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Part-III
Treatment of Data
1
OVERVIEW
(1) Units of measurement
(a) must be indicated in tables/graphs.
(b) use scientific notation
Examples
2.2 µV
71
83.4 kJ
0.21 MW
73 pF
m
m
m
2.2x10-6 V
7.1x10-2
8.34x104 J
2.1x105 W
7.3x10-11 F
Tabulation of Data
Whether one is measuring the distance to the moon using laser
interferometry or measuring the soil strength using a
penetrometer or bond angles/lengths using XRD, carefully made
(and recorded) observations are the cornerstone of a good
experimental work.
1
x2
x1
x
(i) Repeat measurements
These are made even when the quantity (or the value) is
NOT varying.Timing the fall of an object through a given
distance or measuring the wavelength of light emitted from a
lamp containing helium gas.
(ii) Relationship between two variables : y = f (x) so assign
different values to as , ...... and measure y , y 2 ......
Table: Fall times (t) for an object to fall in air through 25 m
(ambient temperature 298 K, 3 pm, 17-09-2013).
t(s) 2.2 2.0 2.6 2.1 1.9 2.2 2.4 2.0 2.3 2.3
(3) Use Scientific Notation
For example, 2 mg
2x10-3 g ≅ 2x10-6 kg
Pressure values (Pa) :
1.03x105 1.01x105 9.9x104 9.83x104 1.01x105 1.05x105
Table : Measured values of pressure.
Pressure(x105Pa ) 1.03 1.01 0.99 0.983 1.01 1.05
or
Pressure(kPa)
103 101 99 98.3 101 105
(4) Uncertainties in measurements
Despite our best efforts and/or the quality of
instruments
“Variability”/”uncertainty” in
experimental data.
graduation of a thermometer
∴ there is nothing called “Exact” measurement.
(One can only do the best possible !)
Fluctuating nature of the mercury in a thermometer
∼ ± 0.5 o C .
There is an uncertainty of 20 ± 0.5o C
What does it mean ?
What you mean is that the temperature could be
o
o
anything from 19.5 C to 20.5 C , with the
possible average value of 20o C !
6
One must always include uncertainty estimates.
Table : Dependence of electrical resistance on temperature of a Copper wire
(Kirkup : Experimental Methods,Wiley 1994).
T
(K) ± 0.5K
281
289.5
296.5
305.0
313.5
327.5
Electrical Resistance
Ω ± 0.001Ω
0.208
0.213
0.222
0.229
0.232
0.243
OTHER FACTORS
Compounding of uncertainties?
Does uncertainty depend upon the magnitude of
the measurement ?
8
(5) Significant Figures
If a value is recorded as 5.1 2
your experiment is able to
distinguish between 5.1 1 and
5.1 3
Similarly, 5.123
5.122 & 5.124
5.12
3 significant digits
5.123
4 significant digits
Resist the temptation to record all the figures given
by an instrument display or a calculator or a
computer.
Let us look at the value of
0.0020409
how many significant figures are there ?
between the first non-zero and the last digit
inclusive
five significant figures
Exercise
2.564
0.00489
64000
1.20
1.2
0.20000878
four
three
?
three
two
eight
Rounding off Numbers
In calculations of derived variables, rounding off is needed.
How to do it ?
11
1.8671132
Reduce to
2 SF
1.8 or 1.9
Reduce to
3 SF
1.87
Reduce to
4 SF
1.867
CALCULATION & SIGNIFICANT FIGURES
Let us say, for a cylinder, D = 7.9
L = 1.5
mm
mm
πD
A=
=
4
4
A = 49.016699mm 2
2
X- Sectional area,
2
(
)
π 7.9
Is this sensible ? Obviously not.Why ?
Well, if D is known to 2 significant figures, area can not be
calculated to 8 significant figures !!
A = 49mm 2
2
π
D
Volume of cylinder =
L
4
π
2
= × ( 7.9) ×1.5
4
= 73.525049mm3
∴ V = 73 mm3
Guidelines for rounding off
Rule # 1 :Addition/Subtraction
Round off the result to the same number of decimal
places = least number of decimal places of the
constituents
Ex.
11.39 – 7.897 + 12.3538 = 15.8468
Correct answer is 15.85
14
Rule # 2 : Multiplication /Division
The number of significant digits in
the answer = least number of significant
digits in the primary numbers.
