Unit 2 Evaluation Of Analytical Data I 3w3e68
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Basic Aspects
UNIT 2 EVALUATION OF ANALYTICAL DATA I Structure 2.1
Introduction Objectives
2.2
Error and Types of Errors Error Types of Errors
2.3
Detection and Minimization of Errors Calibration of Equipment and Purification of Reagents etc Use of Blanks Addition of a standard Independent Analysis Improvement in Methods
2.4
Accuracy and Precision Accuracy Precision
2.5
Reporting of Results Chemical Expression of Results Numerical Expression of Results
2.6 2.7 2.8 2.9
2.1
Significant Figures Summary Terminal Questions Answers
INTRODUCTION
A chemical analysis is usually more than a simple measurement. In a measurement the operator mainly takes care of three components: the system, some property being measured, and the instrument. Errors originate in all three components and need to be considered in a measurement. It is a real fact that no single physical measurement is perfectly accurate. The question of accuracy must, in general, receive attention both before and after an analysis. The aspect which can answer the quality assurance is the evaluation of analytical data. The purpose of this unit is to provide sufficient information to the student to enable him to examine the factors affecting the reliability of results and understand the contributions of errors, their types, their minimization for accuracy and precision of the measurement and the proper use of significant figures.
Objectives After studying this unit, you should be able to: •
understand error and types of errors in measurements,
•
know the distinction between accuracy and precision,
•
express the final result of a measurement, and
•
find the proper use of significant figures in measurement and calculations.
2.2
ERROR AND TYPES OF ERRORS
The data obtained by a physical measurement should always raise the question of errors and their nature. Various types of errors are caused in quantitative chemical
26
analysis. It is, therefore, worthwhile to for these errors. Now first understand what are error and its types.
2.2.1
Evaluation of Analytical Data I
Error
The error is an inverse measure of the accuracy of a result. Less the error, more accurate the result is. Error is mathematically defined as the difference between the observed value and the true value: E=O–T
… (2.1)
where E is the error (absolute error), O is the observed value of a measurement, and T is the true value. It is with regard to sign, and it is reported in the same units as the measurements. Let us consider, for example, for the capacity of a measuring flask whose true value, as given by standard measurements, is 250 ml. For a series of 5 measurements done by an analyst the error is represented in Table 2.1. Table 2.1: Expression of Error in Measurements in Volume of a Flask Serial Number of observations
Observed value (ml)
True value (ml)
Error (E) (ml)
1
249
250
–1
2
247
250
–3
3
250
250
0
4
248
250
–2
5
251
250
+1
Total
– 5 ml
Average
– 1 ml
The error represented in the above table is the absolute error and the average − 5 ml Total error error = = = – 1 ml Number of Observations 5 However, the absolute error is of little practical significance for a quantitative analysis. It is the relative error, that is, the error relative to the true value (E/T) expressed in suitable units, is of the practical importance as a measure of inaccuracy (or as an inverse measure of accuracy). It is convenient to express relative error in of percentage (parts per hundred), or parts per thousand (ppt), preferably. To understand the importance of relative error let us consider the measurement of the capacities of three standard flasks of 10, 100 and 1000 ml by three analysts A, B and C respectively represented as follows: Analyst
A
B
C
T=
10 ml
100 ml
1000 ml
O=
11 ml
101 ml
1001 ml
E=
1 ml
1 ml
1 ml
Relative Error = E/T
1/10 = 0.1
1/100 = 0.01
1/1000 = 0.001
% R.E. = (E×100)/T
10%
1%
0.1%
27
Basic Aspects
You see that the absolute error in all the three cases is the same (1 ml), but the comparison of the relative errors tells that the error (inaccuracy) in case of the analyst C is the least and hence his result is the most reliable out of all the three. However, for a finite measurement the true value is, usually, not known and the scatter is measured in of deviation which is the difference between the observed value and the mean of the given set of data. You will study about the deviation in detail in the next unit (Unit 3) of this block.
