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Assoc. Prof. Dr. M. Tevfik ÖZLÜDEMİR
Lecture 2
Lecture 2
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Due to errors, repeated measurements will
often vary
Precision is the degree to which
measurements are consistent –
measurements with a smaller variation are
more precise
Good precision generally requires much skill
Precision is directly related to random error
Lecture 2
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Accuracy is the nearness to the true value
Since the true value is unknown, true
accuracy is unknown
It is generally accepted practice to assess
accuracy by comparison with measurements
taken with superior equipment and
procedures (the so-called test against a
higher-accuracy standard)
Lecture 2
Observation
pacing
taping
EDM
1
571
567.17
567.133
2
563
567.08
567.124
3
566
567.12
567.129
4
588
567.38
567.165
5
557
567.01
567.144
average
569
567.15
567.133
Which is more precise? Which is more accurate?
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(a) Accurate and precise
(b) Accurate on average, but not
precise
(c) Precise but not accurate
(d) Neither accurate nor precise
Questions:
Can one shot be precise?
Can a group of shots be
accurate?
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Redundant measurements are those taken
in excess of the minimum required
A prudent professional always takes
redundant measurements
Mathematical conditions can be applied to
redundant measurements
Examples – sum of angles of a plane
triangle = 200g, sum of latitudes and
departures in a plane traverse equal zero,
averaging measurements of the length of a
line
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Can apply least squares adjustment which is a
mathematically superior method
Often disclose mistakes
Better results through averaging (adjustment)
Allows one to assign a plus/minus tolerance
to the answer
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Most rigorous of all adjustment procedures
Enables post-adjustment analysis
Gives most probable values
Can be used to perform survey design for a
specified level of precision
Can handle any network configuration (not
limited to traverse, for example)
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Lecture 2
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Note that in this section, we are talking about
multiple measurements of the same quantity.
Numerical analysis – computation of
statistical quantities (mean, variance, etc.)
Graphical analysis – construction of bar
charts, scatter diagrams, etc.
Lecture 2
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Most often, we collect a small data sample
from a much larger population.
For example, say we wanted to determine
the ratio of ITU students speaking French.
Theoretically we could visit every ITU
student and collect this information – then
compute the ratio.
This would be an assessment of the
population, which gives the actual ratio.
Lecture 2
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Visiting every ITU student would take a
very long time, so we might collect a
smaller sample.
If we compute the ratio from this sample,
we would get an estimate of the actual
ratio.
It is important to be unbiased.
If we based our sample on students in this
room, we would get a biased estimate.
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The population size for measurements is
infinite.
Thus, we are always dealing with samples
when analyzing measurements.
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The range (sometimes called dispersion) is
the difference between the largest and
smallest values.
Generally, a smaller range implies better
precision.
The median is the middle value of a sorted
data set.
When the number of data elements is even,
take the mean of the two middle values.
Lecture 2
20.1
21.9
22.5
22.8
23.1
23.5
23.8
24.2
24.8
25.4
20.5
22.0
22.6
22.9
23.2
23.6
23.9
24.3
25.0
25.5
21.2
22.2
22.6
22.9
23.2
23.7
24.0
24.4
25.2
25.9
21.7
22.3
22.7
23.0
23.3
23.8
24.1
24.6
25.3
25.9
21.8
22.3
22.8
23.1
23.4
23.8
24.1
24.7
25.3
26.1
Range = 26.1-20.1 = 6.0
Median is 23.45 (the average of
23.4 and 23.5)
Note that the difference between
the lowest value and the median
is 3.35 and between the highest
and the median is 2.65
Lecture 2
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The frequency histogram (or simply
histogram) is a graphical representation of
data
A histogram is a bar graph that illustrates the
data distribution
To produce a histogram, the data are divided
into classes which are subranges that are
usually equal in width
The number of classes can vary depending on
the number of values, but odd numbers like 7,
9, or 11 are often good choices
Lecture 2
Say we want to construct a histogram of the previous data set using 7
classes spanning the range.
The class width will be 6.0/7 = 0.857143 = 0.86
Therefore the first class subrange will be 20.10 – 20.96, the second
subrange will be 20.96 – 21.81, the third will be 21.81 – 22.67, etc.
We then count the number of values falling within those classes and
compute the fraction of the total.
Lecture 2
Class
interval
20.10 20.96 21.81 22.67 23.53 24.39 25.24 -
20.96
21.81
22.67
23.53
24.39
25.24
26.1
Class
frequency
2
3
8
13
11
6
7
Class relative
frequency
2/50 = 0.04
3/50 = 0.06
8/50 = 0.16
13/50 = 0.26
11/50 = 0.22
6/50 = 0.12
7/50 = 0.14
Σ= 50/50 = 1
Lecture 2
Lecture 2
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Symmetry
Range
Frequencies
Steepness indicates precision, but only if the
histograms have the same class intervals and
scales.
Lecture 2
Lecture 2
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Measures of central tendency
Measures of data variation
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Arithmetic mean or average
Median (mentioned previously)
Mode
Lecture 2
n
y
y
i
i 1
n
y
is the sample mean
n
is the number of values
yi
are the individual values
Lecture 2
n
y
y
i 1
n
i

