Errors and uncertainties

Lecture note on Error(uncertainty)
• This is totally out of the format of
presentation. In this slide I have tried to have
slides of lecture notes. It is useful for As level.
Believe me it is not the waste of time going
through this slide.

Errors and Uncertainties
 understand and explain the effects of systematic
errors (including zero errors) and random errors
in measurements
 understand the distinction between precision
and accuracy
 assess the uncertainty in a derived quantity by
simple addition of absolute, fractional or
percentage uncertainties (a rigorous statistical
treatment is not required)

Understand and explain the effects of systematic errors
(including zero errors) and random errors in
measurements
• Every students in their beginning class of practical class are found to get
worried when they find that they obtain error while performing
experiments. It is expected; because prior to being exposed to this lab
class they have the notion error is the mistake or blunder. So thinking that
they are wrong in their experiments they are worried they have to redo it
to eliminate it. They are relaxed when they happen to know that error in
practical experiments is inevitable and actually error is not the mistake
but the essential part in measurement. They are correct in their
performance by actually getting error in their measurements. They are
pleased to find it is not the error that need to be eliminated but it is their
responsibility to know how different types of error occur in their
measurement and how to reduce all these sorts of error to minimum.
Key Concepts: Error is not the blunder or mistake; instead it is inevitable in
measurements. Error is not to be eliminated but its nature is to be known
and it is to be reduced to the minimum.

Inevitable Error
• Let us understand through an experiment the concept of Inevitable Error.
We are to perform an experiment where we investigate the height to
which a ball rises when it has been released from a stretched piece of
rubber. Experiment is illustrated in fig below.
Fig 1. Illustration showcasing an experiment to investigate the height
to which a ball rises when it has been released from a stretched piece
of rubber.

Contd…..
In this experiment we come across many sources of error. Any four are mentioned
below:
1.We performed readings only twice.
2.We used hand to release the ball hence uneven force acts on the ball the time it is
released.
3.Not correct readings of the meter rule (parallax error)
4.Difficult to measure the maximum height due to the motion of the ball.
Now let us choose any one source of error . we choose the 3rd
one i.e the error
related to the readings of the meter rule(parallax error)
Parallax error is caused as one does not observe perpendicularly to the meter
rule(ie place oneself directly in front of meter rule). We did our best to adjust our
height and the height of the experimental apparatus so that we are able to read the
meter rule perpendicularly. Despite our best effort to eliminate parallax error we are
still not left by another sort of error obtained in measurement(through observation).
That error is due to difficult posed by the scale itself as it is distributed over space of
the meter rule.
Fig 1 Error in the readings through the
instrument itself.

Contd….
As seen in fig 1. If we are bit careful we find the marking related to 15 cm has been
spaced to a certain distance. You notice the left end marking of 15 cm and right end of
marking 15 cm separated by certain distance though it is a negligible distance. This
discrepancy is to be taken into consideration for finding out result nearest to true
value ( though as suggested above it is usually neglected). Hence, though we do our
best to minimize or eliminate completely parallax error there is an error as mentioned
above which force us to have measurements less than the true value. From the
above example we conclude that the error is inevitable.
Key concept: Error is inevitable. Our duty is to minimize these errors not to eliminate
them.

Error and its nature
When we measure physical quantities , we compare this measured value with the
theoretically accepted value. Theoretically accepted value is regarded to be the true
value. This discrepancy between so called true value(since true value actually does not
exist or else experiments won’t be done of that quantity whose true value had been
already found) and the measured value is known as error. There are basically two
types of error namely systematic error and random error. To clarify it more, let us
recall using stop watch for the measurement of time in our lab. We are to start and
stop the watch to find out the time interval between the occurrence of any event. We
started the stop watch as to indicate the beginning of the event and stopped the
watch to indicate the completion of the event. We are very much careful in starting
and stopping the stop watch . However, you still cannot start the watch exactly at the
instant the event is started . There is some difference in time between you start the
watch and the start of the event. This difference is known to be human reaction time.
Let we give it name Hi. Similarly, you cannot stop the watch at the instant the event is
stopped. This brings another discrepancy in the time .Let us name it as the human
reaction time for stop Hff. Now if Hf=Hi, these two human reactions cancel and it does
not affect the actual time duration in between the event. But in the case Hf ≠ Hi,
there is error while measuring the actual time duration in between the event. Now if
Hf>Hi we overestimate the time and if Hi>Hf we underestimate the time. Hence in the
repeated trial this measurement has error because of both underestimate and
overestimate .The error is not in a systematic order and give rise to a random error.