2.2.2 Types of Errors For a practical point of view, it is useful to classify errors into two categories: (i) determinate errors, and (ii) indeterminate errors. i)
Determinate Errors
As the name implies, determinate errors are those whose magnitude can be determined after asg a definite cause and thereby they can be corrected for. For example, weighing of a hygroscopic salt like calcium chloride. Its weight will vary according to water absorbed by it from atmosphere if it were weighed in open. The error caused due to absorption of water by the salt can be corrected for if the salt were weighed after drying and keeping in a desiccator. The determinate errors may be constant or variable. When the determinate error possessed the same value from one measurement to another under a variety of conditions, is called a constant error, for example, error due to uncalibrated weights. On the other hand, in certain cases the determinate errors may vary in magnitude with conditions, for example, the errors caused due to expansion or contraction of volumetric solutions with a change in temperature. The magnitude of the change in volume can be determined by noting the temperature. These variable determinate errors are sometimes called systematic errors. Commonly people do not use this designation of systematic error only for variable errors but frequently call both types of determinate errors (constant or variable) as systematic errors and we shall also follow the same nomenclature. Sources of Determinate Errors Determinate errors may be caused by numerous sources. It is not needed to enumerate them all but more important ones are given as follows:
28
a)
Errors due to equipment: These types of errors arise due to faulty or uncalibrated devices, for example inequality of the balance arms or insufficient accuracy of the balance, uncalibrated measurement vessels, or uncalibrated weights etc.
b)
Errors due to reagents: The quality of the reagents is very important in the quantitative analysis. Certain reagents may possess impurities that will interfere in a particular quantitative analysis. Errors are also caused by the use of incorrectly standardized solutions for titration.
c)
Personal errors: These errors are caused due to constitutional inability of an analyst to make certain observations accurately, that is, they are caused due to some natural weakness of the analyst. For example, some persons always detect the end point a little past in titration because their inability to judge colour changes exactly. Personal errors also include the so called psychological errors, due to certain bias often met within students, for example, some students often tend to choose in a burette reading the division which is closer to the previous determination or even to those found by his fellow students rather than the actual one. Obviously, this makes the results less accurate.
d)
Operational errors: The operational errors are associated with the operation of an analysis. These errors are independent of the instrument and the apparatus employed, also these errors are not related to the chemical properties of the system in hand. Their magnitude depends more upon the analyst himself than on any other factor. They are mainly caused by carelessness of the operator in a quantitative work, for example, loss in bumping of uncovered solution while heating, failure to remove precipitate quantitatively from vessels, underwashing or overwashing of precipitate, etc.
e)
Methodic errors: Sometimes a particular method for the determination of a particular constituent in the given sample may not be accurate because of improper selection of the procedure in the required range and will give the inaccurate result. For example, in the determination of iron (present in traces) in water, the gravimetric method will not give the correct result, and a method suitable for trace contents, say, a spectrophotometric method should be selected. The methodic errors are inherent in the method, and cannot be corrected unless the correct method is applied.
ii)
Indeterminate Errors
Evaluation of Analytical Data I
Indeterminate errors are the errors for those no exact cause can be assigned, hence they cannot be corrected. The sources of these errors may be similar to those for the determinate errors but no definite causes out of these can be assigned for indeterminate errors. Even after applying every correction for the possible determinate errors the replicate observations may vary. Such a variation in observations is due to the indeterminate errors. These errors follow the rules of chance or the laws of probability and are also known as Random Errors. These errors, which accompany every determination, are quite irregular and generally small. Indeterminate errors cannot be prevented or eliminated by corrections. However, they can be considerably reduced by increased care in work, and increase of the number of replicate determinations. Their pattern of occurrence can be analyzed by the techniques of statistics in order to secure a worthwhile insight into their magnitudes, frequencies of occurrence, and effects on the final expression of results (see Unit 3). A plot of the normal distribution of occurrence of indeterminate errors can be prepared.
SAQ 1 In an analysis the observed value is 5.24 g compared with the accepted (true) value of 5.28 g. What is the relative error in parts per thousand? …………………………………………………………………………………………... …………………………………………………………………………………………... …………………………………………………………………………………………... …………………………………………………………………………………………...
2.3
DETECTION AND MINIMIZATION OF ERRORS
It has been shown in the preceding section that the results are affected by various types of errors. Now, we will se how the various types of errors are detected and minimized. Correction of errors of the determinate type can be accomplished through increased care in the mechanics of analysis, application of theoretical considerations so as to eliminate inherent weakness of method, and checking of measurements under varying conditions and with different accepted techniques. However, no correction can be made in the case of indeterminate errors, but it is possible to use statistical methods to deduce the most probable value of a series of measurements and the degree of its reliability.
29
Basic Aspects
The most important thing for analyst is to give a careful thought to the selection of a method for the particular analysis keeping in mind the relative amount of analyte and the composition of the sample. The different means of minimizing errors and improving the accuracy can be discussed in the following paragraphs.
