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ERROR ANSLYSIS AND
GRAPH DRAWING
Measurement Techniques­I
by Anshuman Dalvi
Navin Singh
To err is human, to evaluate and analyze is scientific
➢ Error analysis
➢ Graphical analysis
➢ Significance of digits
Measurement
Henry­I (1100­1135) who decreed that the yard should be "the distance from the tip of the King's nose to the end of his outstretched thumb". Error in a measurement
Length of the object = 128.9 cm ± ??
This is uncertainity or error
Error in a measurement
Error in the primary measurements cause uncertainty in the final measurement
2
g=
4π L
T
2
Nomenclature of errors
✔
Blunder
✔
Systematic errors
✔
Random errors
Blunders
➔ Experimenter makes a genuine mistake in reading an instrument wrongly.
➔ This can be avoided by taking large number of data points, discarding an entirely different value.
Systematic error
✔ This is instrumental error
✔ Constant error which occurs all the time
✔ Difficult to detect
✔ Examples: The scale itself is incorrect, causes error in length measurement
To avoid such kind of errors, each and every experimental instrument must be calibrated against a well established standard set up. Random error
✔ Caused by unknown and unpredictable changes in the experiment.
✔ Unpredictable changes – environmental conditions, electronic noise in the instrument.
Random error
✔ You may reduce it by performing experiment many times (statistical analysis).
✔ This error can not be eliminated and must be estimated and quoted as the uncertainty of the final result.
Precision ✔ Random error is small, precision is high.
✔ High precision means minimum error.
random Accuracy
The accuracy of a measurement is how close the measurement is to the true value of the quantity being measured.
High accuracy means minimum systematic error.
Accuracy & Precision
✔ Higher the accuracy – min is the systematic error.
✔ Higher the precision – min is the random error.
Accuracy with precision
Measurement with high accuracy and precision is highly RELIABLE !!
Estimation of errors
Least count
Maximum error in any primary measurement
Instrument
least count
Effective
least count
Effective least count is 0.3 cm
Effective least count appears as = 0.3 cm
To be in the safer side Length = 3.3 cm
Effective Length = (3.3) cm
Effective Length = (3.35) cm
✔ Always go for high least count apparatus.
✔ Higher the Accuracy, systematic error is min
ERROR ANALYSIS
(STATISTICAL)
Error analysis (Statistical)
✔ One data point measurement.
✔ One variable measurement.
✔ Two variable measurement.
One Variable measurement
Example
Measurement of Diameter of a ball.
Diameter of a ball
x1
x2
x3
xN
Take large number of data points
Mean
1
N
N
x = ∑ x i
i=1
Standard deviation

N
1
2
σ Δx =
x i − x 
∑

N −1 i=1
What is Standard Deviation?
✔ Measure of dispersion of set of values.
✔ Defined as root mean square deviation from the mean value.
✔ If many data points are close to mean SD is small
✔ Data points are equal to means, SD = 0
Diameter of a ball
Diameter of a ball = (x ± Δx) units
% Error
Δx
Error=
×100
x
Linear fit of data between two variable
Let there are n points of measurements (x1, y1), (x2, y2)….(xN, yN)
You obtain
y = mx+c
You measure
For the best fit line, the quantity
S= ∑  yi −mxi −c 
i
2
Will be a
minimum
Hence, the name Method of least square
∂S
=−2 ∑ x i  y i −mx i −c =0
∂m
i
∂S
=−2 ∑  y i −mx i −c =0
∂c
i
Slope and intercept
∑  x i − x  y i
m=
2
∑  x i − x 
c= y −m x
Δ m≈
2
Si
∑
D N −2 
2
Si
∑
1 x
Δ c≈

N D  N −2 

2

where,
D= ∑  x i − x 
2
S i =yi −mxi −c
GRAPHICAL METHOD FOR BEST FIT LINE S
y
S1
BEST FIT LINE
CENTROID
( x, y )
S2
BISECTOR OF THE TWO BOUNDING LINES
x
Qualitative best fit method (Graphical)
✔ Plot all the data points.
✔ Plot centroid (x, y).
✔ Draw limiting lines (S1 and S2)
✔ Draw a best fit line (S)
✔ Get Δ S = (S1~S2)/2
PROPAGATION OF ERRORS
If ∆f is difference in a quantity f (x,y,z) from accurate value then,
     
2
Δf=
2
∂f
∂f
∂f
Δx 
Δy 
Δz
∂x
∂y
∂z
Change in f for small error in x
2
Rule 1
then
Rule 1
Suppose we have measured the starting
position as x1 = 9.3 ± 0.2 m and the
finishing position as x2 = 14.4 ± 0.3 m.
Then the displacement is
∆d = x2 – x1 = 14.4 m - 9.3 m = 5.1
and the error in the displacement is
(0.22 + 0.32 )1/2 m = 0.36 m
Rule 2
Example
2
g=
4π L
T
2
Δg
=
g
     
2
2ΔT
ΔL

L
T
2
We have measured a displacement of x in time t. What is the measured velocity and the error in the velocity?
x= 5 .1±0 . 4 m
t= 0 . 4±0 .1 s
5.1
v=
=12 . 75 m/s
0.4
Δv
=
v

2
  
Δx
Δt

x
t
2
Δv=3 .34 m /s
Rule 3
Example
Error in volume of a sphere
ΔV
Δr
=3
V
r
Graphical Analysis
✔ Use Sharp Pencil.
✔ Draw on full page of graph paper. Use appropriate scale.
✔ Plot dependent variable on the vertical y­axis and the independent variable on the x­axis.
Graphical Analysis
✔ Label the axes.
✔ Label the data points properly.
✔ Title the graph.
✔ Indicate error bars.
Significant Figures
The digits required to express a number to the same accuracy as the measurement it represents are known as significant figures
Understand the difference between 1 and 1.00
Recapitulate BITSAT knowledge
Significant figures
22
2
0.046
2
8.200 x 103
8.20 x 103
4
3
Be careful……….
25 . 2×1374
≠1039 . 7838 . .. .
33 .3
25 . 2×1374
=1040
33 .3
Least counts
Reading is 3 mm from upper scale while 0.58 mm as 29th VSD is coinciding with main