Measurement Error

Unit: - 9    Measurement error and accuracy

The difference between a measured value and the true value is known as true error. The true error cannot be determined as the true value is never known. However, using the theory of probability, it is possible to determine a value called the most probable value which is close to the true value. The most probable error and the standard error can also be determined.

A discrepancy is the difference between two measured values of the same quantity, it is not an error. A discrepancy may be small, yet the error may be great if each of the two measurements contains errors that may be larger. It does not reveal the magnitude of systematic error. 

Significant figure and rules:

In surveying, an indication of accuracy attained is shown by number of significant figures. Each such quantity, expressed in n number of digit in which n-1 are the digit of definite value while the last digit is the least accurate digit which can be estimated and is subject to error.

For example, a quantity 425.65 have five significant figures, with four certain and last digit 5, uncertain.

Sources of errors

Depending upon the sources, the errors can be classified under the following three types:

1.    Instrumental errors

2.    personal errors 

3.    Natural errors

Instrumental errors:

Error which arises due to imperfection or faulty adjustment (maladjustment) of the instrument with which measurement is being taken is known as Instrumental error. It can be eliminated or minimized by adopting Suitable procedures and by applying proper correction & adjustment. for e.g. If the tape used in measurement is of distance is actually 29.90 m long whereas nominal length is 30m, the error occurs because of the imperfect tape called Instrumental error.

Personal error:

The personal error occurs due to human limitation such as sense of sight and touch. The errors occur for want of perfection of human sight while taking observation or for want of perfection of touch while manipulating the instruments. for e.g. A distance of 1.576m may be estimated as 1:577 m or 1.575 m.

 

Natural error:

The error may arise due to variations in natural phenomena such as temperature, humidity, gravity, wind, refraction and magnetic declination. If they are not properly observed while taking measurements, the results will be incorrect. Generally, it is not possible to remove the cause of natural errors. However, the natural errors can be minimized by using good Judgement and applying corrections. for e.g. if a tape has been calibrated at 20°c, but the field temperature is 30°C, there will be natural error due to temperature variation.

 

Types of errors: -

On the basis of nature.

1.    Mistake / Gross error / Blunder

2.    Systematic error / cumulative error / persistent errors

3.    accidental error / Random error / compensation error

 

 

Mistake:

Mistakes occur in measurements due to carelessness, inattention, in experience, poor Judgement or confusion of the surveyor. Mistakes are quite common in a careless work done by the inexperienced surveyor. It is also called as blunder or Gross error. Mistakes are random in nature, as they do not follow any fixed pattern (mathematical rule) and may be larger or small, positive or negative. It can be detected by repeating the whole operation and can be eliminated by adopting standard methods of observation, booking & checking. The work should be done very carefully to avoid mistakes.

 

 Systematic error:

 

An error, that under the same condition it will always be of the same size and sign, and such error is called as constant systematic error. Whereas, if the conditions change the magnitude of error changes and is known as variable systematic error. It always follows some definite mathematical and physical law and correction can be determined and applied. It may arise due to humidity, pressure, and current velocity, and curvature, refraction, etc. and faulty setting or improper levelling of any instrumental and personal vision of an individual. These errors are also known as cumulative error or persistent error. The surveyor must have the full knowledge of various systematic errors in the survey being conducted by him.

For e.g. If the tape has been standardized at a temperature of 20 c but the field temperature is 25 c, the tape will about too long. It will measure less than the actual distance to be.

The error occurs in both signs, If the error is -ve than the correction is positive and If the error is +ve than the correction is negative.

 

Accidental error: -

Accidental error is random in nature therefore it is also known as random error.  Accidental errors are those which remain after mistakes and systematic errors and caused by a combination of reasons beyond the ability of the observer to control. It does not follow any fixed pattern or law. These errors can be positive or negative. Generally, it is of small magnitude and they tend to distribute themselves equally on both sides of the true value. It tends to cancel them in a series of measurement, therefore it is also known as compensating errors. 

It occurs due to:

(i) Imperfection in the instrument,

 (ii) Human limitation,

(iii) Change in atmospheric conditions.

