data handling l classification of errors v systematic v random

Data HandlingData Handling

Classification of ErrorsClassification of Errors

SystematicSystematic

RandomRandom


Random ErrorRandom Error Affects precisionAffects precision Within run precision - Within run precision -

repeatabilityrepeatability Between run precision - Between run precision -

reproducibiltyreproducibilty Indeterminate errorIndeterminate error

Systematic ErrorSystematic Error Affects accuracyAffects accuracy Proximity to the truthProximity to the truth Determinate error or Determinate error or

bias.bias.


Systematic ErrorsSystematic Errors

These are errors which can be avoided, These are errors which can be avoided, or whose magnitude can be determined.or whose magnitude can be determined.

Three important types of systematic Three important types of systematic error.error.


Systematic ErrorsSystematic Errors

Operational and personal errors.Operational and personal errors. Factors for which the analyst is responsible.Factors for which the analyst is responsible.

Instrumental and reagent errors.Instrumental and reagent errors. Uncalibrated equipment, unexpected reactions.Uncalibrated equipment, unexpected reactions.

Errors of method.Errors of method. Incorrect sampling, incompleteness of a reaction.Incorrect sampling, incompleteness of a reaction.

Data HandlingData Handling Random ErrorsRandom Errors

The slight variations which occur in successive The slight variations which occur in successive measurements.measurements.

Due to causes over which the analyst has no Due to causes over which the analyst has no control.control.

If a sufficiently large number of measurements If a sufficiently large number of measurements are taken it can be shown that these errors lie are taken it can be shown that these errors lie on a Gaussian curve.on a Gaussian curve.


Minimising ErrorsMinimising Errors Calibration of apparatusCalibration of apparatus Running a blank determinationRunning a blank determination Running a control determinationRunning a control determination Use of independent methods of analysisUse of independent methods of analysis Running parallel determinationsRunning parallel determinations Standard additionStandard addition Internal standardsInternal standards Amplification methodsAmplification methods Isotopic dilutionIsotopic dilution


Propagation of errorsPropagation of errors

It is important to note that the procedure for It is important to note that the procedure for combining random and systematic errors are combining random and systematic errors are completely different.completely different.

Random errors to some extent cancel one another Random errors to some extent cancel one another out whereas every systematic error occurs in a out whereas every systematic error occurs in a definite and known sense.definite and known sense.



If a and b have a systematic error of +1 then If a and b have a systematic error of +1 then the systematic error in x given by x = a + b is the systematic error in x given by x = a + b is +2.+2.

If however, a and b have a random error of If however, a and b have a random error of the random error in x is not the random error in x is not



Two typesTwo types

Linear combinationsLinear combinations sums and differencessums and differences

Multiplicative expressionsMultiplicative expressions


Linear CombinationsLinear Combinations

In the case of the final value y calculated from In the case of the final value y calculated from the linear combination of measured quantities the linear combination of measured quantities a, b, c, etca, b, c, etc

sy = sa2 + sb

2 + sc2 + ...


Multiplicative ExpressionsMultiplicative Expressions

In the case of the final value y calculated from In the case of the final value y calculated from an expression of the type y = ab / cdan expression of the type y = ab / cd

2222

ds

cs

bs

as

y

sdcbay


Multiplicative Expressions 2Multiplicative Expressions 2

In the case of the final value y calculated from In the case of the final value y calculated from an expression of the type y = ab / cdan expression of the type y = ab / cd

222

100100100100

cs

bs

as

y

scbay


Calibration CurvesCalibration Curves

When carrying out an analysis it is often When carrying out an analysis it is often necessary to carry out a calibration procedure necessary to carry out a calibration procedure by using a series of samples (standards) each by using a series of samples (standards) each having a known concentration.having a known concentration.

A calibration curve is constructed by plotting A calibration curve is constructed by plotting the response of the standards against the the response of the standards against the concentration.concentration.



There are two statistical tests which should be There are two statistical tests which should be applied to a calibration curve.applied to a calibration curve.

To ascertain the linearity of the curve.To ascertain the linearity of the curve.

To evaluate the best straight line through the To evaluate the best straight line through the data points.data points.


Correlation CoefficientCorrelation Coefficient In order to establish whether there is a linear In order to establish whether there is a linear

relationship between two variables xrelationship between two variables x11 and y and y11 the the

Pearson’s correlation coefficient is used.Pearson’s correlation coefficient is used.

nx1y1 - x1y1

[nx12 - (x1)2] [ny1

2 - (y1)2]

r =

Data HandlingData Handling Linear RegressionLinear Regression

Once a linear relationship has been shown to have a Once a linear relationship has been shown to have a high probability by the value of the correlation high probability by the value of the correlation coefficient, then the best straight line through the coefficient, then the best straight line through the data points has to be estimated.data points has to be estimated.

This can often be done by visual inspection but it is This can often be done by visual inspection but it is better to evaluate it by linear regression - the method better to evaluate it by linear regression - the method of least squares.of least squares.


Linear RegressionLinear Regression

The equation of a straight line is y = ax + b where The equation of a straight line is y = ax + b where y the y the dependentdependent variable is plotted as a result of variable is plotted as a result of changing x, the changing x, the independentindependent variable. variable.

To obtain the regression line y on x the slope of the To obtain the regression line y on x the slope of the line “a” and the intercept on the y-axis “b” are line “a” and the intercept on the y-axis “b” are given by the following equations.given by the following equations.


Linear RegressionLinear Regression

nx1y1 - x1y1

nx12 - (x1)2

a =

b = y - ax



y = 1.2469x - 0.0219

0

1

2

3

4

0 1 2 3

Copper Concentration (mg/l)

Ab

s R2 = 0.9992


Significant FiguresSignificant Figures

There are a number of rules for There are a number of rules for computations which you need to be computations which you need to be aware of and familiar with.aware of and familiar with.

Data HandlingData Handling Significant FiguresSignificant Figures

Retain as many significant figures in a result or Retain as many significant figures in a result or in any data as will only give one uncertain in any data as will only give one uncertain figure.figure.

A volume which is known to be between 30.5ml A volume which is known to be between 30.5ml and 30.7ml should be written as 30.6ml, but not and 30.7ml should be written as 30.6ml, but not 30.60ml as the latter would imply that the value 30.60ml as the latter would imply that the value lies between 30.59ml and 30.61mllies between 30.59ml and 30.61ml


Significant FiguresSignificant Figures

In rounding off quantities to the correct In rounding off quantities to the correct number of significant figures add one to the number of significant figures add one to the last figure retained if the following figure last figure retained if the following figure (which has been rejected ) is 5 or over.(which has been rejected ) is 5 or over.


Significant FiguresSignificant Figures In addition or subtraction, there should be In addition or subtraction, there should be

in each number only as many significant in each number only as many significant figures as there are in the least accurately figures as there are in the least accurately known number.known number.

168.11 + 7.045 + 0.6832168.11 + 7.045 + 0.6832 Should be written as:Should be written as: 168.11 + 7.05 + 0.68168.11 + 7.05 + 0.68


Significant FiguresSignificant Figures In multiplication or division retain in each In multiplication or division retain in each

factor one more significant figure than is factor one more significant figure than is contained in the factor having the largest contained in the factor having the largest uncertainty.uncertainty.

1.26 x 1.236 x 0.68341.26 x 1.236 x 0.6834 Should be written as:Should be written as: 1.26 x 1.236 x 0.6831.26 x 1.236 x 0.683

data handling l classification of errors v systematic v random

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