Error, their types, their
measurements
What is an error?
Some are due to
human error…
For example,
by not using the
equipment correctly
Let’s look at
some examples.
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?
Human error
Answers:
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.
How many did you find?
Human error
Example 2
Reading a scale:
Discuss the best position to
put your eye.
your
eye
Human error
2 is best.
1 and 3 give the
wrong readings.
This is called a
parallax error.
It is due to the gap here,
between the pointer and
the scale.
Should the gap be wide or narrow?
your
eye
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 . . .
Anomalous results
Look at this graph:
x
x
x
Which result do you
think may be
anomalous?
x
x
x
A result like this should be taken again, to check
it.
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.
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
Types of Errors
What is systematic error?


Systematic error is caused by any factors that
systematically affect measurement of the
variable across the sample.
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.
Systematic errors
These errors cause readings to be shifted
one way (or the other) from the true reading.
Your results will be systematically wrong.
Let’s look at some examples . . .
Systematic errors
Example 1
Suppose you are
measuring with a ruler:
If the ruler is wrongly
calibrated, or if it expands,
then all the readings will be too
low (or all too high):
Systematic errors
Example 2
If you have a parallax
error:
with your eye
always too high
then you will get a systematic error
All your readings will be too high.
Systematic errors
A particular type of systematic error
is called a zero error.
Here are some examples . . .
Zero errors
Example 3
A spring balance:
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?
Zero errors
Example 4
Look at this
top-pan balance:
It has a zero error.
There is nothing on it,
but it is not reading zero.
What effect do you think this will have
on all the readings?
Zero errors
Example 5
Look at this
ammeter:
If you used it like this,
what effect would it have
on your results?
Zero errors
Example 6
Look at this
voltmeter:
What is the first thing to
do?
Use a screwdriver here
to adjust the pointer.
Zero errors
Example 7
Look at this
ammeter:
What can you say?
Is it a zero error?
Or is it parallax?
Zero error, Parallax error
Example 8
Look at this ammeter:
It has a mirror
behind the pointer,
near the scale.
What is it for?
When
theyou
image
pointer
in theerror?
mirror
How can
useofit the
to stop
parallax
is hidden by the pointer itself,
then you are looking at 90o, with no parallax.
TYPES OF ERROR
Random error
- due to unknown causes
- can be avoid by
(a) increasing number of reading
(b) use statistical means to obtain best
approximation of true value
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.
Random error does not have consistent effects
across the entire sample.
The important property of random error is that it
adds variability to the data but does not affect
average performance for the group.
Random errors
To reduce the random error,
take a lot of readings,
and then calculate the average
(mean).
Other types of Error
Three other ways of defining the error are:
Absolute error
Relative error
Percentage error
Calculating the Absolute Error
Absolute error.
ea = |True value – Approximate value|


ea = X  X ' = Error 
Calculating the Error
Absolute error:
The difference between the measured value
and the true value is referred to as the absolute
error.
ea = |True value – measured value|


ea = X  X = Error 
'
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:
12.1% - 11.1% = 1.0%
29
Relative Error
The relative error is the ratio of the absolute
error to the true value.
Relative error = (absolute error/true value)x100%
= 1.0/12.1) = 0.083
Absolute
Error
X

X




=

er =
 True Value   X 
'
30
Percentage error
Percentage error is defined as the relative error times
100.
X

X



e p = 100er = 100
 X 
'
Examples
Suppose 1.414 is used as an approx to 2 . Find the
absolute, relative and percentage errors.
2 = 1.41421356
ea = True value – Approximat e value 
(absolute error)
∴ ea = 1.41421356 -1.414
= 0.00021356
Examples
Suppose 1.414 is used as an approx to 2 . Find the
absolute, relative and percentage errors.
2 = 1.41421356
Error



er =
True Value
0.00021356



∴ er =
2


(relative error)
= 0.15110
3
Examples
Suppose 1.414 is used as an approx to 2. Find the
absolute, relative and percentage errors.
∴ e p =e r × 100
= 0.15110 1
(percentage error)
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
UNCERTAINTY
The degree of doubt that exists about a
measured value
Range of Uncertainty
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
nominal value
tolerance
range of uncertainty
Range of Uncertainty
Reported value 120 mm ±2% = 117.6 mm to 122.4 mm
nominal value
tolerance
range of uncertainty
Note: 2% of 120 = 2.4, 120 - 2.4 = 117.6, 120 +2.4 = 122.4
PERFORMANCE
CHARACTERISTICS

Accuracy – the degree of exactness
(closeness) of measurement compared to
the expected (desired) value.

Precision – a measure of consistency or
repeatability of measurement, i.e
successive reading do not differ.
Precision – Target 1
Measurement precision must be interpreted in light of
measurement accuracy. Let’s use a target practice
example:
The best situation, the
shots are tightly
clustered (high
precision) on the
center circle (high
accuracy).
Precision – Target 2
:
The next situation,
shots are near the
center (high
accuracy), but not
tightly clustered (low
precision).
Precision – Target 3
In the next situation,
a tight cluster (high
precision) is far off
center (low
accuracy).
Precision – Target 4
Finally, widely
scattered shots
(low precision)
appear away from
the center (low
accuracy).
Precision Comparison
Which is the best and which is worst?
Best
Worst
Most Insidious
Why?
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
Solution (Example)
Given that , expected value = 80V
measurement value = 79V
Y n− X n
i. Absolute error, e =
= 80V – 79V = 1V
Yn  X n


100 = 80  79  100 = 1.25%
ii. % error =
 Yn 
80
iii. Relative accuracy,
Yn  X n


A = 1 
 Yn 
iv. % accuracy, a = A x 100%
= 0.9875 x 100%
= 0.9875
= 98.75%
Example
From the value in table 1.1 calculate
the precision of 6th measurement?
Table 1.1
Solution
the average of measurement value
98 +101+ ....+ 99 1005
Xn =
=
= 100.5
10
10
the 6th reading Precision =
100  100.5
0.5



1
= 1
100.5
 100.5 
= 0. 995
No
Xn
1
98
2
101
3
102
4
97
5
101
6
100
7
103
8
98
9
106
10
99
THE END