error analysis in measurement course.ppt

analysis of error

1. Error, their types, their
measurements

2. What is an error?
Some are due to
human error…
For example,
by not using the
equipment correctly
Let’s look at
some examples.

3. Human error
Example 1
Professor Messer
is trying to
measure the length of a
piece of wood:
Discuss what he is doing wrong.
How many mistakes
can you find? Six?

4. Human error
1.Measuring from 100 end
2.95.4 is the wrong number
3.‘mm’ is wrong unit (cm)
4.Hand-held object, wobbling
5.Gap between object & the rule
6.End of object not at the end of the rule
7.Eye is not at the end of the object (parallax)
8.He is on wrong side of the rule to see scale.
Answers:
How many did you find?

5. Human error
Example 2
Reading a scale:
Discuss the best position to put
your eye.
your
eye

6. Human error
2 is best.
1 and 3 give the
wrong readings.
This is called a
parallax error.
your
eye
It is due to the gap here,
between the pointer and
the scale.
Should the gap be wide or narrow?

7. Anomalous results
When you are doing your practical work, you may
get an odd or inconsistent or ‘anomalous’ reading.
This may be due to a simple mistake in reading a
scale.
The best way to identify an anomalous result is to
draw a graph.
For example . . .

8. Anomalous results
Look at this graph:
Which result do you
think may be
anomalous?
A result like this should be taken again, to check it.
x
x
x
x
x
x

9. ERRORS
If we are making physical measurements, there
is always error involved. The error is notated by
using the delta, Δ, symbol followed by the
variable representing the quantity measured.
For example, if we are measuring volume, the
error in measuring the volume would be
symbolized ΔV.

10. Calculating the Error
A simple way of looking at the error is as the
difference between the true value and the
approximate value.
i.e:
Error (e) = True value – Approximate value

11. Example:
Find the truncation error for at x= if the first 3
terms in the expansion are retained.
Sol: Error = True value – Approx value
e
x

















!
x
+x++
!
x
+
!
x
+x+=
2
1...
32
1
232
...
543
543
+
!
x
+
!
x
+
!
x
=

12. TYPE OF ERRORS
Type of errors
1)Gross error/human Errors
2)Random Errors
3)Systematic Errors
4)Constant Errors
5)Absolute Errors
6)Relative Errors
7)Percentage Errors
Static Errors

13. 1) Gross Error
cause by human mistakes in reading/using instruments
may also occur due to incorrect adjustment of the instrument and the
computational mistakes
cannot be treated mathematically
cannot eliminate but can minimize
Eg: Improper use of an instrument.
This error can be minimized by taking proper care in reading and
recording measurement parameter.
In general, indicating instruments change ambient conditions to some
extent when connected into a complete circuit.
Therefore, several readings (at three readings) must be taken to minimize
the effect of ambient condition changes.
TYPES OF STATIC ERROR

14. TYPES OF STATIC ERROR (cont)
2) Systematic Error
-due to shortcomings of the instrument (such as
defective or worn parts, ageing or effects of the environment
on the instrument)
In general, systematic errors can be subdivided into static and dynamic
errors.
Static – caused by limitations of the measuring device or the
physical laws governing its behavior.
Dynamic – caused by the instrument not responding very fast
enough to follow the changes in a measured variable.

15. -3 types of systematic error :-
(i) Instrumental error
(ii)Environmental error
(iii)Observational error
TYPES OF STATIC ERROR (cont)

16. TYPES OF STATIC ERROR (cont)
(i) Instrumental error
- inherent while measuring instrument because of their
mechanical structure (eg: in a D’Arsonval meter, friction in the
bearings of various moving component, irregular spring tension,
stretching of spring, etc)
- error can be avoid by:
(a) selecting a suitable instrument for the particular
measurement application
(b) apply correction factor by determining
instrumental error
(c) calibrate the instrument against standard

17. TYPES OF STATIC ERROR (cont)
(ii)Environmental error
- due to external condition effecting the
measurement including surrounding area condition
such as change in temperature, humidity,
barometer pressure, etc
- to avoid the error :-
(a) use air conditioner
(b) sealing certain component in the instruments
(c) use magnetic shields
(iii) Observational error
- introduce by the observer
- most common : parallax error and estimation error (while reading the scale)
- Eg: an observer who tend to hold his head too far to the left, while reading
the position of the needle on the scale.

18. TYPES OF STATIC ERROR (cont)
3) Random error
- due to unknown causes, occur when all systematic
error has accounted
- accumulation of small effect, require at high degree of
accuracy
- can be avoid by
(a) increasing number of reading
(b) use statistical means to obtain best approximation
of true value

19. What is systematic error?
Systematic error is caused by any factors that systematically affect
measurement of the variable across the sample.
For instance, if there is loud traffic going by just outside of a
classroom where students are taking a test, this noise is liable to
affect all of the children's scores -- in this case, systematically
lowering them.
Unlike random error, systematic errors tend to be consistently
either positive or negative -- because of this, systematic error is
sometimes considered to be bias in measurement.

