Introduction

Compiled by: suprapto van plaosan

Major topics covered in this chapter

  • Errors in analytical measurements
  • Gross, random and systematic errors
  • Precision, repeatability, reproducibility, bias, accuracy
  • Planning experiments
  • Using calculators and personal computers

Analytical Problems

Define The Problem

Powerpoint Kuliah

Analytical Chemistry:

  • qualitative
  • quantitative analysis.

Example,

  • ‘Does this distilled water sample contain any boron?’
  • ‘Could these two soil samples have come from the same site?’
  • ‘How much albumin is there in this sample of blood serum?’
  • ‘What is the level of lead in this sample of tap-water?’
  • ‘This steel sample contains small amounts of chromium, tungsten and manganese – how much of each?’

Errors in Quantitative Analysis

  • Analysts commonly perform several replicate determinations. Suppose we perform a titration four times and obtain values of 24.69, 24.73, 24.77 and 25.39 ml. (Note that titration values are reported to the nearest 0.01 ml)

  • All four values are different, because of the errors inherent in the measurements, and the fourth value (25.39 ml) is substantially different from the other three. So can this fourth value be safely rejected, so that (for example) the mean result is reported as 24.73 ml, the average of the other three readings? In statistical terms, is the value 25.39 ml an outlier?

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  • Another frequent problem involves the comparison of two (or more) sets of results. Suppose we measure the vanadium content of a steel sample by two separate methods. With the first method the average value obtained is 1.04%, with an estimated error of 0.07%, and with the second method the average value is 0.95%, with an error of 0.04%. Several questions then arise. Are the two average values significantly different, or are they indistinguishable within the limits of the experimental errors? Is one method significantly less error-prone than the other? Which of the mean values is actually closer to the truth?

Types of error

Experimental scientists make a fundamental distinction between three types of error:

  • gross,
  • random and
  • systematic errors.

Gross errors

are readily described: they are so serious that there is no alternative to abandoning the experiment and making a completely fresh start.

Examples: a complete instrument breakdown, accidentally dropping or discarding a crucial sample, or discovering during the course of the experiment that a supposedly pure reagent was in fact badly contaminated.

Random errors

  • Affect precision – repeatability or reproducibility
  • Cause replicate results to fall on either side of a mean value
  • Can be estimated using replicate measurements
  • Can be minimised by good technique but not eliminated
  • Caused by both humans and equipment

Systematic errors

  • Produce bias – an overall deviation of a result from the true value even when random errors are very small
  • Cause all results to be affected in one sense only, all too high or all too low
  • Cannot be detected simply by using replicate measurements
  • Can be corrected, e.g. by using standard methods and materials
  • Caused by both humans and equipment

Four analysis of exactly 10.00 ml of exactly 0.1 M sodium hydroxide is titrated with exactly 0.1 M hydrochloric acid. Each student performs five replicate titrations, with the results shown in Table 1.1.

Mean and standard deviation

Tabel 1.1

A= [10.08 10.11 10.09 10.1  10.12]
B= [ 9.88 10.14 10.02  9.8  10.21]
C= [10.19  9.79  9.69 10.05  9.78]
D= [10.04  9.98 10.02  9.97 10.04]
A B C D
0 10.08 9.88 10.19 10.04
1 10.11 10.14 9.79 9.98
2 10.09 10.02 9.69 10.02
3 10.10 9.80 10.05 9.97
4 10.12 10.21 9.78 10.04
A B C D
count 5.000000 5.000000 5.000000 5.000000
mean 10.100000 10.010000 9.900000 10.010000
std 0.015811 0.171756 0.210476 0.033166
min 10.080000 9.800000 9.690000 9.970000
25% 10.090000 9.880000 9.780000 9.980000
50% 10.100000 10.020000 9.790000 10.020000
75% 10.110000 10.140000 10.050000 10.040000
max 10.120000 10.210000 10.190000 10.040000 
Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Sequential sampling (2 chains in 1 job)
NUTS: [sigma_D, sigma_C, sigma_B, sigma_A, mean_D, mean_C, mean_B, mean_A]
Sampling chain 0, 0 divergences: 100%|██████████| 2000/2000 [00:14<00:00, 138.21it/s]
Sampling chain 1, 0 divergences: 100%|██████████| 2000/2000 [00:11<00:00, 169.04it/s]

Summary

mean sd hpd_3% hpd_97% mcse_mean mcse_sd ess_mean ess_sd ess_bulk ess_tail r_hat
mean_A 10.099 0.014 10.076 10.122 0.001 0.000 419.0 419.0 918.0 451.0 1.00
mean_B 10.015 0.133 9.749 10.265 0.004 0.003 898.0 890.0 1079.0 600.0 1.00
mean_C 9.897 0.149 9.623 10.167 0.005 0.003 943.0 943.0 990.0 913.0 1.00
mean_D 10.012 0.027 9.966 10.062 0.001 0.001 571.0 569.0 784.0 613.0 1.00
sigma_A 0.025 0.018 0.009 0.049 0.001 0.001 374.0 374.0 980.0 478.0 1.00
sigma_B 0.264 0.147 0.085 0.549 0.008 0.006 327.0 276.0 553.0 299.0 1.01
sigma_C 0.299 0.134 0.118 0.550 0.005 0.004 676.0 557.0 968.0 681.0 1.00
sigma_D 0.052 0.033 0.016 0.104 0.001 0.001 575.0 575.0 905.0 666.0 1.00

Posterior Plot of Mean

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Forest Plot of Mean

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Posterior Plot of Sigma

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Forest Plot of Sigma

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Data Precision

Repeatability describes the precision of within-run replicates.

