFTai-戴明鳳
Accuracy (correctness) & Precision (number of significant figures)
Uncertainty, error, or deviation -- the synonymous terms to represent the
variation in measured data.
Two types of errors are possible:
1. Systematic error:
2. Random errors
PhysicsNTHUMFTai-戴明鳳
Systematic Errors The result of 1. A mis-calibrated device, or2. A measuring technique which always makes the
measured value larger (or smaller) than the "true" value.
Example: Using a steel ruler at liquid nitrogen temperature to measure the length of a rod.
The ruler will contract at low temperatures and therefore overestimate the true length.
Careful design of an experiment will allow us to eliminate or to correct for systematic errors.
PhysicsNTHUMFTai-戴明鳳
Random ErrorsThese remaining deviations will be classed
as random errors, and can be dealt with in a statistical manner.
This document does not teach statistics in any formal sense.
But it should help you to develop a working methodology for treating errors.
PhysicsNTHUMFTai-戴明鳳
2. Determining random errors Several approaches are used to estimate the
uncertainty of a measured quantity. (a) Instrument Limit of Error (ILE) and Least Count Least count: the smallest division that is marked on
the instrument. -- A meter stick will have a least count of 1.0 mm, -- A digital stop watch might have a least count of 0.01 s. Instrument limit of error (ILE): the precision to which a measuring device can be read,
and is always equal to or smaller than the least count.-- Very good measuring tools are calibrated against
standards maintained by the National Institute of Standards and Technology.
PhysicsNTHUMFTai-戴明鳳
Instrument Limit of Error, ILE Be generally taken to be the least count or some fraction
(1/2, 1/5, 1/10) of the least count. Which to choose, the least count or half the least count,
or something else. No hard and fast rules are possible, instead you must be
guided by common sense. If the space between the scale divisions is large, you may
be comfortable in estimating to 1/5 or 1/10 of the least count.
If the scale divisions are closer together, you may only be able to estimate to the nearest 1/2 of the least count, and
if the scale divisions are very close you may only be able to estimate to the least count.
PhysicsNTHUMFTai-戴明鳳
For some devices the ILE is given as a tolerance or a percentage.
-- Resistors may be specified as having a tolerance of 5%, meaning that the ILE is 5% of the resistor's value.
Problem: For each of the following scales (all in centimeters) determine the least count, the ILE, and read the length of the gray rod.
PhysicsNTHUMFTai-戴明鳳
Problem: to determine the least count, the ILE, and read the length of the gray rod for each of the following scales (all in centimeters).
Least Count (cm)
ILE (cm)
Length(cm)
(a) 1 0.2 9.6(b) 0.5 0.1 8.5
(c) 0.2 0.05 11.90
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Average Example 1Problem Find the average, and average deviation for the 5 following data on the length of a pen, L.
Length (cm) |
12.2 0.02 0.000412.5 0.28 0.078411.9 0.32 0.102412.3 0.08 0.006412.2 0.02 0.0004
Sum 61.1 Sum 0.72 Sum 0.1880
Average61.1/5 = 12.22
Average 0.14
PhysicsNTHUMFTai-戴明鳳
Average Example 2 Problem: Find the average and the average deviation of the
following measurements of a mass. This time there are N = 6 measurements, so for the standard
deviation we divide by (N-1) = 5.
Mass (grams)
4.32 0.0217 0.0004714.35 0.0083 0.0000694.31 0.0317 0.0010054.36 0.0183 0.0003354.37 0.0283 0.0008014.34 0.0017 0.000003
Sum 26.05 0.1100 0.002684Average 4.3417
Average0.022
The mass is(4.342 + 0.022) g or (4.34 + 0.02) g [using average deviations] or
(4.342 + 0.023) g or (4.34 + 0.02) g [using standard deviations].