3.17×3.393×3.3937 = 36.501992
3 significant
figures
8 significant figures
36.5
CAVEAT
Volume = Area × L
= 49×1.5 = 73.5mm3
3
One might be tempted to round it up to 74 mm !
Do not round off the intermediate results
PROBLEM
In an experiment, the density (ρ) of a metallic sphere
is to be measured. Density ρ is defined as mass per
unit volume , i.e., m/v.
πd 3
V =
6
Lab notebook has the following entries
Mass of sphere = 0.44 g
Diameter of sphere = 4.76
mm
πd 3
π ( 4 .7 6 )
V =
=
= 4 5 1 .7 6 1 7 6 1m m 3
6
6
3
∴
0.44
ρ=
= 9.739647 × 10 -4
451.761761
Does it make sense ?
PRESENTATION OF DATA
(Graphs, bar/ pie charts)
1. Purpose
Graphs: our visual ability is very strong to detect trends as
opposed to tabulated information.
x
y
9.3
19.0
7.3
15.0
9.8
20
1.8
4
5.3
11
Graph/plot can be used to show the:
a) Range of data
b) Uncertainty in each measurement
c) Trend or the absence of a trend
d) Data which do not fall in line with the majority
of data points.
DEPENDENT/INDEPENDENT VARIABLES
Horizontal axis (Independent variable)
2-D plots
Vertical axis (Dependent variable)
Independent variables : controlled or which is varied
systematically
P
P can be varied through a high pressure gas line
P
Q
Q
Independent variable
Dependent variable
Effect of temperature on solubility of salt in water
Amount of water = 1 m3
Mass of salt which can be dissolved = M kg
∴ solubility, S = M/1 (kg/m3)
T (K)
Independent variable
S (kg/m3)
298
S1
315
S2
325
S3
Dependent variable
Example: Effect of temperature on the length of a wire
T (K)
L (m)
273
1.1155
298
1.1164
323
1.1170
348
1.1172
373
1.1180
398
1.1190
423
1.1199
448
1.1210
473
1.1213
498
1.1223
523
1.1223
(3) ORIGINS
Is it necessary to have a (0,0) point on both axes ?
Re-plot the T-L data
This is not a good graph ! Why ?
(4) Error bars
These show the uncertainties in both variables:
For the independent variable:
For the dependent variable:
for any ( x, y) data point
Uncertainty in y- value
Uncertainty in x- value
Cooling of an object
Time (s) ± 5 s Temperature
(°C) ± 4 °C
10
125
70
116
125
104
190
94
260
87
320
76
370
72
NOTE 1:
If uncertainties are not constant, the size of the
error bars must also vary.
NOTE 2:
% error is not same for each data point.
(5) Types of Graphs
•
Linear x-y graphs
•
Semi-log graphs (one scale is linear and one is logarithmic)
•
Log- Log graphs (both scales are logarithmic)
(6) Linear x-y plots
Dependence of relative density on sugar concentration
at 298 K.
Concentration
(kg/m3)
Relative
density (-)
0
1.005
50
1.034
100
1.066
150
1.095
200
1.122
250
1.150
Draw the best line
by naked eye ?
For RD = 1.053, calculate the sugar concentration
CAVEATS
(1) Do not ignore outliers - investigate them further
(2) Interpolation of results is justified when there is a sufficient number of data
points
?
?
(3) Extrapolation should always be avoided.
What are the two characteristics of a linear graph/plot?
y = mx + C
slope
intercept
For two data points, one can only draw one line and this is also the
best line :
(x1,y1), (x2,y2)
m=
y 2 − y1
and C = value of y at x = 0
x 2 − x1
But when we have a large number of data points
(xi ,yi)
(i=20 say)
which pair of data points should be used to calculate the values of m
and C?
BEST FIT
What are the uncertainties in the best fit values of m and C?
Example
T (h)
Size (mm)
0.5
1.5 ± 0.3
1
2.3 ± 0.3
1.5
3.3 ± 0.3
2
4.3 ± 0.3
2.5
5.4 ± 0.3
Linearization of Equations
Prior knowledge about the expected form of
dependence.
Original data might exhibit a non-linear relationship.
Transform one of the variables in such a fashion that
transformed variable leads to a linear relationship.
