how to do error analysis in laboratories

1. ERROR/UNCERTAINTY ANALYSIS
Manohar Nyayate
Emeritus Professor
UM DAE Centre for Excellence in Basic Sciences,
Kalina, Mumbai.

2. Outline of the Talk:
Types of Errors
Precision Vs Accuracy
Resolution Vs Sensitivity
Importance of error
Error/Uncertainty Estimation
Expressing the Results
Estimation of Uncertainty from Graphs
Laboratory Manners

3. Types of Errors
• Procedural Errors
– Instrumental Errors
– Environmental Errors
– Approximations
• Human Errors
• Random Errors

4. Instrumental Errors
• Causes
– Least Count Errors
– Contact Friction, Wear and Tear
– Component Nonlinearities
– Calibration Errors
– Transmission Losses (not important in small setups)
• Estimation
– By Comparison with standards
• Reduction / Elimination
– Careful Calibration
– Proper Maintenance
– Applying Correction Factor
– Using more than one methods for same measurements

5. Environmental Errors
• Causes
– Changes in laboratory condition which can affect the physical
parameters
• Estimation
– Careful monitoring of the environmental conditions
• Reduction / Elimination
– Isolating the apparatus from unstable environment.
– Shielding
– Choice of proper apparatus

6. Approximations
• Causes
– Simplified theory
– Experimental constraints
• Estimation
– Knowing how a certain approximation affects the answer
• Reduction / Elimination
– Devising a method which can bypass the constraints.
– Treating the affect due to approximation as an extra unknown

7. Human Errors
• Causes
– Misreading the instrument
– Improper choice of instrument
– Neglecting zero errors, loading effects etc.
– Erroneous calculations, wrong carry over, incorrect adjustments etc.
• Estimation
– NOT POSSIBLE
• Reduction / Elimination
– Careful attention to the detail
– Awareness of the limitations of the setup

8. Random Errors
• Causes
– Any other small scale random events which may change the reading
– Statistical Randomness
• Estimation
– Cursory look at the data is enough
• Reduction / Elimination
– Avoiding unwanted interference
– Evaluating the statistical randomness
– More readings

9. Precision Vs. Accuracy
• Accuracy
– Difference between measured and the true value of a quantity
• Precision
– The reproducibility of the reading with the same instrument
– Repeatability and reproducibility
• Example
– Suppose an instrument has large zero error but is in fine working
condition otherwise. The multiple readings taken by such instrument will
show it's precision but the answer won't be accurate.
– E.g. If a pencil has length of 10cm
– Instrument A measures it as 10.3, 10.3, 10.3, 10.3, 10.3 (precise but not
accurate)
– Instrument B measures it as 10.1, 9.9, 10.0, 10.0, 9.9 (accurate but not
precise)

10. Accurate and precise
Not accurate, not precise Accurate, not precise
Precise, not accurate

11. Resolution Vs. Sensitivity
• Resolution
– Significance of the least significant digit
– For e.g. If instrument shows reading of 19.22 cm then the resolution of
the instrument is 0.01 cm.
– In other words, smallest incremental quantity displayed by the
instrument
• Sensitivity
– Smallest change in the reading the meter can detect internally.
• which should be higher?
– If resolution is higher (in value and not number of digits) than the
sensitivity then we lose out on accuracy
– On the other hand it suppresses the flickering of the least significant
digit due to random errors
– If the instrument is not as sensitive as the number of digits on display
(apparent higher resolution) then last few digits on the display won't
have any physical meaning

12. Why error is important?
Foundations of Physics depends on it
Verification of general theory of relativity
1916 -- Bending of light from a star by sun was estimated 1.8”
1919 – Dyson, Eddington, Davidson --- 2” +- 0.15” (with 95% confidence)
(paper on error analysis !!?? – controversy on error estimate)
Recent Particle Faster than Light (serious error analysis made it vanish)
Value of Fine Structure Constant (a measure of strength of electromagnetic
Interaction between elementary charged particles) will decide future course of
universe
a = (1/2eo)*(e2/hc) = 7.29735257×10-3−
(a)-1 = 137. 035999074(44)
Are constants, really constant ?? (we need precision, example of c at the end)
Position of Pole star with respect to earth – shifting over 1000 years ?