PhysicsNTHUMFTai-戴明鳳
ExampleOne make several measurements on the mass of an object. The balance has an ILE of 0.02 grams. The average mass is 12.14286 grams, the average deviation is 0.07313 grams. What is the correct way to write the mass of the object including its uncertainty? What is the mistake in each incorrect one? Answer
– 12.14286 g – (12.14 ± 0.02) g – 12.14286 g ± 0.07313 (lack of unit)– 12.143 ± 0.073 g – (12.143 ± 0.073) g – (12.14 ± 0.07) – (12.1 ± 0.1) g – 12.14 g ± 0.07 g
The correct answer is (12.14 ± 0.07) g.
PhysicsNTHUMFTai-戴明鳳
....
yy
xx
zz
....22
yy
xx
zz
The same rule holds, namely add all the relative errors to get the relative error in the result.
For multiplication, division, or combinations
Using simpler average errors
Using standard deviations
PhysicsNTHUMFTai-戴明鳳
Examplew = (4.52 ± 0.02) cm, x = (2.0 ± 0.2) cm.Find z = wx and its uncertainty. (1) z = wx = (4.52) (2.0) = 9.04 cm2
(2) Average error:
z = 0.1044 (9.04 cm2) = 0.944 round to 0.9 cm2, z = (9.0 ± 0.9) cm2. (3) Standard deviation: z = 0.905 cm2 z = (9.0 ± 0.9) cm2
The uncertainty is rounded to one significant figure and the <z> is rounded to match. To write 9.0 cm2 rather than 9 cm2 since the 0 is significant.
PhysicsNTHUMFTai-戴明鳳
Example x = ( 2.0 ± 0.2) cm, y = (3.0 ± 0.6) sec.
Find z = x/y with a dimension of velocity.
(1) z = 2.0/3.0 = 0.6667 cm/s.
(2) Average error:
z = 0.3 (0.6667 cm/sec) = 0.2 cm/sec
(3) Using average error: z = (0.7 ± 0.2) cm/sec
(4) Using standard deviation: z = (0.67 ± 0.15) cm/sec
Note that in this case we round off our answer to have no more decimal places than our uncertainty.
PhysicsNTHUMFTai-戴明鳳
(c) Products of powers: z =xmyn
Using simpler average errors
Using standard deviations
PhysicsNTHUMFTai-戴明鳳
w = (4.52 ± 0.02) cm, A = (2.0 ± 0.2) cm2, y = (3.0 ± 0.6) cm.,
Find z = wy2/A0.5 and Δz .
Example
Awyz
2
49.0cm .02cm 2.05.0
cm .03cm 6.02
cm 5.4cm 2.0
cm 638.28
cm 638.28cm 0.2
cm) (3.02cm 5.4
2
2
2
2
2
22
z
Awyz
The second relative error, (y/y), is x 2 because y2. The third relative error, (A/A), is x 0.5 since xA0.5. z = 0.49 (28.638 cm2) = 14.03 cm2 rounded to 14 cm2
z = (29 ± 14) cm2 (for AE) or z = (29 ± 12) cm2 (SD)Because the uncertainty begins with a 1, we keep two
significant figures and round the answer to match.
PhysicsNTHUMFTai-戴明鳳
-- This is best explained by means of an example.Example: w=(4.52 ± 0.02)cm, x=(2.0 ± 0.2) cm, y=(3.0 ± 0.6)cm Find z = w x +y2, z = wx +y2 = 18.0 cm2
Solution:(1) compute v = wx to get v = (9.0 ± 0.9) cm2. (2) compute
(3) compute Δz = Δv + Δ(y2) = 0.9 + 3.6 = 4.5 cm2 4 cm2
z = (18 ± 4) cm2 for considering average error. For standard deviation, to have v = wx = (9.0 ± 0.9) cm2. The calculation of the uncertainty in y2 is the same as above. get z = 3.7 cm2, z = (18 ± 4) cm2.
(d) Mixtures of multiplication, division, addition, subtraction, and powers
222
2
2
cm 6.3cm 00.940.0
40.0cm 0.3cm) 6.0(22
y
yy
yy
PhysicsNTHUMFTai-戴明鳳
(e) Other Functions: e.g.. z = sin x The simple approach:
For other functions of our variables such as sin(x) we will not give formulae.
However you can estimate the error in z = sin(x) as being the difference between the largest possible value and the average value.