20. These errors cause readings to be shifted one way (or
the other) from the true reading.
Systematic errors
Your results will be systematically wrong.
Let’s look at some examples . . .

21. Example 1
Suppose you are
measuring with a ruler:
Systematic errors
If the ruler is wrongly calibrated,
or if it expands,
then all the readings will be too
low (or all too high):

22. Example 2
If you have a parallax error:
Systematic errors
with your eye
always too high
then you will get a systematic error
All your readings will be too high.

23. A particular type of systematic error
is called a zero error.
Systematic errors
Here are some examples . . .

24. Example 3
A spring balance:
Zero errors
Over a period of time,
the spring may weaken,
and so the pointer
does not point to zero:
What effect does this have on all the readings?

25. Example 4
Look at this
top-pan balance:
Zero errors
There is nothing on it,
but it is not reading zero.
What effect do you think this will have
on all the readings?
It has a zero error.

26. Example 5
Look at this
ammeter:
Zero errors
If you used it like this,
what effect would it have
on your results?

27. Example 6
Look at this
voltmeter:
Zero errors
What is the first thing to
do?
Use a screwdriver here
to adjust the pointer.

28. Example 7
Look at this
ammeter:
Zero errors
What can you say?
Is it a zero error?
Or is it parallax?

29. Example 8
Look at this ammeter:
Zero error, Parallax error
What is it for?
How can you use it to stop parallax error?
It has a mirror
behind the pointer,
near the scale.
When the image of the pointer in the mirror
is hidden by the pointer itself,
then you are looking at 90
o
, with no parallax.

30. What is random error?
Caused by any factors that randomly affect measurement of the
variable across the sample.
Each person’s mood can inflate or deflate their performance on
any occasion. In a particular testing, some children may be in a
good mood and others may be depressed. Mood may artificially
inflate the scores for some children and artificially deflate the
scores for others.
Random error does not have consistent effects across the entire
sample. If we could see all the random errors in a distribution,
the sum would be zero.
The important property of random error is that it adds
variability to the data but does not affect average performance
for the group.

31. Random Errors
Random errors are “not inherent to the measuring process”. 
Frequently they are introduced by external factors that cause a
scattering of the measured data.
When the scattering is distributed equally about the true value, the
error can be mitigated somewhat by making multiple
measurements and averaging the data. 
• Vibration in mechanical devices produces random errors.
•  In electronic devices, noise produces random errors.   

32. Random errors
These may be due to
human error,
a faulty technique,
or faulty equipment.
To reduce the error,
take a lot of readings,
and then calculate the average (mean).

33. Constant Error
•When the results of observation are in error by the
same amount, the error is said to be a constant error.
e.g. if a scale of 15 cm actually measures 14.8 cm.
Then it is measuring 0.2 cm more in every
observation. This type of error will be same in all
measurements done by the scale.

34. Another types of Error
Three other ways of defining the error are:
Absolute error
Relative error
Percentage error

35. Calculation the Absolute Error
Absolute error.
e
a = |True value – Approximate value|
 Error=XX=e
'
a 

36. Calculating the Error
Absolute error:
e
a = |True value – Approximate value|
Relative error is defined as:
 Error=XX=e
'
a 












X
XX
=
ValueTrue
ErrorAbsolute
=e
'
r

37. 37
Absolute Error
The difference between the measured value and the
true value is referred to as the absolute error.
Assume that analysis of an iron ore by some
method gave 11.1% while the true value was
12.1%, the absolute error is:
11.1% - 12.1% = -1.0%

38. 38
Relative Error
The relative error is the percentage of the absolute
error to the true value. For the argument above we
can calculate the relative error as:
Relative error = (absolute error/true value)x100%
= (-1.0/12.1)x100% = -8.3%

39. 39
Relative Accuracy
The percentage of the quotient of observed result to
the true value is called relative accuracy.
 
Relative accuracy = (observed value/true
value)x100%
For the abovementioned example:
Relative accuracy = (11.1/12.1)x100% = 91.7%

40. Calculating the Error
Percentage error is defined as:






X
XX
=e=e
'
rp
100100

41. Examples
Suppose 1.414 is used as an approx to . Find the
absolute, relative and percentage errors.

2
  valueeApproximat– valueTrue=e
a
1.414213562=
 1.414-1.41421356∴=e
a
=0.00021356
error)(absolute

42. Examples
Suppose 1.414 is used as an approx to . Find the
absolute, relative and percentage errors.

2
1.414213562=






2
0.00021356
∴=e
r
3
100.151

=
error)(relative






ValueTrue
Error
=e
r

43. Examples
Suppose 1.414 is used as an approx to . Find the
absolute, relative and percentage errors.

2
e∴
p
=e
r
×100=0.151×10
−1
error)e(percentag

44. Example:
True value = 122 mm
expected value = 120 mm
Then:
a. absolute error = True value - expected value
absolute error = 122 mm – 120 mm = 2 mm Ans

b. relative error = absolute error / expected value
relative error = 2 mm / 120 mm =  0.017 Ans

