The weight of the arithmetic mean of the measurements of unit weight is equal to the number of obs

**LAW OF WEIGHTS**

From
the method of
least squares the following laws
of weights are established:

(i) The weight of the arithmetic mean of the
measurements of unit weight is equal to the number of observations.

For example, let an angle A be measured six times,
the following being the values:

A Weight A Weight

30 o 20? 8' 1 30 o 20? 10' 1

30 o 20? 10' 1 30 o 20? 9' 1

30 o 20? 7' 1 30 o 20? 10' 1

Arithmetic mean

=
30 o 20? + 1/6 (8' + 10' + 7' + 10' + 9' + 10')

= 30 o 20? 9'.

Weight of
arithmetic mean = number of observations = 6.

(2) The weight of the weighted
arithmetic mean is equal to the sum of the individual weights.

For example, let an angle A be measured six times, the
following being the values :

A Weight A Weight

30 o 20? 8' 2 30 o 20? 10' 3

30 o 20?
10' 3 30 o 20? 9' 4

30 o 20? 6' 2 30 o 20? 10' 2

Sum of weights = 2 + 3 + 2 + 3 + 4 + 2 =16

Arithmetic mean = 30 o
20? + 1/16 (8'X2 + 10' X3+ 7'X2 + 10'X3 + 9' X4+ 10'X2)

= 30 o 20? 9'.

Weight of
arithmetic mean = 16.

(3) The weight of algebric sum of
two or more quantities is equal to the reciprocals of the individual weights.

For Example angle A = 30 o 20? 8', Weight 2

B
= 15 o 20? 8', Weight 3

Weight of A
+ B =

(4) If a
quantity of given
weight is multiplied by a
factor, the weight
of the result is obtained by
dividing its given weight by the square of the factor.

(5) If a
quantity of given weight is divided by a factor, the weight of the result is
obtained by multiplying its given weight by the square of the factor.

(6) If a
equation is multiplied by its own weight, the weight of the resulting equation
is equal to the reciprocal of the weight of the equation.

(7) The
weight of the equation remains unchanged, if all the signs of the equation are
changed or if the equation is added or subtracted from a constant.

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Civil Surveying : Survey Adjustments : Law of Weights |

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