Theory of errors

1. THEORY OF ERRORS

2. Contain
• Introduction
• Types of Errors
• Definitions
• The Law of Accidential Errors
• Law of Weights
• Theory of Least Square
• Determination of Probable Errors

3. • Introduction
A typical survey measurements may
involve such operations like centering , pointing ,
setting and reading. Due to some human
limitations, imperfection in instruments ,
environmental changes or carelessness ; certain
amount of errors are produced.
The error in measured quantities should
be eliminated before they used in computing other
quantities.

5. TYPES OF ERRORS
1. Mistakes :
Mistakes are errors that arise from
inexperience , inattention , carelessness on the
part of the observer.
For example, if observer read horizontal
circle of a theodolite 180* for 179* it is a mistake.

6. 2. Systematic errors :
It is an error, that under the same
conditions, will always be of the same sign and
magnitude. They always follows some definite
mathematical or physical law.
For example, The error in the length of
the steel tape due to change in temperature in a
systematic error.
Systematic errors are also known as
cumulative errors.

7. 3. Accidential errors :
Accidential errors are those which
remain after mistakes and systematic errors have
been eliminated. It occure due to the lack of
perfection in the human eye.
They do not follow any specific law and
may be positive or negative.

8. Definitions
 Direct observation :
A direct observation is the one made
directly on the quantity being determined.
For example , distance between two pint is
measured as 45.62 m with tape in field.
 Indirect observations :
Value of quantity derived indirectly from
direct observations. Like , The value of angle at
triangulation station found from value of angle
measured at satellite stations.

9.  Conditioned quantity :
If the value of observed quantity
depends on the value of other quantity , it is called
conditioned quantity.
For example , A + B + C = 180
 True value of quantity :
True value of quantity is the value
which is absolutely free from all the errors.It may
be classified as : 1. independent quantity
2. Dependent quantity

10.  Most probable value (MPV) :
The MVP of the quantity is the value
which has more chances of being true than any
other value. It can be found when the quantity is
measured a number of times.
 True error :
The difference between the observed
value of a quantity and its true value.
 Residual error :
The difference between the observed
value of a quantity and its most probable value.

11. Statistical Formula’s
A. Probable Error of Single
Observations, E
B. Probable Error of the Mean, Em
C, Standard Deviation, S.D.
D. Standard Error, S.E.
E. Precision
Where;
V = x – x
x = observed/measured value of a
quantity
x = mean value
n = number of measurements
1
6745.0
2

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n
V
E
)1(
6745.0
2
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V
Em
1
..
2
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n
V
DS
)1(
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2
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nn
V
ES
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x
E
P m


12. Law of Accidential Error
• Follow a Definite law, the law of probability.
• Defines the occurance of errors and can be
expressed in the form of equation which is used
to compute the probable value of quantity.
I. Probable error(Es ):
II. Probable error of the mean (Em)
III. Probable error of a sum
IV. Mean square error (m.s.e)

13. LAWS OF WEIGHTS
1. The weight of arithmatic mean of number of
observations of unit weights is equal to the
number of observation.
Ex. A is measured 3 times ,
1. 40 ˚ 30’ 20” weight 1
2. 30 ˚ 30’ 10” weight 2
3. 40 ˚ 30’ 15” weight 3
sum = 121 ˚ 30’ 45”

14. Arithmatic mean = 121 ˚ 30’ 45” / 3 = 40 ˚ 30’ 15”
The arithmatic mean has the weight of 3 ,
equal to the number of observations.
2. The weight of the weighted arithmatic mean is
equal to the sum of the individual weights.
1. 40 ˚ 30’ 20” weight 2
2. 40 ˚ 30’ 10” weight 1
3. 40 ˚ 30’ 15” weight 3
Sum of individual weights = 2 + 1 + 3 = 6

15. • Weighted arithmetic mean
= (40 ˚ 30’ 20”)*2 + (40 ˚ 30’ 10’)*1 + (40 ˚ 30’
15”)*3 / 6
= 40 ˚ 30’ 15.83”
The weight of weighted arithmetic mean is 6
3. The weight of algebric sum of two or more
quantities is equal to the reciprocal of the sum of
reciprocal of individual weights.
4. If a quantity of a given weight is multiplied by a
factor the weight of the result is obtained by
dividing its given weight by the square of that
factor.

