sug533 kuliah 2a - analysis of error in observations
TRANSCRIPT
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
1/21
SURVEY OBSERVATIONS/
MEASUREMENTS
Direct observations apply instrument
directly to the unknown quantity and get
readings (distance or angle)
Indirect observations unknownquantity derived from mathematical
relationship to direct observation (distance
or bearings from coordinates)
How to analyse errors in direct and
indirect observations?
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
2/21
WAYS OF ANALYSIS OF MEASURED
DATA SET
Numerical method (mean, median, mode,
standard deviation)
Graphical representation (scatterplot,
frequency histogram)
Median is the middle value of a data set arranged in ascending or descending
order
Mode is the value that mostly occurs in a data set
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
3/21
Precision
The degree of closeness between measured values of repeated
measurements
It is dependent on environmental stability, quality of equipment used and
observers skill with equipment and measurement procedures
Accuracy
The degree of closeness between measured and true values of a quantity
The true values are based on standardized measurement (i.e equipment &procedures)
The difference between measured and true values is termed systematic error in
the measured value
WHAT TO ANALYSE FROM MEASURED DATA SET
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
4/21
A B C
533 547.97 547.573
540 547.08 547.594
504 547.43 547.523
513 547.76 547.568
572 547.51 547.528
min 504 547.08 547.523
max 572 547.97 547.594
range 68 0.89 0.071
A B C
524 547.42 547.549
585 547.39 547.567
515 547.35 547.562
543 547.76 547.517
538 547.50 547.513
547.89 547.586
547.24 547.554
547.87 547.588547.05 547.564
547.94 547.583
547.522
547.567
547.568
547.557
547.51
min 515 547.35 547.513
max 585 547.76 547.567
range 70 0.41 0.054
Range is an indication of
precision
Which is the highest precision
in Table 1?
Comparing the precision in
Table 2 is meaningless, why?
Table 1 Table 2
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
5/21
A B C
533 547.97 547.573
540 547.08 547.594
504 547.43 547.523
513 547.76 547.568572 547.51 547.528
n 5 5 5
mean 532 547.55 547.557
std dev 27 0.34 0.031
n
Z
Z
n
i
i== 1
1
1
2
=
=
n
S
n
i
i
Mean
data set
Standard deviation
data set
C v v2
547.573 -0.016 0.0002
547.594 -0.037 0.0014
547.523 0.034 0.0012
547.568 -0.011 0.0001
547.528 0.029 0.0009
sum 0.0037
n 5
mean 547.557
std dev 0.031
Using statistics descriptors
v = residual = mean - obs
n = no of observation
n 1 = degree of freedom (redundancy)
S2 = variance of obs (precision)
Standard deviation
of mean
n
SSx
=
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
6/21
mean calibrated
547.557 547.500
EFFECT OF SYSTEMATIC ERRORS ON
ACCURACY OF DATA SET
Observed value corrected for systematic error
Cerror
(systematic) corr obs v v2
547.573 0.057 547.516 -0.016 0.0002
547.594 0.057 547.537 -0.037 0.0014
547.523 0.057 547.466 0.034 0.0012
547.568 0.057 547.511 -0.011 0.0001
547.528 0.057 547.471 0.029 0.0009
sum 0.000 0.0037
n 5
mean 547.500
std dev 0.031
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
7/21
C v v2
547.573 -0.016 0.0002
547.594 -0.037 0.0014547.523 0.034 0.0012
547.568 -0.011 0.0001
547.528 0.029 0.0009
sum 0.0037
n 5
mean 547.557std dev 0.031
C
error
(systematic) corr obs v v2
547.573 0.057 547.516 -0.016 0.0002
547.594 0.057 547.537 -0.037 0.0014
547.523 0.057 547.466 0.034 0.0012
547.568 0.057 547.511 -0.011 0.0001
547.528 0.057 547.471 0.029 0.0009
sum 0.000 0.0037
n 5
mean 547.500
std dev 0.031
Removing systematic
error makesobservation more
ACCURATE and does
not change the size of
random errors in the
observations
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
8/21
The lower the systematic errors, the higher is the accuracy
The lower the random errors, the higher is the precision (i.e
through adjustment process)
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
9/21
COMPARISON BETWEEN ACCURACY & PRECISION
