** 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 x_{1} is a which taken weight w_{1} & measurement, x_{2} is a which taken weight w_{2 }than weight of result & k is factor, for various operation are found as:

Result | Weight of Result |

x | \(\frac{1}{\frac{1}{w_1}+\frac{1}{w_2}}\) |

\(\frac{x_1}{k}\) | w1·k^{2} |

x_{1}·k | \(\frac{w_1}{k^{2}}\) |

x1 + k x _{1} -kk-x _{1} | w_{1} |

** Probable Error **

If for a single quantity multiple measurement are taken as x_{1}, x_{2}, … x_{n} with weight w_{1}, w_{2}, …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

1. | x = 34.55 | ∂x= ±0.005 ex= ±0.0025 |

2. | y= 56.5 | ∂y= ±0.05 ey= ±0.025 |

3. | 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 :

∂a= 0.05m

∂A=?

A= a^{2}

∂A= 2·a·∂a = 2×150×0.05= 15m^{2}