13. Fig. 1. Distribution of the four-lepton events selected in the CMS analysis of H→ZZ(*)→4l. A clear accumulation of
events is responsible for the excess 6.7σ above the background-only expectation at 125.8 GeV. Despite the large
significance, the signal is 0.91+0.30–0.24times the expected amount for the Standard Model Higgs boson. The bottom
panel provides further information on the individual events entering the analysis, including the final-state type and the
per-event estimate of the mass and mass resolution.

15. Estimating errors in individual
measurements or observations

16. (i) The case of least count: Any measurement
has a least count and to start with, the least
count or half of the next digit becomes the
uncertainty in that measurement. If we are
measuring length, with a conventional meter
scale, the least count is 1mm. One can take 1
mm as the uncertainty in any length
measurement by this scale. If the length is
measured as 63 mm, then, the uncertainty in
that would be 1 mm and the observation could
be reported as 63 ± 1 mm.

17. (ii) The case of fluctuating DMM: If one were
using a DMM, and the meter reads 0.34* mV
with third significant digit fluctuating between
5 and 9, one could take the uncertainty as
0.002 mV and the observation could be
recorded as 0.347 ± 0.002 mV.

18. (iii) The case of steady DMM: If the DMM is
measuring a steady 0.347 mV, then the
uncertainty has to be taken to be the next
significant digit 0.0005 and the observation
could be reported as
0.3470 ± 0.0005 mV

19. (iv) The case of extrapolating a reading: It is
also possible to extrapolate the reading of an
observation, if the instrument would allow it
and the observer is willing. In the above
example of length measurement, one can
observe the length carefully, report a value
with a judged extrapolated value for the
further significant digit. For example, the
length could be 63.3 ± 0.5 mm or 63.3 ± 0.3
mm (it is in this error judgment, the subjective
element comes in. ± 1 mm is the maximum
possible error. ± 0.3 mm is the least possible
error in the observer’s judgment.

20. (v) The case of inability to judge: In some cases, like,
Vernier calipers, some times, it may be difficult to judge
which line of the Vernier scale is coinciding with the main
scale (such a thing could happen, either because, the scale
markers are not thin enough or because, there is parallax
between the main scale and the Vernier scale.). In such case,
one way to decide on the uncertainty could be as follows:
observe each of the Vernier line from the least value and
decide whether it could be a possible reading or not. Where
you are not able to decide, take that as the least possible
value. Then similarly decide for the upper value. The
difference between the two values divided by two gives the ±
uncertainty and the average of the two gives the value for
the observation.

21. For example: The 0 of the Vernier scale lies
between 4.3 and 4.4 cm. We observe from the
first line of the Vernier scale and find that the
marker lines 0, 1, 2, 3, are not coinciding. We
are not sure whether 4, 5, 6, 7, 8 are
coinciding. We are sure that 9 is definitely not
coinciding. Then, the observations imply that
4.34 cm is the least possible reading, 4.38 cm is
the maximum possible reading. Then we can
take the observation as: 4.36 cm ± 0.02 cm. (or
4.36 ± 0.02) cm.

22. (vi) The case of stop watch: In this case, the
least count of the stop watch is NOT the
uncertainty. The uncertainty has to be decided
by repeated measurement. For example, in the
case of pendulum experiment, measurement
of the time period has to be repeated a certain
number of times, get the average and
determine the uncertainty from the maximum
spread of the observed values with respect to
the average value.