Contd…
Similarly let us recall the experiment of measuring physical quantity by a micrometer
screw guage. In some of the micrometer screw guage, when we enable spindle meet
anvil with no substance in between by rotating thimble(usual trend is by rotating
ratchet knob, but in some books I read it is a wrong practice) some observation is
against our usual thought. We think- when spindle and anvil are made to touch each
other with the ratchet knob making just a single tick sound and with no substance in
between-the zero scale in the circular scale has to exactly coincide with the datum
line( a straight line on the main scale). But sometimes we find the datum line on the
main scale slightly above or below the zero marking on the thimble( circular scale).
When zero reading on the thimble is below the datum line then it overestimates the
measurement (positive zero error)and when zero reading on the thimble is above the
datum line then it underestimates the measurement(negative zero error).If repeated
trial is performed using micrometer screw guage you find the error either due to
overestimate or underestimate but not due to both overestimate or underestimate in
a single repeated experiment. This error is systematic hence known as systematic
error.
Fig1 labeled micrometer screw gauge

Contd…
Random error: Indefiniteness of result introduced by finite precision of measurement or
statistical variations. Measure of fluctuation after repeated experimentation.
Systematic error: Reproducible inaccuracy introduced by faulty equipment, calibration or
technique.
Reference: Reduction and error analysis for the physical sciences,3rd
edition Philip
R.Bevington , D.Keith Robinson)
Zero error in micrometer screw gauge is an example of systematic error. If the error is
due to overestimate of the actual reading, every readings done with that micrometer
screw gauge reproduces error(error which is due to overestimate of the actual reading).
Similarly if error is due to underestimate of the actual reading, every readings done with
that micrometer screw gauge reproduces error(error which is due to underestimate of the
actual reading). The systematic error we discussed above is due to faulty equipment.
Similarly we have watch that is running slow compared to the actual time. The error in
the time after every measurement with this watch is slow . This is a systematic error
caused due to miscalibration of watch.
On performing trial many times with the same instrument, systematic error cannot be
reduced. To reduce this sort of error we need to change the instrument or even the
method..

Contd….
Similarly, as has been discussed above ,due to the error in human reaction time of the
observer the stop watch overestimates or underestimates the actual value. In many trials
done for the same experiment, the reading fluctuates . Some readings are more than the
actual value and some are lesser than the actual value. This error is the random error
and it can be minimized by the repeated trial. Random errors are due to parallax error
and the error in the micrometer screw gauge as we press harder(uneven pressure
applied) during measurements etc.
Key concepts: Random errors are known in a greater detail through repeated trials in
a single experiment whereas, systematic errors are unknown through repeated trials
in a single experiment using the same instrument or same method. Systematic errors
are errors in a systematic order(errors are constant and does not fluctuate in a
repeated trials in a single experiment) whereas, random errors are not errors in a
systematic order they are random in nature and the readings fluctuate in repeated
trials in a single experiment.
Note: Parallax error can be both systematic and random error. For detail refer section
4.1 Random and Systematic Errors of the book entitled An Introduction to Error Analysis
by John Taylor

Understand the distinction between precision and accuracy
Accurate measurement: Any observed measurement is an accurate measurement if it
is close to the true value. The accuracy of a measurement determines the discrepancy
or difference between the observed value and the true value . Lesser the discrepancy,
more accurate is the measurement, and if more the discrepancy lesser is the accuracy.
Percentage error determines the accuracy of the measurement. Greater the
percentage error, less accurate is the measurement.
Precision determines how close are the readings of different trials carried out repeatedly
in a single experiment. Precision alone is not sufficient for measurement to be accurate.
For example on using micrometer screw guage to measure the diameter of the spherical
ball , we became very careful to reduce errors and the readings of repeated trials are
close to each other. We thought we have found accurate readings .Only later we found
that our micrometer screw guage is prone to zero errors and we did not take that into
account in our measurement. Hence, though our measurement was precise but it is not
accurate.
The accuracy of an experiment is a measure of how close the result of the experiment is
to the true value; the precision is a measure of how well the result has been determined,
without reference to its agreement with the true value. The precision is also the measure
of the reproducibility of the result in a given experiment. Reference : Reduction and
error analysis for the physical sciences,3rd
edition Philip R.Bevington , D.Keith Robinson

Pictorial distinction between accuracy and precision
Courtesy(: Reduction and error analysis for the physical sciences,3rd
edition Philip
R.Bevington , D.Keith Robinson)
Fig(a) depicts
precise but
inaccurate
data.Fig(b) depicts
accurate but
imprecise data.