Period of oscillation (T)
~ mass of the body (M)
M
M (kg)
T (s)
0.02
0.7
0.05
1.11
0.10
1.6
0.20
2.25
0.30
2.76
0.40
3.18
0.50
3.58
0.60
3.97
0.70
4.16
0.80
4.60
Dependence is not linear
Let us recall our high school physics:
M k
π
2
=
M
∝
T
M
∝
T
∴
(k : spring constant)
T =
2π
k
y = m
M
x + C
We expect C = 0
Linear graphs
great advantage
EXAMPLE - 1
1 2
s = ut + at
2
Relationship between
“s” and “t” is quadratic
s
1
= u + at
t
2
y
C
mx
Constant acceleration (a)
Initial Velocity @ t = 0
u =u
s : distance travelled in time t
a = 9.81 m/s2
u = 1 m/s
s/t
m = a/2
C=u
0
t
5
Linearization has come about at a price?
We should now estimate uncertainty in (s/t)
NOT in s.
EXAMPLE-2
For a radioactive material,
N = N0 exp(-λt)
N : undecayed nuclei @ time t
N0: Initial value of N @ t = 0
λ : characteristic constant of material
How to linearize it?
ln N = ln N0 + (-λt) ln e
ln e = 1
ln N = (-λ)t + ln N0
y
= mx + C
C = ln N 0
“Semi-log” plot
−λ
ln N
0
t
Now we must estimate uncertainity in lnN?
Finally, let us come to log-log graphs:
Motivation
Sometimes no matter what we do, it is not possible to choose
suitable scales for linear graphs.
Table : Current-Voltage relationship
for a silicon diode
Voltage (V)
I (Amperes)
0.35
9 x 10-7
0.40
3 x 10-6
0.45
5 x 10-5
0.50
2 x 10-4
0.55
1.7 x 10-3
0.60
1.5 x 10-2
0.65
7.5 x 10-2
0.70
0.55
0.75
3.5
4
3.5
3
I
2.5
Linear
2
1.5
1
0.5
0.4
0.5
V
0.6
0.7
0.8
Semi-Log
1
10
0
10-1
I
0
0.3
10
10
-2
10
-3
10
-4
10
-5
10-6
10-7
0.3
0.4
0.5
V
0.6
0.7
0.8
When both variables entail several
orders of magnitudes
Use double log
coordinates
y = axb
ln y = ln a + b ln x
ynew
C
m
xnew
log-log scale
40
NATURE OF UNCERTAINTIES
Uncertainty is an inevitable evil, both in experimental and numerical studies.
Let us look at a simple test:
Same object, constant value of S, same operator/equipment.
Time(s)
0.74
0.71
0.73
0.63
0.69
0.75
0.70
0.71
0.74
0.81
s
water
What do you make of these measurements?
Uncertainty is an inherent part of experiments
Two Questions
1) Identification of sources: temperature, tube not being vertical, object
is not being dropped at the same location/different initial condition/air
bubbles attached to it….
2) Quantification of the uncertainty
Practice problems
• Fill a coffee making kettle with 1L of water and record the time it
takes for the water to boil? What factors will contribute to the variability
of the results?
(1) Uncertainty
(i) Single measurement:
• No method to establish the extent of uncertainty. Repeat the test at least
one more time.
• There are situations when it is not possible to repeat a test: biology, radioactivity, CERN, on the surface of the moon, etc.
• Test itself is varying with time.
(2) Uncertainty stems from:
(a) Resolution of instruments: What is the minimum value the
instrument can measure?
Length: 1 mm graduations
0.5 mm
For better resolution, one can use a micrometer or
Vernier callipers, but these also have their least count.
375 ± 1.2mm → 373.8 ≤ true value ≤ 376.2 mm
This is what is used for a single test, i.e., the uncertainty introduced by the
instrument.
(b) Reading uncertainty:
N=0
T
Corresponds to the value at a fixed point.
N≠0
Thermometer shows wild
fluctuations.
T
water
So, if you make a single measurement, we can not
evaluate the uncertainty arising from the heating
process.
(c) Calibration uncertainty:
All instruments require benchmarking or calibration which can
change over a period of time !!
heating
What is the way forward?
The mean or average comes in handy-returning to our earlier example.
