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INSTRUMENTATION &
MEASUREMENT
Chapter 2:
Error InMeasurement
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CHAPTER OBJECTIVES
In this chapter, student should be able to:
a) Define termsrelated to the error in
measurements.
b) Use basic concept of measurement in error
analysis.
c) Perform basic statistic analysis calculations.
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ERROR IN MEASUREMENT
a) Absolute Error = Expected ValueMeasured Value
a) Percent Error =
nn XYe
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Relative accuracy and percent
accuracy
c) Relative Accuracy, A=
d) Percent of accuracy
Accuracy can be expressed as percent of
accuracy, a, is:
n
nn
Y
XY 1
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Question 1
1. The expected value of the voltage across a
resistor is 50V; however, measurement yields
a value of 49 V. Calculate:
a) The absolute error
b) The percent of error
c) The relative accuracyd) The percent of accuracy
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ACCURACY
Accuracyrefers to how close a measurement
is to the correct/accepted value.
Precisionrefers to the degree of agreement
with a group of measurement or instruments.
n
nn
X
XX
precision
1
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ACCURATE VS PRECISE
Accurate-> after taking a lot of
measurements, you find they agree with the
true value
Precise-> after taking a lot of measurements,
you notice that they are all very close to each
other.
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Precision
Calculate the precision of the fourth
measurement Measurement Output Voltage,V
2 12.11
2 11.99
3 13.004 11.88
5 11.99
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TYPES OF ERROR
Types of Error
Gross Error
SystematicError
InstrumentalError
ObservationalError
EnvironmentalError
Random Error
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Gross Errors
Caused by the fault of the person using the
instrument, incorrect recording of
experimental data, incorrect adjustment and
improper application of instruments and
computational mistakes.
How to avoid gross errors???
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Avoiding Gross Error
The following actions may be necessary to
reduce the effects of gross errors:
Great care should be taken in reading and
recording the data.
Different experiments should take two or more
readings of the same measured quantity.
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Systematic Errors
a) Instrument errors: Caused by the construction, calibration or operation
of mechanical structure in the instruments.
Misuse of the instruments. For example, these may
be caused by failure to adjust zero of the
instruments.
Loading effect of the instruments
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Systematic Errors
b) Observational errors.
Observational errors are those errors introduce by
the observer. The two most common
observational errors arise due to two mainreasons:
Type of instrument display, whether it is analog or
digital
Parallax (eye should be directly in-pine with the
measurement point).
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Systematic Errors
c) Environmental errors.
Due to condition external to the measuring
device such as the area surrounding the
instrument.
These conditions may be caused by the changes
in pressure, humidity, dust, vibration or external
magnetic or electrostatic fields.
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Avoiding Systematic Errors
These errors can be eliminated or at least
reduced by using the following methods:
The procedure of measurement must be
carefully planned
Correction factors should be applied after
detection of these errors
Re-calibration the instrument carefully
Use the instrument intelligently
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Random Errors
Due to unknown causes.
In some experiments, the results show
variation from one reading to another; even
after all systematical and gross errors have
been accounted for.
The best way to offset the errors is by
increasing the number of readings and use
statistical means to obtain the approximation
of the true values of the quantity under
measurement.
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STATISTICAL ANALYSIS IN
MEASUREMENT
The evaluation based on a statistical analysis a
Allows us to obtain such information as the
mean value, average deviationand standard
deviationof our data.
Such information allows us to make
quantitative judgments on the variations or
error in our data.
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STATISTICAL ANALYSIS IN
MEASUREMENT
Mean value / Arithmetic mean
n
xxxxx n
....321
Deviation
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STATISTICAL ANALYSIS IN
MEASUREMENT
Average Deviation
n
dddD n
....21
Standard Deviation
n
ddd n22
2
2
1 ...
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Example
a) The arithmetic mean
b) Deviation of each value
c) Sum of the deviations
d) Average Deviation
e) Standard Deviation
,
,
,
1.501 x 7.49
2 x 6.49
3 x 2.50
4 x
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Question 2
Table 1 showsthe output voltage of an amplifier
was measured by the student. Calculate the:
a) Arithmetic mean.
b) Algebraic sum of deviation.c) Average deviation.
d) Standard deviation.
Measureme
nt
Output
Voltage, V
1 12.11
2 11.99
3 13.00
4 11.885 11.99
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LIMITING ERROR
Most manufactures of measuring instruments
state that an instrument is accurate within a
certain percentage of a full-scale reading.
The limits of these deviations from the
specified values are known as limiting errors
or guarantee errors.
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LIMITING ERROR
For example, the manufacture of a certain
voltmeter may specify the instrument to be
accurate within with full-scale deflection.
This specification is called limiting error and
means that a full-scale reading is guaranteed
to be within the limits of 2%of a perfectly
accurate reading
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LIMITING ERROR
Another example, if the resistance of a resistor
is given as 5000 10%, the manufacturer
guarantees that the resistance is between
4500 and 5500 (limiting error is 10% of 5000i.e. 500).
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Question 1
1. A 300V voltmeter is specified to be accurate
within 2% at full scale. Calculate the limiting
error when the instrument is used to
measure a 120V source.
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Question 2
1)The resistor has three different ratings:
R1 = 15 5%
R2 = 33
2%R3 = 75 5%
Determine the limiting errors for all the three
resistors.
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Assignment 1
Given the resistors with following rating:
R1=505%
R2=60
5% Determine:
i. Magnitude of error in each resistor
ii. If the resistors are connected in series, calculate the
resistance and limiting error in ohms and in percentage.
iii. If the resistors are connected in parallel, calculate the
resistance and limiting error in ohms and in percentage.
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EXERCISE 2
The current in a certain circuit was measured and the
results are tabulated as shown in following data.