SUG533 Kuliah 2a - Analysis of Error in Observations

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?


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


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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


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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


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

=


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


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


8/21

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)


9/21

COMPARISON BETWEEN ACCURACY & PRECISION


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


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


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%


13/21


14/21

More exact percent probable error

checking for blundersE68% = 1*(sigma)

E95% = 1.960 * (sigma)

E99.7% = 2.965 * (sigma)


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


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)


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


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 +=


19/21


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


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 ===

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SUG533 Kuliah 2a - Analysis of Error in Observations