2.3.1 Calibration of Equipment and Purification of Reagents etc The determinate errors due to equipment and reagents can be minimized in a quantitative estimation by the self evident means of reducing errors as (i) calibration of weights, (ii) calibration of volumetric glasswares (iii) use of purified reagents (iv) use of samples of proper size etc. After calibrating the devices the correction should be properly applied. The equipment should be recalibrated frequently. The reagents of desired purity should be taken and purification should be performed whenever necessary. A comparison should be made with high purity marked reagents, for example, AnalaR reagents. Determinate errors are frequently caused in the use of measuring devices powered by electricity due to the variation in voltage, change in resistance by dirty electrical s and temperature effect. These errors are detectable and correctable. The use of instrument of adequate sensitivity and reproducibility is essential. Special methods of minimizing errors are applied in instrumental methods, for example, in spectrophotometry by measuring absorbance between 0.1 to 1.0 on scale and in X-rays by proper counting, etc. The size of the sample and the relative amount of the analyte is also worth to be considered. Trace determinations are almost always carried out by instrumental methods. At present, modern instruments of high sensitivity and low detection limit are being developed where many errors are automatically omitted. Sometimes, the constant errors may be decreased by variation in sample size.
2.3.2 Use of Blanks The errors caused due to certain sources as reagents, vessels, solubility of precipitate etc., can be minimized by the proper use of blanks. Of course in refined work a blank run is always preferred. A blank contains all other reagents required for a particular determination except the material to be analyzed and a blank run involves all steps of the actual analysis as of the sample. The correction is then made in the results known as blank correction. For example, in a volumetric determination the titrant reading is often corrected for the volume of the titrant required to cause an indicator (indicator blank) to change the colour. The blank run is best carried, simultaneously, along with the sample. The use of blank is not necessary in every determination. In some a blank can safely be omitted, while in others it is essential. For example, a gravimetric determination of chloride with silver nitrate hardly requires a blank, on the other hand the volumetric determination of chloride by silver nitrate using potassium chromate as indicator essentially requires the correction due to indicator blank.
2.3.3 Addition of a Standard This method involves addition of a known amount (standard) of constituent to a sample, which is then analysed for the total amount (sample + standard) of constituent present. The measurement (of sample + standard) is compared with the reading taken on the sample (alone) to be analyzed. The correct amount of analyte in the sample is determined by simple calculation. It has been proved to be very useful in a few instrumental methods (e.g., radiometric analysis, polarography, etc.). An important use of standard addition is made in isotopic dilution method. The technique is suitable where a compound can be isolated in a pure state but with only a poor yield. A known amount of the same substance (as analyte) containing an active isotope is added to the unknown (sample) and thoroughly mixed with it. The desired
30
constituent its isolated in a pure form from the mixture (active + inactive sample) and is activity is measured. The quantity of the desired substance in the sample can be determined by simple calculation. This technique has the great advantage that the standard and the unknown are measured under identical conditions.
2.3.4
Evaluation of Analytical Data I
Independent Analysis
Detection of methodic errors is rather difficult. There may be more than one way to cause these errors. In one way the determinate errors of method can be detected by analysis of a synthetic sample whose overall composition is known and closely resembles that of the analyte. In an other way, to minimize the methodic errors often analysis of the sample is done by a method of established reliability. Here the results of one method may be compared by the results of an entirely different method. If the results obtained by the two radically different methods are concordant, there is high probability of values being correct within small limits of error. In some instances the determinate error of the method may be minimized by carrying out the analysis by a method of high accuracy. For instance, for gravimetric determination of aluminium, the old procedure involves the precipitation of aluminium as hydrous aluminium oxide and ignition of precipitate at high temperature to aluminium oxide and weighed. In the newer procedure aluminium is precipitated as aluminium 8-hydroxyquinolinate, precipitate is dried at (low temperature) 130°C and weighed as hydroxyl quinolinate. The newer method requires a much lower temperature heating as compared to the older method. The high molecular weight weighing form of the quinolinate precipitate reduces the weighing error increasing the accuracy as is evident by the calculation ratios: 2 Al = 0.5291 Al 2 O 3
2.3.5
and
Al = 0.05872 Al (C 9 H 5 ON ) 3
Improvement in Methods
You can recall about the determinate errors where we considered that such errors can be minimized by proper control of conditions but requires an understanding of the effect of variables on the analytical systems. We can, therefore, make the improvements in the analytical methods in which determinate errors can be reduced in magnitude. Such work is chiefly the task of analytical chemist to modify the methods in the light of demands. Especially in applied analysis, methods are essentially needed to be modified to minimize the errors. For example, suitable separations may be adopted in the given procedure for the determination of analyte. The proper combination of methods of separation and determination can do much to minimize the errors due to interferences.
