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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




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

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart1.html#ile

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart1.html#estimate

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart1.html#estimate

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart1.html#conflicts

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart1.html#stdError

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart1.html#range

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart1.html#relabs


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)

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart2.html#propagation%23propagation

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart2.html#addsub%23addsub

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart2.html#muldiv%23muldiv

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart2.html#powers%23powers

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart2.html#mixtures%23mixtures


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


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.


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.)


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


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.


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.


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.


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.


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.


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


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


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].


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.


....

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


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.


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.


(c) Products of powers: z =xmyn




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.


-- 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


(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)


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


(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


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



...22

22

22

2

yyfx

xfw

wfz


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)


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.


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.


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!


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


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.


8. Seven Problems on Uncertainties and Error

Propagation

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