How to determine which set of results could be described as precise but not
accurate ?
The set with more precision is with least mean deviation.
Values
x
d=
|x-xm|
9.81 0.01
9.79 0.03
9.84 0.02
9.83 0.01
Set A
Mean value xm=9.82
Mean
deviation=0.07/4
=0.02
Values
x
d=
|x-xm|
9.81 0.41
10.12 0.72
9.21 0.19
8.46 0.94
Set B
Mean value=9.4
Mean
deviation=2.26/4
=0.56
Values
x
d=
|x-xm|
9.45 0.35
9.21 0.11
8.99 0.11
8.76 0.34
Set C
xm=9.10
Mean
Deviation=0.91/4
=0.23
Values
x
d=|x-xm|
8.45 0.01
8.46 0
8.50 0.04
8.41 0.05
Set D
Xm=8.46
Mean
Deviation=0.1/4=0.02
Here it is observed set A and set D with the least mean deviation is more precise but we
find set D is not accurate. Hence D is more precise but not accurate.

Precision, significant figures and Uncertainty
We have observed in the practical lab, scale measures up to nearest millimeter(least
count is 1 mm), vernier caliper to the one tenth of a millimeter(least count is 0.1mm)
and micrometer screw gauge to the one hundredth of a millimeter(least count is
0.01mm). Suppose we are to measure the length of pencil as illustrated in fig. As
shown , with scale we are uncertain whether length is 9.5 cm or 9.6cm.Out of two
significant figures, the first digit is known to be correct while the last digit is uncertain.
The last digit is in the unit place. Hence the uncertainty is 0.1 cm or 1mm.On
measuring the length of the same pencil by vernier caliper we measure the length as
9.55cm. We are certain upto the first two digits but the last digit is uncertain. The
uncertainty is 0.01cm or 0.1 mm. Similarly using micrometer screw gauge we obtained
the measurement as 9.555cm ie we are certain to first three digit but the last digit is
uncertain. Hence uncertainty is 0.001 cm ie 0.01mm. We observed that uncertainty is
expressed by significant numbers. Precision is in turn expressed by uncertainty. .
Hence , we conclude that significant numbers are used for determining the precision of
the measurements.

Significant figures and round off
1.The leftmost nonzero digit is the most significant digit.
2.If there is no decimal point, the rightmost non zero digit is the least significant
digit.
3.If there is a decimal point, the rightmost digit is the least significant digit even
if it is zero.
4.All digits between least and the most significant digits are counted as
significant digits.
5.Zeros that set only the decimal point are not significant. Both 0.000555 and
0.555 contain three significant figures.
6.Zeros not used to hold the decimal point are significant . For example 5.000
has four significant digits.
7.Zeros that follow a number may be significant. Generally in this case it is
preferred to write the number in a scientific notation
Rules for rounding off significant digits
1.If the fraction is greater than ½ , increment the new least significant digit.
2.If the fraction is less than ½ ,do not increment.
3.If the fraction equals ½, increment the least significant digit only if it is odd.
Reference: Reduction and error analysis for the physical sciences,3rd
edition Philip R.Bevington , D.Keith Robinso

Rules for significant numbers
1)When adding or subtracting, the sum or difference is rounded to the last decimal
place in the least precise number .
For example(Addition)
1.02
5
8.6
—
14.62=15
2)When multiplying or dividing, the product or quotient is the rounded to the number
of significant digits in the number with the least number of significant figures.
5.3224*5.03=26.771672=26.8
4.5565/2.13=2.139201878=2.14