23. In this case of stop watch, one hopes that the
reaction time of starting and stopping the stop watch
might cancel on the average and need not be
explicitly taken into account. Even if one takes only
one reading, the reaction time is likely to cancel out,
because, one is likely to have the same delay both in
starting and stopping the stop watch. Even then, one
cannot take the least count as the uncertainty
because, modern stop watches determine and show
time up to an accuracy of 0.01 s. For single readings,
it may be safe to take the maximum possible error as
0.5 s. (once again, the subjective element in the
uncertainty analysis comes in)

24. It is necessary that one estimates the errors BEFORE the experiment, the
uncertainties involved in the observables. This will enable one to decide how
accurately the measurements have to be done for the expected accuracy. For
example, if for want of time, one decides to have a result which is uncertain
to 10%, then accordingly one can relax the accuracy needed in the
observations. On the other hand, if one knows that one of the quantities is
going to dominate (in the propagation of the error) the uncertainty, then
more care can be given to that quantity alone. The case of thickness in
determination of Young’s modulus by bending method is an example, where,
the thickness occurs as a power. (square or cube ?).
Estimate Error Before Starting the Experiment

25. • Knowing the uncertainty and propagation
uncertainty also enables one to decide the
significant digits up to which the observations
have to be recorded or reported.
• DO NOT REPORT ALL THE NUMBERS THAT EXCEL SPREAD SHEET
SHOWS ON THE SCREEN. THIS IS A VERY IRRITATING HABIT.
Experiment on Density of water
Mass (gm) Vol (ml) Density
10.162 10.1 1.0061386138613900000000000
• MAKE USE OF THE FORMAT CELLS FACILITY AVAILABLE IN EXCEL.

26. Propagation of errors
When result is from a formula and the formula
has many terms (observables), then the way
the uncertainties in the observables get
reflected in the final result is propagation of
errors.

27. For sum and difference
(addition and subtraction):
• Addition and subtraction: Uncertainties of the
individual parameters add.
• Emphasis: Even in the case of subtraction, the
uncertainties of the individual parameters add
• z = x + y or z = x – y
• 𝛿𝑧 = (𝛿𝑥2 + 𝛿𝑦2)

28. The case of
Multiplication and Division
• The fractional uncertainties of individual
parameters add.
• z = x*y or z = x / y
• 𝛿𝑧/𝑧 =
𝛿𝑥
𝑥
2 +
𝛿𝑦
𝑦
2

29. Power
• z = (x)n
• (δz / z) = n × (δx / x)
• One has to be very careful while taking
readings of physical parameter which occurs
with higher powers in the formula.

30. Error is not precise – It is an estimate
PROBABLE error and MAXIMUM error
• Uncertainty, by its very nature, cannot be precise, and
has an element of subjectivity.
• For the same measurement, different observers may
prefer to take different uncertainty. But the difference
would not be of the kind to make big difference in the
conclusion.
• For example, some may like to quote the
probable error, whereas some may like to quote
maximum possible error.
(This in itself may depend on the context).
Later we will see an example on this aspect.

31. Reporting of error – the implication of 68%
probability.
• First we comment on the reporting of result including the error. After that
we comment on evaluation of the error or the uncertainty in the result.
• Any result should be quoted along with its uncertainty, in the following
manner:
• Result ± uncertainty (usually denoted by σ and called standard deviation –
where the ‘normal distribution’ value falls to 𝑒−1
2 .
• Note that the ± uncertainty does NOT mean that the result will definitely
be within the ± σ band.
• It means that the result will lie within ± σ band with a probability of 0.68
(this value comes from error integral).
• In other words, if a value is within ± σ band, one can say with 68%
confidence that the result represents the true value.