2.4
ACCURACY AND PRECISION
You have experienced that a single measurement cannot be taken as an accurate result. A single result could be in error of one kind or the other. Our confidence in an analytical result is increased by increasing the number of parallel determinations (replicate determinations). When assessing the final results it is necessary to judge (i) their accuracy, and (ii) their precision. It will be worth while to understand the meanings of these two while evaluating the analytical data.
2.4.1
Accuracy
The term accuracy is defined as the nearness of a measurement to its true value (or accepted value). It is expressed in of error. Error (defined in section 2.2) is an inverse measure of accuracy. Less the error greater is the accuracy. Thus, after knowing the relative error the loss in accuracy can be estimated.
31
Basic Aspects
There are various ways and units to express the accuracy of a measurement. The most common being either in of percent relative error or in of relative accuracy in percentage. Consider the illustration. Example 2.1 A sample was analyzed for desired constituent having 2.62 g as the true value. The results of three measurements were 2.50 g, 2.54 g, and 2.52 g. Find the error of the mean (mean error), the percent relative error and the relative accuracy of the mean of the measurements. Solution Measurement (O)
True Value (T)
Error (O-T)
2.50 g
2.62 g
– 0.12 g
2.54 g
2.62 g
– 0.08 g
2.52 g
2.62 g
– 0.10 g
Total 7.56
Total – 0.30 g
Mean 7.56/3 =2.52g
Mean Error = –0.10g
% Relative Error =
Mean Error − 0.10 ×100 = ×100 = – 3.8% True Value 2.62
Relative Accuracy (%) =
Mean 2.52 ×100 = ×100 = 96.2% True Value 2.62
Also it can be calculated from % R.E. as = 100 – 3.8 = 96.2%
2.4.2 Precision Precision is defined as the reproducibility of measurements. It tells an agreement between the numerical values of replicate measurements. The magnitude of random errors determines the precision of the analytical results. It follows that the closer the results of replicate determinations are to each other, the more precise is the analysis considered to be. Precision in a common way is expressed in of deviation. Less the deviation more precise the result is. Deviation or apparent error is defined as the difference between the measured value and the mean (average) of the series of measurements. The deviation bears a relationship to the mean value of a series similar to that which exists between the absolute error and the true value. Mathematically,
d = Xi − x
… (2.2)
where, d is the deviation, xi is the observation, and x is the mean of series of xi x + x + ...+ xn measurements, x = 1 2 = where symbol ∑ represents summation n n (add all). Deviation is, generally, taken without regard to sign.
∑
It is more informative to express the precision in of relative deviation which is deviation relative to the mean expressed in suitable units. ( xi − x) d + d2 + dn Thus, average deviation (a.d.) = 1 = n n
∑
32
and % a.d. =
a.d.
×100
x The most important measures of precision are the standard deviation and the variance. The standard deviation s of a measurement is theoretically given by: s=
d12 + d 22 + ... + d n2 n −1
Evaluation of Analytical Data I
… (2.3)
where, n is the number of observations and (n – 1) is known as the degree of freedom. Variance V is the square of the standard deviation, V = s2 =
∑d
2
… (2.4)
n −1
Distinction between Accuracy and Precision
The accuracy should not be confused with the precision. Good agreement in parallel determinations signifies that the determinations have been made under closely similar conditions, it does not guarantee the accuracy of the results. A method may be precise but may not be accurate if a large systematic error is made. On the other hand it is nearly impossible to have accuracy without good precision. The difference of the accuracy and precision can be illustrated by considering the shooting of series of bullets on the targets by three riflemen (A, B, C), shown in Figure 2.1
..... ... A Both Accurate and Precise
B Precise but not accurate
. .
.
.
.
..
C Neither precise nor accurate
Fig. 2.1: Shooting by three Riflemen.
