Theory of Errors

Introduction
Error can occur
Due to some human limitations or carelessness.
Imperfection in instruments.
Environmental changes.

Types of Errors
1. Mistakes.
2. Systematic errors.
3. Accidental errors.

Mistake
 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
1800 for 1790 it is a mistake.

Systematic errors
 It will always be of the same sign and magnitude.
 Error 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.

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

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

 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

 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.

Laws of Weights
1. The weight of arithmetic mean of number of observations of unit
weights is equal to the number of observation.
E.g. Angle is measured 3 times ,
1. 40 ˚ 30’ 20” weight 1
2. 30 ˚ 30’ 10” weight 2
3. 40 ˚ 30’ 15” weight 3
Solution: Sum of Angle = 121 ˚ 30’ 45”
Arithmetic mean = 121 ˚ 30’ 45” / 3 = 40 ˚ 30’ 15”
The arithmetic mean has the weight of 3, equal to the
number of observations.

2. The weight of the weighted arithmetic mean is equal to the sum of the
individual weights.
E.g. Angle is measured 3 times
1. 40 ˚ 30’ 20” weight 2
2. 40 ˚ 30’ 10” weight 1
3. 40 ˚ 30’ 15” weight 3
Solution: Sum of individual weights = 2 + 1 + 3 = 6
Weighted arithmetic mean
= (40˚30’20”)x 2 + (40˚30’10”)x1 + (40˚30’15”)x3 / 6
= 40˚30’15.83”
The weight of weighted arithmetic mean is 6.

3. The weight of algebraic sum of two or more quantities is
equal to the reciprocal of the sum of reciprocal of individual
weights.
E.g. Angle is measured 3 times ,
1. 40 ˚ 30’ 20” weight 1
2. 30 ˚ 30’ 10” weight 2
3. 40 ˚ 30’ 15” weight 3
Solution: Weight of algebraic sum of quantities
=
1
1
1
+
1
2
+
1
3
=
6
11

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.
E.g. A = 40 ˚ 30’ 20” has a weight of 6,
Weight of 2A =
6
22 =
3
2
Weight of 3A =
6
33 =
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.
E.g. A = 40 ˚ 30’ 20” has a weight of 6,
Weight of A/2 = 6 x 22 = 24
Weight of A/3 = 6 x 33 = 54

Theory of Least 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 ,
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.

Law of Accidental Error
 Follow a Definite law, “the law of probability”.
Defines the occurrence of errors and can be expressed in the
form of equation which is used to compute the probable value of
quantity.
1. Probable error.
2. Probable error of the mean.
3. Probable error of a sum.
4. Mean square error.

Determination of Probable Error
1 . Direct observation of equal weights
(a) Probable Error of single observation of unit weight
v = residual error
n = Number of observations

(b) Probable error of single observation of weight w
(c) Probable error of single arithmetic mean

2. Direct observation of unequal weights
(a) Probable error of single observation of unit weight
(b) Probable error of single observation of weight w
(c) Probable error of weighted arithmetic mean

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

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Theory of Errors

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