32. Confidence percent
• A result within 1σ gives a confidence of 68%
• A result within 2σ gives a confidence of 95.4%
• A result within 3σ gives a confidence of 99.7%
• A result within 4σ gives a confidence of 99.9%
• These numbers come from the integral of
Gaussian distribution over σ intervals:
• 𝑦 =
1
2𝜋 𝜎
𝑒
− 𝑥 − 𝑥
2
2𝜎2

33. Estimating errors
• For a set of N measurements {xi}
• Mean 𝑥 = ∑ xi /N
• Standard deviation 𝜎𝑥 =
1
𝑁−1
𝑑𝑖
2
∑(di)2) (σx)2 is called the variance,
• Note: N-1; --- for large N, N or N-1 does not make
significant difference
• Concept of standard deviation of one measurement
(σx) and standard deviation of MEAN
• 𝜎𝑥 =
𝜎𝑥
𝑁
.

34. Significant figure
• Error should be rounded off to ONE significant digit
(in some special cases, might be two significant
digits). The result should be quoted only up to the
significant digit of the error.
• e.g., if the evaluated error in e/m is 0.0275 x 1011
units, then the error should be quoted as
• ± 0.03 x 1011 units.
• Accordingly, if the e/m value is calculated as
• 1.7526 x 1011 units, then the result should be
reported as:
• (1.75 ± 0.03) 1011 units.

35. • There is NO theoretical value for results. We
only have expected value, or reported value or
literature value.
• Error is NOT the difference between the
experimentally obtained value and the
expected or reported in literature value

36. • The difference between the experimentally
obtained value and the expected or reported in
literature value is ‘DICREPENCY’
• One of the aims of the error analysis is to show
that the discrepancy is less than the error or
uncertainty in the result obtained in the
experiment. If this is so, then
THE EXPERIMENT HAS BEEN DONE
SUCCESSFULLY.

37. • Error or uncertainty exists, even if we do not
know what is the value of the result to be
expected.
• Even one does not know what the expected
value would be (such as a given glass prism
whose refractive index we may not know), we
still have an error to report

38. Error Analysis
• Average value
• Deviation of each value from
average
• Average / Mean deviation
• Standard Deviation and Variance
• Chebyshev’s Inequality
• For any random distribution, (1 –
1/k2)x100% values are within k
standard deviations from the mean.
 )

=

=
N
i
i x
x
N 1
2
1

σ
68.26894921371
% =68.27%
2σ
95.44997361036
% =95.45%=====
3σ
99.73002039367
% =99.73%
4σ
99.99366575163
% =99.99%
5σ
99.99994266969
% =99.99%
6σ
99.99999980268
% = 99.99%
7σ
99.99999999974
%

39. ANALYSIS OF DATA
• Measurement of Diameter of a given ball using
two different length measuring instruments,
Vernier Calliper and Micrometre Screw gauge.

40. Year Experimenter method Speed (km/s) Uncertaint
y (km/s)
% Uncertainty
1600
(?)
Galileo Lanterns and
Shutters
Extraordinarily
rapid
1676 Roemer Moons of Jupiter “finite”
1729 Bradley Aberration of star
light
304,000
1849 Fizeau Toothed Wheel 313,300
1862 Foucault Rotating mirror 298,000 500 0.2 %
1880 Michelson Rotating mirror 299,910 50 0.02 %
1923 Mercier Standing waves
on wires
299,782 15 0.005 %
1926 Michelson Rotating mirror 299,796 4 0.001 %
1958 Froome Microwave
interferometer
299,792.50 0.10 0.00003 %
1967 Grosse Geodimeter 299,792.5 0.05 0.000015 %
1973 Evenson et al. Laser technique 299,792.4574 0.0012 0.0000004 %
1978 Woods et al. Laser technique 299,792.4588 0.0002 0.00000007 %
1987 Jennings et al. Laser technique 299,792.4586 0.0003 0.0000001 %
Progress in “certainty” in value of Speed of light (electromagnetic radiation)
(taken from Physics – Halliday, Resnik, and Walker vol 2 p888)