Time(s)
t min = 0.63 s
0.74
0.71
0.73
0.63
0.69
0.75
0.70
0.71
0.74
0.81
tmax =0.81s
1 n
0.74 + 0.71 + 0.73 + ....
t min = ∑ t i =
n i =1
10
t = 0.721s
On average, this is the result we can expect.
No. of significant figures?
What is the uncertainty in the mean value?
range (spread) = x max − x min
Uncertainty =
For our example:
Uncertainty =
x max − x min
n
0.81 - 0.63
= 0.018
10
Now we should round off, the mean value sensibly to xmean= 0.72 s
∴ P ro b a b le v a lu e o f x (o r t in o u r c a s e ) = 0 .7 2 ± 0 .0 1 8 s
One can also quote % uncertainty
uncertainty × 100
mean value
0.018
=
×100 = 2.5%
0.72
% uncertainty =
Therefore, t mean = x = 0.72s with ±2.5% uncertainty
Without repeating measurements, one can not estimate the uncertainty.
(3) True value, accuracy & precision
Aim of an experiment
true value
But this is impossible to do.
On the other hand, we are trying to approximate the true value by an
average or mean value, i.e., x true ≈ x
How many times we must repeat the measurements?
Recall,
x=
1
xi
∑
n
Larger the value of n, closer will be
If x ≈
x
tru e
Our measurements are accurate.
x
to its true value.
For example, the charge of an electron is known to be
(−1.6021773 ± 0.0000005) ×10−19 C
−5
i.e., an uncertainty of 3 × 10 %
What is precision?
Uncertainty is small
accurate! How?
range is small, but it does not mean that the results are
Let us look at an example:
Boiling point of water @ 1 standard atmosphere
T
(oC)
102.4
102.6
102.3
102.6
102.4
102.4
102.5
102.6
102.4
102.7
Tmean = x = 102.49 o C
102.7 − 102.3
= 0.04 o C
10
∴ Boiling point of water = 102.49 ± 0.04 o C
Uncertainty =
This looks very impressive in terms of precision except that it is not very
accurate.!
Let us use a different thermometer (+0.5 oC)
T
(oC)
101.0
101.0
100.5
99.0
100.5
101.0
100.5
99.0
x = 100.2 o C
101 − 99
= 0.2 o C
10
∴ Boiling point of water = 100.2 ± 0.2 o C
uncertainty =
Less precise, but more accurate experiments.
99.5
In summary,
Accurate
Precise
close to the true value
Low uncertainty, but not necessarily close to the true value
Accurate & Precise
close to the true value, with a small uncertainty
TYPES OF UNCERTAINTIES
(1) Systematic
Difficult to detect and deal with.
Offset uncertainty
Melting point of ice
x(o C)
Thermocouple
-7.5, -6.9, -7.3
-7.4, -7.6, -7.4
-7.3, -7.7, -7.6
-7.6
x = −7.43,
uncertainty = 0.08o C
Very precise but inaccurate measurements!
ice+water mixture
The true value is expected to be close to zero!
There is a big offset error here. Check your calibration, electronic gadgets, warm
up period, insensitive thermocouple, etc.
On the other hand, for a plasma furnace (~ 1500 oC), 7.5 oC is not a significant
offset.
Try to develop a feel for the answer you are looking for!
Gain uncertainty: This varies with the magnitude of quantity itself.
Example: Calibration masses and electronic balance
Standard mass (g)
0.00
20.00
40.00
60.00
80.00
100.00
Electronic balance
value (g)
0.00
20.18
40.70
61.00
81.12
101.68
The difference between the two values increases as the
mass increases.
(2) Random Uncertainties:
• These are responsible for scatter in the measurements.
• Environmental factors can also introduce random uncertainty:
electrical interference (switching on/off equipment, vibrations
caused by rotomachinery, power supply fluctuations
(water/air/steam main pressures etc.)
If these are truly random, the averaging of several
measurements will even out this effect.
COMBINING UNCERTAINTIES
So far we have talked about uncertainties when we are
interested in the measurement directly.