Assess the uncertainty in a derived quantity by simple addition of
absolute, fractional or percentage uncertainties (a rigorous statistical
treatment is not required)
Suppose on measurement of time by stop watch, we found different readings for
3 different trials .First trial reads 3.1s, second trial reads 3.2s and third trial reads
3.3s.In this experiment, the best estimate of time=3.2s and the probable range is
from 3.1s to 3.3 s. These readings can be written in a single statement as
:measured value of time =3.2 ± 0.1s. In general the result of any measurement of
a quantity x can be written as (measured value of x)=x(best) ± ᵟ(x)-----(1)
Expression( 1) means that our best estimate for the quantity concerned is the
number x(best) and we are reasonably confident the quantity lies somewhere
between x(best)- ᵟ(x) and x(best) + ᵟ(x).The number ᵟ(x) is called the uncertainty
or error.
Rule for stating uncertainties:
The uncertainty ᵟ in the final result should have at most 2 digits and more
commonly only 1 digit.
ᵟ(x) being an estimate of uncertainty , it should not be stated with too much precisely
For example, it would be absurd to write (measured g)=9.82 ± 0.02385 m/s^2.
Experimental uncertainties should be rounded to one significant figure. Hence, we
write (measured g)=9.82 ±0.02 m/s^2.
The only exception to the rule mentioned above is-if the leading digit in the
uncertainty is 1,then keeping two significant figure in ᵟ(x) may be better.
Rerence:An introduction to error analysis,the study of uncertainties John R .Taylor

Contd…
For example suppose the calculation of some measurements gave the uncertainty
ᵟ(x)=0.14.Rounding this number to ᵟ(x)=0.1 would be a substantial proportionate
reduction, so we could argue that retaining two significant figures must be less
misleading, and quote ᵟ(x)=0.14.The same argument be perhaps be applied if the
leading digit is 2 but certainly not if it is any larger.
Once the uncertainty in a measurement has been estimated ,the significant figures in the
measured value must be considered. A statement such as
Measured speed=6051.78 ± 30 m/s is obviously ridiculous. The uncertainty of 30 means
5 may be as small as 2 or as large as 8.Clearly trailing digits 1, 7, 8 have no significance
at all and should be rounded. That is the correct statement is measured speed=6050
±30 m/s. The general rule is :
The last significant figure in any stated answer should usually be of the same
order of magnitude(in the same decimal position) as uncertainty.
For example the answer 92.81 with uncertainty 0.3 should be rounded as 92.8 ± 0.3
If its uncertainty is 3 , the same answer should be rounded as 93 ±3.
And if the uncertainty is 30, the answer should be rounded as 90 ±30.
Note: The slide is exact copy of chapter 2 How to report and use uncertainties page 16
of the book Introductory Error Analysis by John Taylor. No modification is made and is
exactly copied so as not to loose the original flavor of the topic and not to bring any
misunderstandings in the concept

Absolute, fractional and percentage uncertainty and error propagation
In the expression(measured value)=x(best) ±ᵟ(x)
ᵟ(x) is the absolute uncertainty in x and it carries the same unit as x.
Relative uncertainty is defined as the percentage or fraction of the best estimate value. It
is the ratio of uncertainty’ᵟ(x)’ to the x(best).Hence for expression 9.81 ± 0.02;
0.02 is the absolute uncertainty , 0.02/9.81 (ᵟ(x)/x(best)) is the fractional uncertainty and
(0.02/9.81)* 100 is the percentage uncertainty.
Derived units are units that are derived from fundamental units. Suppose we are to
calculate volume. We know Volume is a derived unit and it is written as
Volume=length(l)*breadth(b)*height(h). In deriving volume as we see from above
expression we need to find out length, breadth and height. All these physical quantities
have uncertainty and this uncertainty is propagated in volume as well. Now we will see
the rules for propagation of error.
a)Addition or subtraction:
Suppose A=B+C; and ᵟA , ± ᵟB and ± ᵟC are the uncertainties in A, B and C
respectively, then ᵟA= ± (ᵟB+ᵟc)
Also if A=B-C, then ᵟA= ± (ᵟB+ᵟc)
b)Product
Suppose V=lbh; and ᵟl,ᵟb and ᵟh are the uncertainties in l, b and h respectively , then
v/v=ᵟ ± (ᵟl/l+ ᵟb /b+ ᵟh /h)

Contd…
Speed(v)=Distance covered(d)/time taken(t) and ᵟd and ± ᵟt are the uncertainties in
distance and time respectively, then ᵟ(v)/v= ± (ᵟd /d+ ᵟt /t)
C) Quotient
D) Multiply with a constant
If s=5x and ᵟx is the uncertainty in x , then ᵟs/s= ± 5*(ᵟx/5x)= ± ᵟx/x
If U=x^r, where r is constant and ᵟu and ᵟx are the uncertainty then maximum uncertainty
then ᵟ(u)/u= ± r*ᵟx/x
Numericals related to error propagation will be done in the class.
E)Power
The End

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