CHEM2017 ANALYTICAL CHEMISTRY Mrs Billing Gate House 8th floor, GH840 Caren.Billing@wits.ac.za 011 717-6768

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STATISTICAL TESTS AND ERROR ANALYSIS

PRECISION AND ACCURACY PRECISION – Reproducibility of the result ACCURACY – Nearness to the “true” value

TESTING ACCURACY TESTING PRECISION

SYSTEMATIC / DETERMINATE ERROR Reproducible under the same conditions in the same experiment Can be detected and corrected for It is always positive or always negative To detect a systematic error: Use Standard Reference Materials Run a blank sample Use different analytical methods Participate in “round robin” experiments (different labs and people running the same analysis)

RANDOM / INDETERMINATE ERROR Uncontrolled variables in the measurement Can be positive or negative Cannot be corrected for Random errors are independent of each other Random errors can be reduced by: Better experiments (equipment, methodology, training of analyst) Large number of replicate samples Random errors show Gaussian distribution for a large number of replicates Can be described using statistical parameters

For a large number of experimental replicates the results approach an ideal smooth curve called the GAUSSIAN or NORMAL DISTRIBUTION CURVE Characterised by: The mean value – x gives the center of the distribution The standard deviation – s measures the width of the distribution

The mean or average, x  the sum of the measured values (xi) divided by the number of measurements (n) The standard deviation, s  measures how closely the data are clustered about the mean (i.e. the precision of the data) NOTE: The quantity “n-1” = degrees of freedom

Other ways of expressing the precision of the data: Variance Variance = s2 Relative standard deviation Percent RSD / coefficient of variation

POPULATION DATA For an infinite set of data, n → ∞ : x → µ and s → σ population mean population std. dev. The experiment that produces a small standard deviation is more precise . Remember, greater precision does not imply greater accuracy. Experimental results are commonly expressed in the form: mean  standard deviation

The more times you measure, the more confident you are that your average value is approaching the “true” value. The uncertainty decreases in proportion to

EXAMPLE Replicate results were obtained for the analysis of lead in blood. Calculate the mean and the standard deviation of this set of data. Replicate [Pb] / ppb 1 752 2 756 3 4 751 5 760

Replicate [Pb] / ppb 1 752 2 756 3 4 751 5 760 NB DON’T round a std dev. calc until the very end.

The first decimal place of the standard deviation is the last significant figure of the average or mean. 754  4 ppb Pb Also: Variance = s2

Motor vehicle emissions Lead plumbing Pewter Lead-based paints Lead is readily absorbed through the gastro intestinal tract. In blood, 95% of the lead is in the red blood cells and 5% in the plasma. About 70-90% of the lead assimilated goes into the bones, then liver and kidneys. Lead readily replaces calcium in bones. The symptoms of lead poisoning depend upon many factors, including the magnitude and duration of lead exposure (dose), chemical form (organic is more toxic than inorganic), the age of the individual (children and the unborn are more susceptible) and the overall state of health (Ca, Fe or Zn deficiency enhances the uptake of lead). European Community Environmental Quality Directive – 50 g/L in drinking water World Health Organisation – recommended tolerable intake of Pb per day for an adult – 430 g Pb – where from? Motor vehicle emissions Lead plumbing Pewter Lead-based paints Weathering of Pb minerals Food stuffs < 2 mg/kg Pb Next to highways 20-950 mg/kg Pb Near battery works 34-600 mg/kg Pb Metal processing sites 45-2714 mg/kg Pb

CONFIDENCE INTERVALS The confidence interval is the expression stating that the true mean, µ, is likely to lie within a certain distance from the measured mean, x. – Student’s t test The confidence interval is given by: where t is the value of student’s t taken from the table.

A ‘t’ test is used to compare sets of measurements. Usually 95% probability is good enough.

Example: The mercury content in fish samples were determined as follows: 1.80, 1.58, 1.64, 1.49 ppm Hg. Calculate the 50% and 90% confidence intervals for the mercury content. Find x = 1.63 s = 0.131 50% confidence: t = 0.765 for n-1 = 3 There is a 50% chance that the true mean lies between 1.58 and 1.68 ppm

x = 1.63 s = 0.131 90% confidence: t = 2.353 for n-1 = 3 90% 50% 1.68 1.48 1.58 1.78 90% 50% 90% confidence: t = 2.353 for n-1 = 3 There is a 90% chance that the true mean lies between 1.48 and 1.78 ppm

Confidence intervals - experimental uncertainty

APPLYING STUDENT’S T: 1) COMPARISON OF MEANS Comparison of a measured result with a ‘known’ (standard) value tcalc > ttable at 95% confidence level  results are considered to be different  the difference is significant! Statistical tests are giving only probabilities. They do not relieve us of the responsibility of interpreting our results!

2) COMPARISON OF REPLICATE MEASUREMENTS Compare two sets of data when one sample has been measured many times in each data set. For 2 sets of data with number of measurements n1 , n2 and means x1, x2 : Where Spooled = pooled std dev. from both sets of data Degrees of freedom = (n1 + n2 – 2) tcalc > ttable at 95% confidence level  difference between results is significant.

3) COMPARISON OF INDIVIDUAL DIFFERENCES Compare two sets of data when many samples have been measure only once in each data set. e.g. use two different analytical methods, A and B, to make single measurements on several different samples. Perform t test on individual differences between results: d = the average difference between methods A and B n = number of pairs of data Where tcalc > ttable at 95% confidence level  difference between results is significant.

Are the two methods used comparable? Example: (di) Are the two methods used comparable?

ttable = 2.571 for 95% confidence tcalc < ttable  difference between results is NOT significant.

F TEST COMPARISON OF TWO STANDARD DEVIATIONS Fcalc > Ftable at 95% confidence level  the std dev.’s are considered to be different  the difference is significant.

Q TEST FOR BAD DATA The range is the total spread of the data. The gap is the difference between the “bad” point and the nearest value. Example: 12.2 12.4 12.5 12.6 12.9 Gap Range If Qcalc > Qtable  discarded questionable point

Arrange in increasing order: EXAMPLE: The following replicate analyses were obtained when standardising a solution: 0.1067M, 0.1071M, 0.1066M and 0.1050M. One value appears suspect. Determine if it can be ascribed to accidental error at the 90% confidence interval. Arrange in increasing order: Q = Gap Range