Engineering experiments
we need to combine
several measurements to
calculate the quantity of
interest
Let us say you are given a cylindrical bar of an unknown
metal and we want to calculate its density.
m
D
ρ=
mass
volume
L
D
L
Uncertainty in the value of ρ depends upon the uncertainties in
the measured values of m, D, L
ρ=
m
D ≡ D ± ∆D
π 2
D L
4
L ≡ L ± ∆L
m ≡ m ± ∆m
ρ ≡ ρ ± ∆ρ
One way to estimate ∆ρ
m ≡ m + ∆m
L ≡ L + ∆L
∆ρ
D ≡ D + ∆D
D ≡ D − ∆D
L ≡ L − ∆L
L ≡ L + ∆L
L ≡ L − ∆L
m ≡ m − ∆m
m ≡ m + ∆m
m ≡ m − ∆m
m ≡ m + ∆m
m ≡ m − ∆m
m ≡ m + ∆m
m ≡ m − ∆m
This looks like a lot of hard work!!
We can be a little smarter than this:
ρ=
m
(π 4) D L
2
ln ρ = ln m − ln π + ln 4 − 2 ln D − ln L
Differentiate it:
∆ρ ∆m
∆D ∆L
=
−2
−
m
D
L
ρ
Multiply this equation by 100 on both sides
% uncertainty in ρ = % uncertainty in m + 2 × % uncertainty in D
+ % uncertainty in L
NOTE:
• Uncertainty in D is multiplied by 2.
• All terms have been added up.
STATISTICAL ANALYSIS OF DATA
Statistics is a science of numbers !
It helps you draw good inferences, but it also gives you confidence
to tell lies !!
“There are three kind of lies – lies, damned lies, and statistics.”
(attributed to Ben Desraeli & Mark Twain).
Naturally, statistics and statistical methods
“large dataset”.
DEFINITIONS
Variance of a dataset
(i)
x
is assumed to be the best estimate of the true
value of the quantity.
Despite
uncertainty
measurement,
x
in
each
individual
, the single value ≈ true value.
62
Example: Time to slide down the plane:
θ
0.74, 0.74, 0.69, 0.68, 0.80, 0.71, 0.78, 0.65, 0.67, 0.73
x=
xi (s)
1
∑ xi = 0.719
10
di = x- xi (s)
(
d 2i = x - xi
)
2
(s2 )
0.74
-0.021
0.000441
0.74
-0.021
0.000441
0.69
0.029
0.000841
0.68
0.039
0.001521
0.80
-0.081
0.006561
0.71
0.009
0.000081
0.78
-0.061
0.003721
0.65
0.069
0.004761
0.67
0.049
0.002401
0.73
-0.011
0.000121
∑di = 0
∑di2 = 0.02089
x = 0.719
∑d
σ =
=
n
Variance,
2
i
2
∑ ( x − xi )
2
n
For our example,
0.02089
σ =
= 0.002089 s 2
10
2
Another related parameter
σ= σ =
2
Standard deviation
∑ ( x − xi )
2
n
σ = 0.002089 = 0.04571s
Usually σ is NOT strongly dependent on n.
Uncertainty in the mean of repeat measurements:
Let us say :- We have repeat data sets.
Series I
Series II
43
-
-
55
-
-
53
-
-
52
-
-
55
-
-
52
-
-
51
-
-
54
-
-
50
-
-
52
-
-
-----
Series VI
I
II
III
IV
V
VI
VII
VIII
x
51
51.7
50.4
51.5
51.7
50.4
52.5
49.5
σ
3.13
3.29
3.07
3.11
3.20
2.94
2.73
3.20
COMMENTS
•
Variability in the “means” < Variability in each set.
 ( x ) − xi
( x ) = 51.1; σ x = 
 n
σx =
1
2

 = 0.893


σ
n
Therefore, the best estimate of x is 51.1 ± 0.893
This, however, only eliminates the role of random uncertainty & NOT
of the systematic uncertainty.
If there are sufficient number of data points without any systematic
uncertainties :
Frequency or distribution
x
This is more or less the universal curve which is encountered
literally in every application relying on numerous data points.
This is called Normal distribution or Bell-shaped curve.
What is so special about this curve?
Two metrics are needed to describe this population?
Mean () (Height of peak)
Standard deviation (σ) (spread of curve)
x-σ ≤ x ≤ x +σ :
Line of
symmetry
Area under the curve between these
limits α number of data points lying
in this range.
x±σ :
∼ 70% of the total
area
x ± 2σ : ∼ 95% of the total
:
σ
6
±
x
area
∼ 99.9999% of the
total area
x-σ
x
x+σ
Formal Treatment of Population & Sample
On one hand, we wish to have data which are reliable, reproducible and with
as small uncertainty as possible, one can not go on making ∞ repeat
measurements.