Reproducibility describes the precision of between-run replicates.

  • The reproducibility of a method is normally expected to be poorer (i.e. with larger random errors) than its repeatability.

Student A:

  • The results are all very close to each other; all the results lie between 10.08 and 10.12 ml. In everyday terms we would say that the results are highly repeatable.
  • They are all too high: in this experiment (somewhat unusually) we know the correct answer: the result should be exactly 10.00 ml.

Evidently two entirely separate types of error have occurred.

  • First, there are random errors – these cause replicate results to differ from one another, so that the individual results fall on both sides of the average value (10.10 ml in this case). Random errors affect the precision, or repeatability, of an experiment. In the case of student A it is clear that the random errors are small, so we say that the results are precise.
  • In addition, however, there are systematic errors –these cause all the results to be in error in the same sense (in this case they are all too high). The total systematic error (in a given experiment there may be several sources of systematic error, some positive and others negative; see Chapter 2) is called the bias of the measurement. (The opposite of bias, or lack of bias, is sometimes referred to as trueness of a method: see Section 4.15.)

Student B:

  • The average of B’s five results (10.01 ml) is very close to the true value, so there is no evidence of bias, but the spread of the results is very large, indicating poor precision, i.e. substantial random errors.

Comparison of these results with those obtained by student A shows clearly that random and systematic errors can occur independently of one another.

Students C:

  • The results have poor precision (range 9.69–10.19 ml) and the average result (9.90 ml) is (negatively) biased.

Student D:

  • The results were both precise (range 9.97–10.04 ml) and unbiased (average 10.01 ml).

Source of Error

  • The tolerance of a top-quality 100 g weight can be as low as 0.25 mg, although for a weight used in routine work the tolerance would be up to four times as large. Similarly the tolerance for a grade A 250 ml standard flask is 0.12 ml: grade B glassware generally has tolerances twice as large as grade A glassware. If a weight or a piece of glassware is within the tolerance limits, but not of exactly the correct weight or volume, a systematic error will arise.
  • Weighing procedures are normally associated with very small random errors. In routine laboratory work a ‘four-place’ balance is commonly used, and the random error involved should not be greater than ca. 0.0002 g (the next chapter describes in detail the statistical terms used to express random errors). Since the quantity being weighed is normally of the order of 1 g or more, the random error, expressed as a percentage of the weight involved, is not more than 0.02%. Systematic errors in weighings can be appreciable, arising from adsorption of moisture on the surface of the weighing vessel; corroded or dust-contaminated weights;and the buoyancy effect of the atmosphere, acting to different extents on objects of different density. For the best work, weights must be calibrated against standards provided by statutory bodies and authorities (see above). This calibration can be very accurate indeed, e.g. to 0.01 mg for weights in the range 1–10 g.
  • Most of the random errors in volumetric procedures arise in the use of volumetric glassware. In filling a 250 ml standard flask to the mark, the error (i.e. the distance between the meniscus and the mark) might be about 0.03 cm in a flask neck of diameter ca. 1.5 cm. This corresponds to a volume error of about 0.05 ml – only 0.02% of the total volume of the flask. The error in reading a burette (the conventional type graduated in 0.1 ml divisions) is perhaps 0.01–0.02 ml.
  • Indicator errors can be quite substantial, perhaps larger than the random errors in a typical titrimetric analysis. For example, in the titration of 0.1 M hydrochloric acid with 0.1 M sodium hydroxide, we expect the end point to correspond to a pH of 7. In practice, however, we estimate this end point using an indicator such as methyl orange. Separate experiments show that this substance changes colour over the pH range ca. 3–4. If, therefore, the titration is performed by adding alkali to acid, the indicator will yield an apparent end point when the pH is ca. 3.5, i.e. just before the true end point. The error can be evaluated and corrected by doing a blank experiment, i.e. by determining how much alkali is required to produce the indicator colour change in the absence of the acid.

Tackling systematic errors:

  • Foresight: identifying problem areas before starting experiments.
  • Careful experimental design, e.g. use of calibration methods.
  • Checking instrument performance.
  • Use of standard reference materials and other standards.
  • Comparison with other methods for the same analytes.
  • Participation in proficiency testing schemes.