Rifleman A has the ideal marksmanship. His all hits are in centre. His results are both accurate and precise. The shooting by rifleman B shows a good grouping of hits which indicate that the marksman is undoubtedly consistent, but in this target the grouping is centered at 3 O’clock hence cannot be considered as representing accurate shooting. His results agree well mutually means precise, but the final result (obtained as their mean value) differs somewhat from the actual value, therefore not accurate. This illustrates the effect of a constant error such as poorly adjusted sights and, since precise marksmanship is evidenced, it is reasonable to assume that if the source of error can be located and corrected, accurate shooting should be forthcoming. The shooting by rifleman C shows that the hits are spotted all over the face of the target in a display of poor reproducibility. It seems that the rifleman has no experience of shooting, his hits result only as an accident. We say that the mean value is of low reliability. Hence, these results are neither precise nor accurate. Also we see that good precision is needed for good accuracy. Of course the most favourable methods are those which give precise and at the same time accurate results. In practice results that are precise but subject to small systematic
33
Basic Aspects
error are often more useful than results with an accurate mean value but low precision, since in first case we can actually find out how much they differ from the true value, while in the second case we know nothing but that the mean value is of low reliability. In the following example you can see the difference of accuracy and precision for the results of burette reading of a titration. Example 2.2
The burette readings of titrations carried out by three students A, B and C are given below. Compare the accuracy and precision of the three students, if the true reading is 22.22 ml. Student
A
B
C
Burette readings
ml
ml
ml
22.22
22.28
22.38
22.24
22.27
22.12
22.23
22.29
22.32
22.21
22.28
22.30
22.20
22.28
22.18
22.22
22.28
22.18
Mean value
You can understand from the observed values and calculated mean values of titrations of three students that the results of student A are reproducible and the mean value resembles the true value. Hence the results of student A are both precise and accurate. A look of the titration results of student B shows that his results are reproducible but the mean value is slightly on the higher side than the true value. May be he might be taking the end point (colour change) on the higher side. Therefore, the results of student B are precise but not accurate. The readings of students C are spread in a wide range with a poor reproducibility. Hence his results are neither precise nor accurate.
SAQ 2 Define accuracy and precision. …………………………………………………………………………………………... …………………………………………………………………………………………... …………………………………………………………………………………………...
2.5
REPORTING OF RESULTS
An analytical result is reported in two parts: (i) chemical, and (ii) numerical.
2.5.1 Chemical Expression of Results As far as possible, the result of the element determined should be reported in the chemical form in which it is present in the sample analyzed. For example, in reporting the result of a determination of nitrogen, it should be reported as nitrate, nitrite or ammonia depending upon the chemical form in which the element (nitrogen) is present. The analysis of a solution of electrolytes is generally expressed in of the ions present (as Fe3+, Ca2+, Cl– , CO32–, etc.).
34
However, when the actual form is not known or some other specific purpose is to be solved, the expression may be modified. Often the purpose for the analysis decides the form in which the constituents are reported. For example, when limestone is used for the purpose of manufacture of lime, its calcium content is expressed as calcium oxide. The hardness of water is usually expressed in of calcium carbonate (although a number of ions other than calcium are present in water).
2.5.2
Evaluation of Analytical Data I
Numerical Expression of Results
In most of the analyses, it is the relative amount of the constituent in a sample that is of importance. Therefore, the numerical expression represents the amount of the desired constituent as parts of the amount of the sample in suitable units. Thus, if Wi is the amount of the constituent of interest, Ws is the amount of the sample, and C is a W factor required to express the results in suitable units, the expression i × C is useful Ws for the numerical expression of results of an analysis. For example, if Wi and Ws are in grams and C is 100, the result is given in percent (%) by weight of the constituent in the sample. If C is set equal to 1000 and Wi and Ws in same units, the answer is in parts per thousand (ppt) by weight of the constituent in the sample. And if C is set equal to 1,000,000 the answer is in parts per million (ppm) by weight of the constituent in the sample, and so on. The quantities Wi and Ws may also be expressed in volume units. Way of numerical expression of the results also depends on the physical state (solid, liquid, or gas) of the constituent and of the sample. Solids
In case of the solid sample, usually, the weight constituent of interest and that of sample are taken in the same weight units and the result is expressed as percentage by weight (weight of constituent × 100/ weight of sample) to give the number of parts of analyte in 100 parts of the sample. Liquids
The percentage in liquid samples is expressed in three ways: i)
Weight percentage: It is expressed in the same way as in solids and Weight of constituen t ×100 gives the number of parts of the desired constituent Weight of sample in 100 parts of the sample. Both weights are taken in same units.
ii)
Weight-volume percentage:
iii)
Weight of constituen t in g × 100 gives the number Volume of sample in mL of parts by weight of constituent in 100 parts by volume of the sample. The temperature should be mentioned.