Using the similar techniques for other functions.
z = (sin x) = sin(x + x) - sin(x) z = (cos x) = cos(x - x) - cos(x)
PhysicsNTHUMFTai-戴明鳳
ExampleConsider S = wcos() for w = (2.0 ± 0.2) cm, =53 ± 2°.Find S and its uncertainty.Solution:(1) S = (2.0 cm)cos 53° = 1.204 cm (2) To get the largest possible value of S: make w larger, (w + w) = 2.2 cm, and smaller, ( - ) = 51°. The largest value of S, namely (S + S), is (S + S) = (2.2 cm) cos 51° = 1.385 cm. (3) The difference between these numbers is S = 1.385 - 1.204 = 0.181 cm round to 0.18 cm. Result: S = (1.20 ± 0.18) cm
PhysicsNTHUMFTai-戴明鳳
(f) Other Functions: Getting formulas using partial derivatives
The general method of getting formulas for propagating errors involves the total differential of a function.
Suppose that z = f(w, x, y, ...) where the variables w, x, y, etc. must be independent variables!
The total differential is then
Treat the dw = w as the error in w, and likewise for the other differentials, dz, dx, dy, etc.
The numerical values of the partial derivatives are evaluated by using the average values of w, x, y, etc. The general results are
...
dyyfdx
xfdw
wfdz
PhysicsNTHUMFTai-戴明鳳
The numerical values of the partial derivatives are evaluated by using the average values of w, x, y, etc.
The general results are
...
yyfx
xfw
wfz
Using simpler average errors
Using standard deviations
...22
22
22
2
yyfx
xfw
wfz
PhysicsNTHUMFTai-戴明鳳
ExampleQuestion: Consider S = xcos () for x = (2.0 ± 0.2) cm, = (53 ± 2)°= (0.9250 ± 0.0035) rad. Find S and its uncertainty. Note: the uncertainty in angle must be in radians!Solution: (1) S = (2.0 cm)(cos 53°) = 1.204 cm
(3)
S = (1.20 ± 0.12) cm (standard deviation)
S = (1.20 ± 0.13) cm (average deviation)
(2)
PhysicsNTHUMFTai-戴明鳳
6. Rounding off answers in regular and scientific notation
A. In regular notation(1) Be careful to round the answers to an appropriate
number of significant figures.
The uncertainty should be rounded off to one or two significant figures.
If the leading figure in the uncertainty is a 1, we use two significant figures,
otherwise we use one significant figure.
(2) Then the answer should be rounded to match.
PhysicsNTHUMFTai-戴明鳳
ExampleRound off z = 12.0349 cm & z = 0.153 cm. Since z begins with a 1
round off z to two significant figures:
z = 0.15 cm Hence, round z to have the same number of
decimal places:
z = (12.03 ± 0.15) cm.
PhysicsNTHUMFTai-戴明鳳
B. In scientific notation When the answer is given in scientific notation, the
uncertainty should be given in scientific notation with the same power of ten.
If z = 1.43 x 106 s & z = 2 x 104 s, The answer should be as
z = (1.43± 0.02) x 106 s This notation makes the range of values most easily
understood. The following is technically correct, but is hard to
understand at a glance. z = (1.43 x 106 ± 2 x 104) s. Don't write like this!
PhysicsNTHUMFTai-戴明鳳
ProblemExpress the following results in proper rounded form, x ± x.
(1) m = 14.34506 grams, m = 0.04251 grams.
(2) t = 0.02346 sec, t = 1.623 x 10-3 sec.
(3) M = 7.35 x 1022 kg, M = 2.6 x 1020 kg.
(4) m = 9.11 x 10-33 kg, m = 2.2345 x 10-33 kg
PhysicsNTHUMFTai-戴明鳳
Problem How many significant figures are there in each
of the following? (1) 0.00042 (2) 0.14700 (3) 4.2 x 106 (4) -154.090 x 10-27
8.Problems on Uncertainties and Error Propagation.
PhysicsNTHUMFTai-戴明鳳
8. Seven Problems on Uncertainties and Error
Propagation