Let us say that a population of measurements with mean µ and σpop:
µ =
∑ xi
n
σ
 ∑ ( µ − x i
= 
n

pop
)
2



1
2
Evidently, µ = true value
We would like our “sample” (small sub-set of population) such that sample
mean ≈ µ and
σ
pop
∑
= s = 

(x
− xi
n −1
)
2



1
2
Confidence Bands
x-σ
x
x+σ
If 70% of the data lie within x±σ, so we can say that there is a 70% probability
to predict the expected outcome within ±σ.
If 95% of the data lie within ±2σ, so we can say that there is a 95% probability
that we can predict the expected outcome within ̅x̅ ±2σ, etc.
REJECTION OF DATA
Some will argue “all data are equal”.
∴ It is not correct to through away any data point.
Other extreme is that “one data set looks like spurious or suspect” and
therefore is less reliable than the other sets.
There are statistical tests to deal with this issue.
Therefore, the automatic filtering by a computer program or another
device should be assessed properly.
The question is:“truly spurious” vs.“new phenomenon”?
Therefore, meticulous recording of data, observations, unusual features,
frequent voltage fluctuations, exceptional temperature, etc. all must be
documented in detail in lab notebooks.
Methodology of rejecting data
One data point strikingly disagrees with all the others.
Fall time (seconds) of an object in a liquid :
3.8 , 3.5 , 3.9 , 3.9 , 3.4 , 1.8
very different from all others !!
Recall that individual data can differ within a band from each other.
However, legitimate discrepancy of this size is highly improbable.
Controversial
Data rejection
Important
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t =
3.8 + 3.5 + 3.9 + 3.9 + 3.4 + 1.8
≃ 3.38
6
σ = 0.8 s
t = 3.4 s
Our suspect measurement of 1.8s deviates ,
3.4 − 1.8 = 1.6 s
i.e., by 2σ
Assuming Normal or Gaussian distribution, we can calculate the
probability of a measurement lying outside ±2σ :
‫ ؞‬Probability (outside 2σ) = 1 – probability (within 2σ)
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95.45 %
-2σ
3.4
2σ
‫ ؞‬Probability (outside 2σ) = 1 – 0.9545 = 0.0455
< 0.05
Thus, there is only 5% chance of a measurement lying outside ± 2σ
, i.e., 1 in 20 measurements could be beyond ± 2σ .
Out of 6 measurements, only 6 x0.05 = 0.3 is likely to be beyond ± 2σ .
Chauvenet’s criterion : If this number < 0.5 , this data can be rejected.
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For N measurements : x1 , x2 ,....., xN
xsus : value in doubt
tsus =
xsus − x
σ
# of standard deviations
Find the probability of (outside tsusσ).
No. of expected deviants , n = N x Prob (outside tsusσ)
If n < 0.5
reject the data point in question and re-calculate
x , σ , etc.
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EXAMPLE
A student makes 10 measurements of mass (g) as follows :
46 , 48 , 44 , 38 , 45 , 47 , 58 , 44 , 45 , 43.
x = 45.8 , σ = 5.1
Our suspect is : 58
‫؞‬
tsus =
58 − 45.8
= 2.4
5.1
i.e., our suspect deviates by 2.4σ.
Prob (outside ± 2.4σ) = 1 – Prob (inside ± 2.4σ)
= 1 – 0.9836 = 0.016
‫ ؞‬In a set of 10 measurements , 10 x 0.016 = 0.16 of data can be
outside ± 2.4σ.
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Since 0.16 < 5 , we can safely reject this data point. New results
are :
x = 44.4, σ = 2.9
Not much change in x , but σ has dropped significantly.
Remember, the choice of n < 0.5 is arbitrary.
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CONCLUDING REMARKS
• Units of measurements, scientific notation, their
representation, uncertainties, significant figures.
• Presentation of data (Graphs, tables, bar/pie
charts).
• Uncertainties: Systematic/Random
• Statistical analysis of data: linear regression/nonlinear regression, adequacy of fit, R2, etc.
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