1175.0
 23.500
50
Lecture 2
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The median is the middle value.
Half of the values are above and half are
below.
It is more effective as a measure of central
tendency when there are outliers (blunders) in
the data set.
Lecture 2
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The mode is the most frequently occurring
value.
It is seldom of use when dealing with
measurements (real numbers).
More useful with integers (e.g. most common
age).
Lecture 2
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True value. A quantity’s theoretically
correct or exact value. In theory, it is the
population mean, μ, which is
indeterminate for measurements
Error (ε). The difference between a
measurement and the true value
 i  yi  
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Most probable value (y ). Derived from a
sample, it is the average of equally weighted
measurements
Residual (v). The difference between the most
probable value and an individual
measurement. It is similar to an error, but
definitely not the same thing
vi  y  yi
Lecture 2
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Degrees of Freedom. The number of
observations that are in excess of the minimum
number necessary to solve for the unknowns –
it equals the number of redundant observations
Population variance (σ2). This quantifies the
precision of the population of a set of data. It
can also be called the mean squared error
n
2
 
2

 i
i 1
n
Lecture 2
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Sample variance (S2). This is an unbiased
estimate of the population variance.
n
2
i
S2 

v
i 1
n 1
Standard error (σ). Square root of population
variance – 68.3% of all observations lie within ±σ
of the true value
Lecture 2
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Standard deviation (S). This is the square root
of the sample variance – it is an estimate of
the standard error. (Do not expect 68.3% of
sample observations to fall within ±S of the
sample mean unless n is large.)
Lecture 2
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Standard deviation of the mean ( S y ) The
mean value will have a lower standard
deviation than any single measurement. As
n →∞, S y →0.
S
Sy 
n
Lecture 2
n
2
2
(
y

y
)

i
i 1
S 
n 1
n
2


  yi 
n
2
 i 1 
y

n

i
n


n
2
2
i 1
y

n
y



i

  i 1
S2 
n 1
n 1
Lecture 2
Week 2, February 14, 2012
20.1
21.9
22.5
22.8
23.1
23.5
23.8
24.2
24.8
25.4
20.5
22.0
22.6
22.9
23.2
23.6
23.9
24.3
25.0
25.5
21.2
22.2
22.6
22.9
23.2
23.7
24.0
24.4
25.2
25.9
21.7
22.3
22.7
23.0
23.3
23.8
24.1
24.6
25.3
25.9
21.8
22.3
22.8
23.1
23.4
23.8
24.1
24.7
25.3
26.1
y  23.50
Lecture 2
n
2
v
i
S
92.36

 1.37
n 1
50  1
i 1
Lecture 2
By alternate form:
2
y
 i  27,704.86
27,704.86  50( 23.50) 2
S
50  1
27,704.86  27,612.50
92.36
S

 1.37
50  1
49
Standard deviation of the mean
 1.37
Sy 
 0.194
50
(Note the higher precision for the mean)
Lecture 2
About 68% of the values should be between ±S of the
mean.
So:
23.50 ±1.37
22.13 to 24.87
34 out of the 50 values fall within that range, which is
68%
Lecture 2
Quote of the lecture (*)
“Experience is the name everyone gives to their mistakes.”
Oscar Wilde (Lady Windermere's Fan, 1892)
(*) http://en.wikiquote.org
Have a nice week 
Lecture 2