Types of Errors

&

How to reduce them

Functional Elements of and Instrumentation System

Errors

1. Gross error

2. Systematic error

3. Random error

Gross error

some gross error can be detected & some others cannot.

complete elimination is not possible.

Types of gross errors

Human Error

due to humans

a) may be due to misreading of

instruments.

b) Incorrect adjustments

c) Improper application of instruments

d) Computational mistakes etc.

common in begineers

Remedy

take care in reading and recording measured data

Take at least three separate readings

(preferably under conditions in which instruments are switched off-on)

needs good practice

Installation error due to

improper applications

faulty insulations

is predominant if device used beyond limit

Or if used in excess temp. , vibration, pressure

Or poor impedance matching

Remedy

use devices according to the specifications recommended by manufactures

Zero error :

If the instrument is not set to zero before taking measurement.

Due to variation in ambient conditions

Due to ageing

Systematic error

relatively constant error

frequency evident in direct observation

Types of Systematic errors

Instrumental error

eg. Irregularity of spring in galvanometers

calibration error

Remedy

selecting a suitable instruments for the particular measurements according to applications

Applying correction factors after determining the amount of error.

Calibrating the instruments against a known standard

Environmental errors

due to surroundings

Eg. Noise from electrical machine. Magnetic field, temperature.

Remedy

By providing proper shielding.

Random Error or Uncertanity error

Errors due to unknown Causes.

Magnitude and direction not known

Expressed as average deviation of probale errors or standard deviation

Remedy

Increase the no. of readings and using statically method

Statistical Analysis

Arithmetic Mean

Deviation from Mean

d1=x1- A.M.

d2=x2- A.M.

etc

Average Deviation

D= |d| / n

Standard Deviation

= sqrt ( |d i| 2/ n)

For finite observations

= sqrt ( |d i| 2/ (n-1))

Variance

v= 2

mean square deviation

Probable error

= +/- 0.6745

(obtained from Gaussian error curve)

Limiting error

specified by manufactures

Precision

a measure of reproductively

a measure of degree to which successive measurements differ from one another.

Higher precision means tight cluster of repeated results

lower precision means scattering of results

Precision composed of conformity &

Significant figures

Significant figures

indication of precision is obtained by no. of significant figures

Accuracy and precision

difference :

Accuracy- telling truth

Precision : repeating same story.

Calibration

to check the instrument against a known value and finding the error and then finally making the instrument more accurate.

Error is determined at no. of points and graph is plotted – this graph is error calibration curve.- used for calibration.

Sensitivity

ratio of deflection of pointer to given change in measured quantity.

or

smallest change in measured quantity to which an instrument responses.

inverse of sensitivity is deflection factor.

Unit depends on type of I/p & o/p.

Obtained from slope of I/p – o/p calibration curve

Value of sensitivity – influenced by requirement of inst. Application.

static sensitivity = qi / qo

Deflection factor = 1/ sensitivity

Range or Scale range.

difference b/w the largest and smallest reading of instrument.

If. X max. is max. value that can be measured an instrument and

X min. Min . Value that can measured using an instrument , then

We say range of that instrument is

b/w X min & X max

Span or scale span.

variation from minimum to maximum value

X max - X min

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1. Basic ConceptF

req

uen

cy

True value, XT Mean of measurement

output, X

Variations, i.e.

random error

Systematic error = X XT

Repeatability (that describes

precision) Standard deviation of

measurement data

Systematic error: Shift in the mean valueRandom error: Standard deviation

Repeat measuring the same physical quality a number of

time, we shall obtain the following:

39

2. Random errorsThey are represented by repeatability described in

earlier lectures. Repeatability (R) is numerically equal to the half range random uncertainty (Ur) of the measurement.

For normal distribution (n), at 95%

confidence level

x mean, population-1.96 +1.96

Repeatability, R = Z

=1.96

z

xmean,population

For normal distribution (n), at % confidence

level

- Z + Z

Repeatability, R = Z

z

40

To estimate the population standard deviation fromthe sample deviation, say, n = 10 or n = 20, we shalluse the Student’s t distribution. The repeatabilitymay be expressed as

Repeatability, R = Z = t s

where the value of t can be found from the tdistribution table based on the sample size n and theconfidence level ,

s is usually referred to as the sample standarddeviation.

11

2

n

xx

s

n

i

41

Random errors can largely be eliminated bycalculating the mean of the measurements,

since in statistical analysis of datawhere is the standard deviation of the

sample mean, is the standard deviation ofthe population, and n is the sample size (i.e.number of repeated measurements for thesame measurand). The standard error of themean is usually expressed by .

n

xx

i

nx

x

x

42

3. Error reduction using intelligentinstrumentsIt also includes

a microcomputer, and one or more transducers (secondary transducers)The secondary transducers monitor the environmentalconditions (modifying inputs). By reading the outputs of theprimary transducer and the secondary transducers, themicrocomputer processes the signals based on a pre-loadedprogramme.* It can be programmed to take a succession ofmeasurements of a quantity within a short period of time(sampling frequency)-1 and perform statistical calculations onthe readings before displaying an output measurement. This

is valid for reducing random errors.

43

The ability of intelligent instruments to reduce systematic errors requires the following pre-conditions be satisfied:• The physical mechanism by which a measurement

transducer is affected by ambient condition changes must be fully understood and all physical quantities which affect the transducer output must be identified.

• The effect of each ambient variable on the output characteristic of the measurement transducer must be quantified.

• Suitable secondary transducers for monitoring the value of all relevant ambient variables must be available for input to the intelligent instrument.

44

The accuracy of a measurement system is a function of its ability to indicate the true value of the measured quantity under specific conditions of use and at a defined level of confidence.

The accuracy A is expressed by

where R is the instrument repeatability and

22

sURA

4. Calculations of accuracy and errors

4.1 Accuracy of a measuring instrument

45

Systematic Error

Random Error( Z)

46

Example. The systematic error of a balanceis estimated to be 5g and the random errorof its measurements is 25g. State therepeatability and calculate the accuracy of theinstrument.

Solution:

Repeatability, R = Ur = 25gSystematic error, Us = 5g

gA 495.25525 ,Accuracy 22

As the systematic error and the repeatability areboth stated in grams, the accuracy of theinstrument is 26g.It is usual to express the accuracy in terms ofits full scale deflection (f.s.d.).

Accuracy = f.s.d.%26.0%100

1010

263

47

The sum of the systematic and random errors of a typical measurement, under conditions of use and at a defined level of confidence.

Error assessment must also include the error of the calibrator itself and take account of the confidence level upon which it is founded.

4.2 Estimation of total error

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4.3 Compound error/uncertaintyIn many instances, the ultimate error (uncertainty) of a

measurement is dependent upon the errors of a number ofcontributory measurements which are combined to determinethe final quantity. The measurement output

M=M(x1, x2, x3,…)

is a function of a number of individual measurements x1, x2,x3, etc. All of these measurements have individual error ofDx1, Dx2, Dx3, etc. Then, the compound error of themeasurement M, DM, can be determined by substituting intothe equation of M=M(x1, x2, x3,…) the maximum andminimum values of x1, x2, x3, etc., and thus finding themaximum and minimum values for M.

49

This would obviously be a laborious process, andthe problem is better solved using partialdifferentiation:

Attentions must be paid to those high valueterms that give dominant contributions to DM.

i

i

dxx

MdM

i

i

xx

MM D

D

i

i

xx

MD