Note: relative error has no units.
c. percent error = relative error · 100%
percent error = 0.017 · 100% =  1.7 % Ans


45. The following are general classifications for errors:
1.For consumer purposes, 5-10% error is acceptable
2.For engineering purposes, 1% error is acceptable
3.For scientific purposes, 0.1% error is acceptable
Classification of Error

46. Range of uncertainty is reported as a nominal value plus or minus
an amount called the tolerance.
Reported value: 120 mm ±1 mm = 119 mm to 121 mm
Range of Uncertainty
nominal value tolerancerange of uncertainty

47. Range of uncertainty is also reported as a nominal value plus or
min Range of uncertainty is reported as a nominal value plus or
minus an amount called the tolerance us an percent tolerance.

Reported value 120 mm ±2% = 117.6 mm to 122.4 mm
Note: 2% of 120 = 2.4, 120 - 2.4 = 117.6, 120 +2.4 = 122.4
Range of Uncertainty
nominal value tolerancerange of uncertainty

48. PERFORMANCE CHARACTERISTICS
Accuracy – the degree of exactness (closeness) of measurement
compared to the expected (desired) value.
Resolution – the smallest change in a measurement variable to
which an instrument will respond.
Precision – a measure of consistency or repeatability of
measurement, i.e successive reading do not differ.
Sensitivity – ratio of change in the output (response) of instrument
to a change of input or measured variable.
Expected value – the design value or the most probable value that
expect to obtain.
Error – the deviation of the true value from the desired value.

49. Measurement precision must be interpreted in light of
measurement accuracy.  Let’s use a target practice example:
Precision – Target 1
The best situation, the
shots are tightly
clustered (high
precision) on the center
circle (high accuracy).

50. Measurement precision must be interpreted in light of
measurement accuracy.  Let’s use a target practice example:
Precision – Target 2
The next situation, shots
are near the center (high
accuracy), but not tightly
clustered (low
precision).  

51. Measurement precision must be interpreted in light of
measurement accuracy.  Let’s use a target practice example:
Precision – Target 3
In the next situation, a
tight cluster (high
precision) is far off
center (low accuracy).

52. Measurement precision must be interpreted in light of
measurement accuracy.  Let’s use a target practice example:
Precision – Target 4
Finally, widely scattered
shots (low precision)
appear away from the
center (low accuracy).

53. Which is the best and which is worst?
Precision - Comparison
Best Most Insidious
Why?
Worst

54. Example
Given expected voltage value across a resistor is 80V.
The measurement is 79V. Calculate,
i.The absolute error
ii.The % of error
iii.The relative accuracy
iv.The % of accuracy

55. Solution (Example)
Given that , expected value = 80V
measurement value = 79V
i. Absolute error, e = = 80V – 79V = 1V
ii. % error = = = 1.25%
iii. Relative accuracy, = 0.9875
iv. % accuracy, a = A x 100% = 0.9875 x 100% = 98.75%
Y
n
−X
n
100
80
7980


100





n
nn
Y
XY







n
nn
Y
XY
=A1

56. Example
From the value in table 1.1 calculate Table 1.1
the precision of 6
th
measurement?
Solution
the average of measurement value

the 6
th
reading Precision =
No X
n
1 98
2 101
3 102
4 97
5 101
6 100
7 103
8 98
9 106
10 99
100.5
10
1005
10
99....10198
==
+++
=X
n
100.5
0.5
1
100.5
100.5100
1 





 = =0.995

57. LIMITING ERROR
The accuracy of measuring instrument is
guaranteed within a certain percentage (%) of full
scale reading
E.g manufacturer may specify the instrument to be
accurate at 2 % with full scale deflection
For reading less than full scale, the limiting error
increases

58. LIMITING ERROR (cont)
Example

Given a 600 V voltmeter with accuracy 2% full scale.
Calculate limiting error when the instrument is used to measure a
voltage of 250V?
Solution
The magnitude of limiting error, 0.02 x 600 = 12V
Therefore, the limiting error for 250V = 12/250 x 100 = 4.8%

59. LIMITING ERROR (cont)
Example
Given for certain measurement, a limiting error for voltmeter at 70V is
2.143% and a limiting error for ammeter at 80mA is 2.813%. Determine
the limiting error of the power.
Solution

The limiting error for the power = 2.143% + 2.813%
= 4.956%

60. Example
What is the relative error in the approximation X = 2.0 to x = 1.98?
Solution
Relative error = = = 0.01010 (to 5d.p.)
X
x
1-
2
1.98
1-

61. In summary
•Human errors can be due to faulty technique.
•Systematic errors, including zero errors, will
cause all your results to be wrong.
•Random errors can be reduced by taking many
readings, and then calculating the average
(mean).
•Parallax errors can be avoided.
•Anomalous results can be seen on a graph.

62. THE END