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
10/21
Standard deviation explains the precision of sample data set. In theory,
68% of all observations in a sample lie within one-standard deviation
about the mean value (most probable value or MPV)
The larger the standard deviation the more dispersed the values in the
data set, and less precise is the data set
68% probability (area
under normal curve)
STANDARD DEVIATION
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
11/21
Random errors in a measurement have certain number of possibilities to
occur
Say, a single random error in a measurement = 1, then there are two
possibilities for the value of the resultant error (i.e +1 or -1) to occur in a
single measurement
The probability of +1 error is and of -1 error is
If the measurement is carried out in two parts, then there would be four
possibilities of resultant error (+1+1=+2), (+1-1=0), (-1+1=0) and (-1-1=-2)
The probability of +2 error is , of 0 error is 2/4 and of -2 is
(P)
0.5
0.4
0.3
0.2
0.1
-1 1
(P)
0.5
0.4
0.3
0.2
0.1
-2 0 2
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
12/21
The plot of error sizes against its probabilities would approach a smooth
curve (bell shaped) as the number of combining measurement increases
The curve is known as NORMAL DISTRIBUTION CURVE of the random
error
Probability of the standard error (or standard deviation) can be derived from
the STANDARD NORMAL DISTRIBUTION FUNCTION as follows;
P(-s < z < +s) = Nz(+s) Nz(-s)
For +/- 1s, the value of z = +1 and z = -1 are obtained from std normal
distribution t-table, where t = 1 is 0.84134. Thus t = -1 is (1 0.084134 =
0.15866)
Hence P(-s < z < +s) = 0.68268 = 68.3%
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
13/21
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
14/21
More exact percent probable error
checking for blundersE68% = 1*(sigma)
E95% = 1.960 * (sigma)
E99.7% = 2.965 * (sigma)
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
15/21
GRAPHICAL ANALYSIS OF DIRECT REPEATED OBSERVATIONS
(FREQUENCY HISTOGRAM)
- Compute range = max min observed values
- Set number of class (odd)
- Compute class width = range/no class
- Estimate class interval
- Compute and arrange the following:
class interval/ class frequency/ relative frequency
- plot frequency histogram
ordinate axis relative frequencyabscissa axis class interval
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
16/21
(a) Normal distribution
histogram (zeroskewness)
(b) more precise
than (a) (zeroskewness)
(c) Positively
skewed distribution
(negative
skewness)
(d) Negatively
skewed distribution
(positive skewness)
Skewness (measure of
normality of distribution)
( )3
3
nS
YY
skewi
=
TYPES OF DATA DISTRIBUTIONS
(HISTOGRAMS)
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
17/21
error (random)
-0.050
-0.040
-0.030
-0.020
-0.010
0.000
0.010
0.020
0.030
0.040
0 1 2 3 4 5 6
Obs no
error
Obs No C v
1 547.573 -0.016
2 547.594 -0.037
3 547.523 0.034
4 547.568 -0.011
5 547.528 0.029
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
18/21
How to analyse errors in both types of observations?- Direct repeated observations (central tendency and spreadness)
- Indirect observation (Law Of Propagation Of Variance LOPOV)
LOPOV
22
22
2
11
2
.........
++
+
= XpPXXZ XZ
X
Z
X
Z
)()( 2211 xxz xxz +=
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
19/21
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
20/21
Analysis of errors in indirect observations - LOPOV
A B C
mm 012.0560.10 mm 015.0370.120
Find length AC and its errors
mLengthAC 93.130370.120560.10 =+=
[ ] [ ] 019.0)015.0)(1()012.0)(1( 22 =+== ACErrorAC
T
yX AA=
[1=A ]1
=
1
1TA
==
ABBC
AB
ACAC
,
2
2
2
,
BC
BCAB
-
8/7/2019 SUG533 Kuliah 2a - Analysis of Error in Observations
21/21
mm 021.0163.37 R =
Find the area of circle and its error
502.233)163.37(22 ===
R
R
A
22
2
2
2
1
1
2......
++
+
= Ynn
YYXY
X
Y
X
Y
X
222 818.4338)163.37( mRAcircleofArea ====
[ ] 22 904.4)021.0)(502.233( mareainError A ===