16. A = 40 ˚ 30’ 20” has a weight of 6,
weight of 2A = 6 / 2*2 = 3/2
weight of 3A = 6 / 3*3 = 2/3
5. If a quantity of given weight is divided by a factor ,
the weight of the result is obtain by multiplying its
given weight by square of that factor.
Weight of A/2 = 6* 2*2 = 24
Weight of A/3 = 6*3*3 = 54

17. Theory of Lest Squares
• The principle of least square can be stated as
“The most probable value of a quantity
evaluated from a number of observations is
the one for which the squares of the residual
errors is minimum”
• We have ,
222
)(  
n
X
Vnrd

18. • As the second term is square of a quantity ,it
is always positive. Therefore, is always
more than . Hence, the sum of squares of
errors from the mean is the minimum.
 2
d
 2
r

19. • Determination of probable error :
1 . Direct observation of equal weights
(a) P.e. Of single observation of unit weight
v = residual error
N = Number of observations
2
1
6745.0



n
V
Es
22
3
2
2
2
1
2
.... nvvvvV 

20. (b)P.e. Of single observation of weight w
(C) P.e. Of single arithmetic mean
2. Direct observation of unequal weights
(a) p.e. Of single observation of unit weight
w
Es
se
E 
n
Es
m
E 
2
1
6745.0



n
wV
Esu

21. (b)P.e. Of single observation of weight w
(c) P.e. Of weighted arithmetic mean
w
Es
suw
E 
2
1
6745.0



n
wV
2
)1(*
6745.0



nw
wV
Esum

22. 3. Indirect observation of independent quantities
4. Indirect observations involving conditional
equations
5. Computed quantities

23. • Example: Following reading of levels were carried
out, 2.335,2.345,2.350,2.315,2.305,2.325 & 2.315
calculate:
(i) Probable error for single observation
(ii) Probable error of mean.
Sr. No. Reading Of Levels Mean V V2
1.
2.
3.
4.
5.
6.
7.
8.
2.335
2.345
2.350
2.300
2.315
2.305
2.325
2.315
2.324
0.011
0.021
0.026
-0.024
-0.009
-0.019
0.001
-0.009
0.000121
0.000441
0.000676
0.000576
0.000081
0.000361
0.000001
0.000081

24. ∑ = 18.59 N = 8
Mean value = 18.59÷ 8 = 2.324
(i) Probable error for single observation (Es)
= ±0.01233 m
(ii) Probable error of mean(Em)
= ± 0.00436 m
2
1
6745.0



n
V
Es
n
Es
m
E 

25. Example :Find most probable value of angles A,B and C of triangle ABC
from Following Values.
A= 52˚24ˡ 30ˡˡ
B= 64˚06ˡ 20ˡˡ
C= 63˚30ˡ 02ˡˡ
Solution :
Equation is A +B+C =180 ˚
C= 180 ˚- (A+B)
(A+B)= 180 ˚- 63˚30ˡ 02ˡˡ
= 116˚29ˡ 58ˡˡ
Hence ,Observation equation are,
A= 52˚24ˡ 30ˡˡ ………(1)
B= 64˚06ˡ 20ˡˡ ………(2)
A+B = 116˚29ˡ 58ˡˡ……..(3)

26. 2A+B = 168˚54ˡ 28ˡˡ ………(4)
A+2B = 180˚36ˡ 18ˡˡ………(5)
Multiplying eq (5) by 2,
2A+4B = 361˚12ˡ 36ˡˡ ………(6)
Subtracting eq (4) from eq (6)
:>B= 64˚6ˡ 2.67ˡˡ
A= 52˚24ˡ 12.66ˡˡ
C = 180 ˚- (A+B)
= 180 ˚- (52˚24ˡ 12.66ˡˡ +64˚6ˡ 2.67ˡˡ )
= 63˚29ˡ 44.67 ˡˡ

27. • Standard Deviation ()
Also known as the Standard Error or Variance
2 = (M-MPV)
n-1
M-MPV is referred to as the Residual
 is computed by taking the square root of the
above equation

28. Error of a Sum
• Independently observed observations
– Measurements made using different equipment, under
different environmental conditions, etc.
2 2 2
...Sum a b cE E E E    
Where E represents any specified percentage error
And a, b and c represent separate, independent
observations

29. Example:
• A line is observed in three sections with the lengths 1086.23 ±
0.05 ft, 569.08 ± 0.03 ft and 863.19 ± 0.04 ft. Compute the total
length and standard deviation for the three sections.
• Solution:
2 2 2
(0.05) (0.03) (0.04) 0.07SumE ft   
1086.23 569.08 863.19 2,518.50Length ft   
Probable length = 2,518.50 ± 0.07 ft

30. Error of the Mean
m
E
E
n

E is the specified percentage error of a single observation and n is the
number of observations
This equation shows that the error of the mean varies inversely as the
square root of the number of repetitions. In order to double the
accuracy of a set of measurements you must take four times as many
observations.

31. THANK YOU