Theory of Errors

Measurements of angles and distances are made in different surveying processes.
It is impossible to determine the actual values of these quantities because some type of errors always creep in every measurement.

Occurrence of Error

Imperfection in Instruments
Environmental Condition
Human Limitations & Carelessness.

Imperfection in Instruments

Due to faulty Instrument.

Example: incorrect length of chain, collimation error.

Environmental Condition

Due to nature error.

Example: error due to curvature, error due to slope, error due to temperature.

Human Limitations & Carelessness

Due to wrong reading/ writing.

Type of Error

The errors can be classified into three types.

Gross errors or mistakes
Systematic or cumulative errors
Accidental or random errors.

1. Gross Errors or Mistakes

Inexperienced or careless surveyors make mistakes.
These errors are unpredictably occurring and can be avoided by following a standard operating procedure.

2. Systematic or Cumulative Errors

The systematic errors are of same magnitude and nature in the same conditions.
The systematic errors can be computed and suitable corrections applied.
For example, the error in the length of the steel tape due to change in temperature is a systematic error.

3. Accidental or Random errors

Accidental errors occur due to lack of perfection in human eye.
The accidental errors obey the law of chance.
These error tends to compensate each other, when large in number.
These can be understood and eliminated with the help of mathematical theory of probability.

Important Terminology

1. Precision

Degree of perfection use at the time of measurement is called precision.
It is adopted by using high quality instrument, skilled surveyor, and using correct manner of measurement.

2. Accuracy

Degree of perfection obtained in measurement is called accuracy.
If any measurement more near to the true value of quality it is considered as more accurate.

3.True Value

Exact value of quantity is called true value.
It is ideal value which can not be obtained.

4.True Error

The difference between the observed value and the true value of a quantity is knows as true error.
True error = Observed value – True value
As the true value of a quantity is never known, the true error can also never be determined.

5.Most Probable Value (M.P.V)

⇒ The most probable value of a quantity is a value which has more chance of being true value than any other value.

⇒ The most probable value is very close to the true value.

⇒ It is calculated on the basic of Principle of Least Square.

⇒ It is equal to the the weighted average mean.

MPV for multiple measurement
MPV = \(\bar{x} = \frac{x_1+x_2….+x_n}{n}\)
MPV for multiple measurement with weight
MPV = \(\bar{x} = \frac{x_1w_1+x_2w_2….+x_nw_n}{w_1+w_2…+w_n}\)

6. Principle of Least Square

MPV is that value of quantity for which sum of square of error is least.

Y = sum of square of error
\(Y=(x_1-\bar{x})^{2}+(x_2-\bar{x})^{2}\)\(+…..(x_n-\bar{x})^{2}\)

7.Residual Error

The difference between the observation value and most probable value of a quantity is called the residual error, residual or variation (v).

Residual Error= Observed Value – Most Probable Value.

Law’s of Weights

⇒ ‘Weight’ is a numerical value a singed to a measurement on the basic of degree of precision & adopted which represent important of reading.

⇒ Weight of measurement is inversely propositional to error.

⇒ The weight of the arithmetic mean of a number of observations of unit weight is equal to the number of observations.

For example, if an angle P is measured three times and the values are obtained as below:

1.	40 30’20”	weight 1
2.	40°30′15″	weight 1
3.	40°30’10”	weight 1
	Sum= 121°30′45′’

Arithmetic mean = (121°30′45′’)/3= 40°30’15”

The arithmetic mean 40°30’15” will have weight 3.

⇒ The weight of the weighted arithmetic mean is equal to the sum of individual weights.

For example, if an angle P has the following values.

1.	40 30’5″	weight 1
2.	40°30′7″	weight 2
3.	40°30′10″	weight 3

Weighted Arithmetic mean ={(40°30’5″)×1 + (40°30′7″)×2 + (40°30′10″)×3}/( 1+2+3)= {243°0’22”}/6 = {40°30’3.6”}

The weighted arithmetic mean 40°30’3.6” will have weight 6.

⇒ If measurement x₁ is a which taken weight w₁ & measurement, x₂ is a which taken weight w₂than weight of result & k is factor, for various operation are found as:

Result	Weight of Result
x₁ +x₂ or x₁ -x₂	\(\frac{1}{\frac{1}{w_1}+\frac{1}{w_2}}\)
\(\frac{x_1}{k}\)	w1·k²
x₁·k	\(\frac{w_1}{k^{2}}\)
x1 + k x₁ -k k-x₁	w₁

Probable Error

If for a single quantity multiple measurement are taken as x₁, x₂, … x_n with weight w₁, w₂, …w_n. and the MVP= \(\bar{x}\) then probable error can be calculated as.

1.Probable Error in Single Measurements

\(E_s=\pm 0.6745\sqrt{\frac{\Sigma w_n(x_n-\bar{x})^{2}}{(n-1)}}\)

2.Probable Error in Mean (MPV)

\(E_m=\pm 0.6745\sqrt{\frac{\Sigma w_n(x_n-\bar{x})^{2}}{(n-1)\Sigma w_n}}\).

\(E_{m}=\frac{E_s}{\sqrt{\Sigma w_n}}\).

3.Probable Error of Single Measurement of Weight

\(E_{sw}=\pm 0.6745\sqrt{\frac{\Sigma w_n(x_n-\bar{x})^{2}}{(n-1)w_o}}\)

\(E_{sw}=\frac{E_s}{\sqrt{w_o}}\).

NOTE: For all measurement have unit weight. \(\Sigma w_n=1\)

Error In Computed Result

If a quantity is expressed in digit then first and -1 digit are certain and last digit is least accurate.

Error in quantity can be assumed as

x= 25.34

then Possible Error or Max error (∂x)= ±0.005

and Probable error (ex)= ±0.0025

For better understanding more example are taken

x = 34.55

∂x= ±0.005

ex= ±0.0025

y= 56.5

∂y= ±0.05

ey= ±0.025

y= 352.5

∂y= ±0.05

ey= ±0.025

Various operation are found as

Result [s]	Possible Error or Max Error [∂s]	Probable Error [es]
s = [x+y]	∂s = ± [∂x +∂y]	es = ±\(\sqrt{ex^{2}+ey^{2}}\)
s = [x-y]	∂s = ± [∂x +∂y]	es = ±\(\sqrt{ex^{2}+ey^{2}}\)
s = [x×y]	∂s = ± [x·∂y +y·∂x]	es = ±s\(\sqrt{(\frac{ex}{x})^{2}+(\frac{ey}{y})^{2}}\)
s = [x/y]	∂s = ± \(\frac{\partial x}{\partial y}+x\frac{\partial y}{\partial y^{2}}\)	es = ±s\(\sqrt{(\frac{ex}{x})^{2}+(\frac{ey}{y})^{2}}\)

Question: The side of a square land was measured as 150 m and is in error by 0.05m. what is the corresponding error in the computed area of the land?

Given :

side of square (a)= 150m

∂a= 0.05m

∂A=?

A= a²

∂A= 2·a·∂a = 2×150×0.05= 15m²

FOR MORE INFORMATION ABOUT SURVEYING

Theory of Errors (Type of Error, Law’s of Weights, Probable Error, Error In Computed Result)