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__Basic Concepts of Surveying (Part 3)__

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__Errors in Measurement __

In practice, we can never measure the true value of any length, angle or any geoinformation because there will always be some error in the measurement.

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__Sources of Error in Measurement:__

- Natural Sources (like temperature, wind, etc.)

- The instrument being used (there may be some error in the measurement of the instrument)

- The person taking the measurement (the person may introduce some error in the measurement)

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__Explanatory Video __

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__Classification of Errors:__

## Blunder or Mistake

- It occurs because of carelessness.

## Systematic Error

- It is caused due to systematic forces.

- These errors follow some physical law (as in the case of temperature, gravity, etc.).

- We can write the mathematical model for these errors.

- These errors can be eliminated.

## Random Error

- These errors are caused by many sources which we cannot account for.

- They are equally distributed in both the positive and negative direction (or their probability of occurrence in both the positive and negative direction is the same).

- Random errors which are larger in magnitude occur less while random errors which are smaller in magnitude occur more.

- After knowing that the observation is not a blunder or mistake and eliminating systematic errors from it, there are still random errors in it.

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__Precision __

- The meaning of Precision is the closeness of observations.

- If the systematic errors are eliminated, the precision of the observations become an indicator of the accuracy.

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__Accuracy__

- The meaning of Accuracy is the closeness of our observation to the true value.

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__True Error __

- True error is defined as the difference between the true value and the observed value.

- True error = True value (L)- Observed Value (l)

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__Relative Error__

- In one of the cases, for a scale, the relative error is defined as the ratio of the error which is being introduced because of the least count to the observed value.

- Relative error = [that error (Δl) / observed value (l)]

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__Percentage Error__

- The relative error multiplied by 100 gives the value of the percentage error.

- Percentage error = Relative error * 100

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__Most Probable Value (MPV)__

- It is defined as the value which has the maximum probability of being nearest to the true value.

- It is a kind of replacement for the true value.

- MPV = Arithmetic Mean

- x̄ = [(x1 + x2 + x3 + .................. + xn) / n)] = MPV

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__Residual __

- For any observation x,

- Residual (r) = MPV (x̄) - Any observed value (x)

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__Proof for MPV= Arithmetic Mean__

Since, r = x̄ - x

So, r1 = x̄ - x1

Similarly, r2 = x̄ - x2 , r3 = x̄ - x3 .....and so on.

and rn = x̄ - xn

On adding the values of r (from r1 to rn), we get;

r1 + r2 + r3 + r4 + ......... + rn = (x̄*n) - (x1 + x2 + x3 + ............. + xn)

Now, in the case when all the observations have only random errors;

For large n, (r1 + r2 + r3 + ...... + rn) = 0 (because the random errors have got equal probability of going into the positive as well as in the negative direction).

Now, (x̄*n) = x1 + x2 + x3 + ............. + xn

Therefore, MPV (x̄) = [(x1 + x2 + x3 + ............. + xn) / n)] = Arithmetic Mean

Hence Proved.

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__Random Error Distribution__

- The random error distribution curve is called the Normal curve or Gaussian curve or bell-shaped curve.

- There is an equal probability of observation or error to go in both the positive and negative direction of the Most Probable Value.

- Our observations or our errors are equally distributed in both the positive and negative side.

- The errors which are large in magnitude have a smaller probability of occurrence and the errors which are small in magnitude have a larger probability of occurrence.

- The probability of occurrence of the Maximum Probable Value (MPV) is maximum.

Gaussian Curve or Normal Curve |

The Probability density function for the Gaussian Curve is given by:-

Probability density function P(x) |

where μ is MPV

x is an observation and a variable

P(x) is the probability of occurrence of the variable x or Probability density function

σ is Standard deviation which is =

*Self Typed

*Source - Internet, Books, Self-Analysis