41. Graph Plotting and Error Estimation from Plots
One efficient tool of analysis is graph plotting.
The independent variable (or secondary data obtained from it) is
plotted on the x-axis and the corresponding data representing
dependent variable on the y-axis.
 Velocity of falling ball measured at different times
 Period of simple pendulum measured for various lengths
 Rate of thermal expansion of a thin metal strip
 Correlation in sprint record times
 Depression of bending of beam measured at different
loads

42. Least squares fit of data (curve fitting)
• If one knows the relationship of the result with
respect to the observable parameters involved in the
experiment, then one can use the Least Square
Fitting procedure to compute the estimated errors in
the parameters and the result.
• The procedure is simple in the case of polynomial
function
• Simplest when the relationship is expected to be a
linear
• General plotting programs gives an option for fitting
curves to data.

43. A few Graphs…

45. Origin (x0,y0)

48. The scales along the x and y axes are chosen in such a way that the
graph occupies maximum part of the graph paper.
The smallest unit on graph is a simple multiple of the unit of data.
The quantities plotted should be written along with their units beside
the axes.
“Do not take scale like 3 cm of graph paper equal to 1 unit of data.”

49. If the unit of graph is smaller than the unit of data, then care has to be
taken to see that the uncertainties in the results obtained from the
graph must be calculated from the uncertainties of the data.
Data: 10.3 sec, 10.8 sec, 11.5 sec
Scale: y-axis: 1cm = 0.1sec
What this means is that error bars representing the uncertainty in
the data must be shown on the graph.

50. When the graph is supposed to be a straight line, it is essential to see
which is the best fit line.
If we believe that there is only one line representing the data, then
there will be no uncertainty in the slope of the line and it will mean
that there is no error in data plotted.
…obviously, this can’t be correct.
If all the points appear to lie on a straight line then it is simple to
identify the line by simply drawing the line passing through all the
points.

51. Generally it is assumed that the uncertainties in the independent
variable are negligible. Hence, error bars are shown to represent the
uncertainties in the dependent variable only.
The position of the point is thus shown to be anywhere along the
line segment shown by the error bar.
The best fit straight line can, therefore, be any line passing through
all the error bars.
The lines crossing all the error bars and with maximum and
minimum slopes will then be giving the extent of uncertainty of the
best fit line.

54. If there is large scatter and the distances of various points from the
best fit are more than the length of the error bar, then showing the
error bars becomes useless because random errors (or systematic
errors) in data are larger and error bars need not be plotted.
If all the points do not lie on the same straight line, visual judgment is
to be used to find the two lines, which appear to represent the best fit
within reasonable limits. According to arguments based on statistical
theory the line with best fit should pass through the point called centre
of gravity with co-ordinates {average values}. So the two lines
may be drawn to pass through ‘the centre of gravity’ and the values of
maximum and minimum slopes may be calculated as above.
)
,
( x
y

55. Laboratory “Manners”
• Before starting the experiment (Planning):
– Be sure about:
• What you are measuring
• Plan of Action
• Apparatus needed and if it’s available
• Time management
• Objectives and are they realistic?
• Proper alignment

56. Laboratory “Manners”
• During the experiment
– Arrangement of the apparatus
– Be honest in recording your observations
– Strictly follow the precautions
– Tidiness and care of the apparatus
– Work ethics
• After the experiment:
– Reporting the results
– Rearranging everything back

57. Last words
• You have successfully completed an experiment when,
– You are crystal clear about the science behind it
– You are confident about your observations
• This does not mean that you must get “right” answer
– You can explain how and why for each part of the experiment to a third
person
– You are aware about limitations of your experiment
– Made an ordered report of your experiment

58. References
1. “An Introduction to Error Analysis” by
John R. Taylor, 2nd edition,
Pub: University Science Books, California, 1997
2. “A practical guide to Data Analysis for
Physical Science Students”, by
Louis Lyons,
Pub: Cambridge Univ. Press, 1991
3. “Manual for Experiments kit of Academies”,
R. Srinivasan and K.R. Priolkar

59. Thanks