Data Handling l Classification of Errors v Systematic v Random

Data Handling l Random Error v Affects precision v Within run precision - repeatability v Between run precision - reproducibilty v Indeterminate error l Systematic Error v Affects accuracy v Proximity to the truth v Determinate error or bias.

Data Handling l Systematic Errors v These are errors which can be avoided, or whose magnitude can be determined. v Three important types of systematic error.

Data Handling l Systematic Errors v Operational and personal errors. n Factors for which the analyst is responsible. v Instrumental and reagent errors. n Uncalibrated equipment, unexpected reactions. v Errors of method. n Incorrect sampling, incompleteness of a reaction.

Data Handling l Random Errors v The slight variations which occur in successive measurements. v Due to causes over which the analyst has no control. v If a sufficiently large number of measurements are taken it can be shown that these errors lie on a Gaussian curve.

Data Handling l Minimising Errors v Calibration of apparatus v Running a blank determination v Running a control determination v Use of independent methods of analysis v Running parallel determinations v Standard addition v Internal standards v Amplification methods v Isotopic dilution

Data Handling l Propagation of errors v It is important to note that the procedure for combining random and systematic errors are completely different. v Random errors to some extent cancel one another out whereas every systematic error occurs in a definite and known sense.

Data Handling l Propagation of errors v If a and b have a systematic error of +1 then the systematic error in x given by x = a + b is +2.  If however, a and b have a random error of  the random error in x is not 

Data Handling l Propagation of errors v Two types v Linear combinations n sums and differences v Multiplicative expressions

Data Handling l Linear Combinations v In the case of the final value y calculated from the linear combination of measured quantities a, b, c, etc s y =   s a 2 + s b 2 + s c

Data Handling l Multiplicative Expressions v In the case of the final value y calculated from an expression of the type y = ab / cd

Data Handling l Multiplicative Expressions 2 v In the case of the final value y calculated from an expression of the type y = ab / cd

Data Handling l Calibration Curves v When carrying out an analysis it is often necessary to carry out a calibration procedure by using a series of samples (standards) each having a known concentration. v A calibration curve is constructed by plotting the response of the standards against the concentration.

Data Handling l Calibration Curves v There are two statistical tests which should be applied to a calibration curve. v To ascertain the linearity of the curve. v To evaluate the best straight line through the data points.

Data Handling l Correlation Coefficient v In order to establish whether there is a linear relationship between two variables x 1 and y 1 the Pearson’s correlation coefficient is used. n  x 1 y 1 -  x 1  y 1  [n  x (  x 1 ) 2 ] [n  y (  y 1 ) 2 ] r =

Data Handling l Linear Regression v Once a linear relationship has been shown to have a high probability by the value of the correlation coefficient, then the best straight line through the data points has to be estimated. v This can often be done by visual inspection but it is better to evaluate it by linear regression - the method of least squares.

Data Handling l Linear Regression v The equation of a straight line is y = ax + b where y the dependent variable is plotted as a result of changing x, the independent variable. v To obtain the regression line y on x the slope of the line “a” and the intercept on the y-axis “b” are given by the following equations.

Data Handling l Linear Regression n  x 1 y 1 -  x 1  y 1 n  x (  x 1 ) 2 a = b = y - ax

Data Handling l Calibration Curves y = x Copper Concentration (mg/l) Abs R 2 =

Data Handling l Significant Figures v There are a number of rules for computations which you need to be aware of and familiar with.

Data Handling l Significant Figures v Retain as many significant figures in a result or in any data as will only give one uncertain figure. v A volume which is known to be between 30.5ml and 30.7ml should be written as 30.6ml, but not 30.60ml as the latter would imply that the value lies between 30.59ml and 30.61ml

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

Data Handling l Significant Figures v In addition or subtraction, there should be in each number only as many significant figures as there are in the least accurately known number. v v Should be written as: v

Data Handling l Significant Figures v In multiplication or division retain in each factor one more significant figure than is contained in the factor having the largest uncertainty. v 1.26 x x v Should be written as: v 1.26 x x 0.683