 

 These errors are small unavoidable errors which cannot be detected by the surveyor because of human limitation. For e.g. While marking a point below a plumb bob may cause an accidental error.

 The error is due to limitation of eye judgment and limited precision. However, with more precise instrument and better methods, the accidental errors may be minimized.

 

Laws of accidental error:

Investigation of observation of various types shows that an accidental error follows a definite law, the law of probability. This law defines the occurrence of errors and can be expressed in the form of equation which is used to compute the probable value or the probable precision of a quantity. 

The most important features of accidental errors which usually occur are:

1.  Small errors tend to be more frequent than the large ones; that’s they are most probable.  
2. Positive and negative errors of the same size happen with equal frequency; that’s they are equally probable. 
3. Large errors occur infrequently and are impossible.

Propagation of error:

The process of an accumulation and cancellation of random error in a measurement, directly or indirectly is known as propagation of error.

Proof:

l= l1 + l2

Let, the function be of the type

z=x+y

Or, z+∆z = ( x+∆x)+(y+∆y). (∆=error)

Or, ∆z = x + ∆x +y + ∆y 1 – z

Or, ∆z = x+ ∆x+ y+∆y -x-y

Or, ∆z = ∆x +∆y

Squaring on both side, we get:

∆z 2 =∆x 2 +∆y 2 +2∆x∆y

Since we have 'n' number of measurement,

Then,

∆z1 2 =∆x1 2 +∆y1 2 +2∆x1.∆y1

∆z2 2 =∆x2 2 +∆y2 2 +2∆x2.∆y2

∆z3 2 =∆x3 2 +∆y3 2 +2∆x3.∆y

∆zn 2 =∆xn 2 +∆yn 2 +2∆xn.∆yn

∆zi 2 =(∆xi 2 +∆yi 2 +2∆xi.∆yi)

Dividing both side by n

Or, ∆zi 2 /n = ∆xi 2 /n + ∆yi 2 /n + 2∆xi.∆yi/n

Or, z 2 =x 2 +y 2 +0

Or, z = x 2 + y 2

Distribution of error of the field measurements:

Whenever observation is made in the field, it is always necessary to check for the closing error, if any.

The closing error should be distributed to the observed quantities. For example, the sum of the angles measured at a central angle should be 360°; if the sum is not equal to 360°, the error should be distributed to the observed angles after giving proper weightage to the observations. 

The following rules should be applied for the distribution of errors:

 

1.     The correction to be applied to an observation is inversely proportional to the weight of the observation.

2.     The correction to be applied to an observation is directly proportional to the square of the probable error.

3.     In case of line of levels, the correction to be applied is proportional to the length.

 

Tolerance and permissible error:

Tolerance and permissible error are closely related concepts, but they have distinct meanings:

Tolerance:

·         Refers to the acceptable range of variation for a measurement or a physical dimension.

·         It's essentially a predetermined limit for how much something can deviate from a specified value and still be considered acceptable.

·         Tolerances are often expressed as a plus-or-minus value around a nominal value.

·         For example, a distance measure by using tape must be +/- 2 millimeters. This means the distance can be 2 millimeters wider or narrower than the nominal size.

Permissible Error:

·         Permissible error can be seen as a synonym for tolerance, especially when referring to the amount of deviation allowed in a measurement.

·         It emphasizes the fact that this error is acceptable within the predetermined limits set by the tolerance.

 

Accuracy:

It denotes the closeness of a measurement to its true values. If the measured value is very close to its true values, it is very accurate. Accuracy therefore, indicates nearness to the true value. It is degree of perfection achieved in measurement.

Degree of accuracy:

The degree of accuracy indicates the accuracy attained in the measurements. It is usually expressed as the ratio of the error and the measured quantity. E.g. A degree of accuracy in 1 in 1000 indicates that there is an error of 1 units in 1000 units.

 

Precision:

Precision of a measurement denotes its closeness to another measurement of the same quantity. If a quantity is measured several times and the values obtained are very close to one another, the precision is high.

 

 

Degree of Precision:

The degree of precision is used to express the precision of the various measurements. It is usually expressed as a ratio of the standard deviation to the mean value of quantity itself.