Volume of constituen t × 100 gives the number of parts of Volume of sample volume of desired constituent in 100 volumes of the sample. Both the volumes should be taken in the same units and the temperature should be specified.
Volume percentage:
Gases
The composition of a gaseous mixture is usually expressed in percentage by volume, Volume of constituen t that is, × 100. Both the volumes should be taken in the same Volume of sample units and the temperature should be specified.
35
Basic Aspects
The percentage representation is very common, but it is useful mainly for major constituents. When the constituent is in traces, it is advantageous to express in parts per million by weight or volume. For further lower amounts, parts per billion or parts per trillion may also be used.
2.6
SIGNIFICANT FIGURES
In the preceding section you learnt how to report the results of measurements of an analysis. In this section you will learn about the use of correct number of significant figures. The correct number of significant figures in measurement and calculations is critical in giving the proper significance to an analysis. You know that a number is a mathematical expression of a quantity. A figure, or digit, is any one of the characters 0, 1, 2,…, 9 which, alone or in combination, serves to express a number. The digits of a number which are needed to express the precision of the measurement from which the number was derived are known as significant figures. Digits from 1 to 9 are always a part of significant figures, while 0 may or may not be a significant figure. A digit signifies the amount of the quantity in the place in which it stands. In case of the number 542, the figures signify that there are five hundreds, four tens, and two units and are therefore all significant. The character zero (0) is used in two ways, it may be used as a significant figure or it may be used merely to locate the decimal place. When zero is the part of the measurement it is significant. For example, the weight of a crucible is found to be 12.610 g. The terminal zero is significant meaning that the weight can be measured correctly upto third place of decimal. The zero after 1 is significant because this is the part of the measurement. Similarly, expressing the concentration of a copper sulphate solution as 0.1000 N, the three zeros after 1 are all significant. Consider the number 107.2 cm. The zero between 1 and 7 is significant because zero placed between two significant figures is significant. This number has four significant figures regardless of where the decimal point is placed, say 1072 mm, 10.72 dm, 1.072 m and 0.001072 km all have four significant figures, they simply represent the result in different units. In the last number 0.001072 km, the zeros before 1 are just to locate the decimal point and therefore are not significant. To write a result with some degree of certainty the correct use of significant figures must be made, which depends on various rules for computation. The student should be familiar about these (the rules have got the limited validity), i.e. i)
Observed quantities should be recorded with one uncertain figure retained. That is, there must be as many significant figures in a result or in any data as will give only one uncertain figure. Thus, in most analyses represent the last retained significant figure by ±1. For example, a value 22.6 ml represented as 22.6±1 means that this is known to be between 22.5 ml and 22.7 ml.
ii)
In rounding off quantities to desired number of significant figures by dropping the superfluous figures, increase the last retained figure by one if the following figure (which is dropped) is greater than 5. For example, the number 46.2368 rounded off to four significant figures becomes 46.24. If the dropped digit is exactly 5 (not … 51, 524, etc. which are treated as greater than 5), the last retained figure is rounded off to the nearest even digit. Thus, 3.55 is rounded off to two significant figures = 3.6 3.65 is rounded off to two significant figures also = 3.6
36
Evaluation of Analytical Data I
14.75 rounded off to one decimal place = 14.8 2.652 rounded off to one decdimal place = 2.7 If the dropped figure is less than 5 the last retained figure is not changed. Thus, 26.4332 ronded off to four significant figures = 26.43. iii)
In addition or subtraction the answer is rounded off to the significant figures in of the least significant unit. It is mainly for decimal places in the numbers and the number having the fewest decimals is thus the least significant unit. Thus, the result of sum or difference should have the number of decimals equal to the number of decimals present in the least significant unit (means the number having the fewest decimals). Although, all numbers being added or subtracted can be rounded off to the least significant unit. But again for the consistency in the answer in practice we keep an extra figure during stepwise calculations and then the final result is rounded off to one less figure. For example, summing the numbers: 26.234 + 3.223 + 143.4 + 2.2260, the third number 143.4 is the least significant unit which contains only one decimal place. Therefore, all other numbers are rounded off to two decimal places and the final result is then rounded off to one decimal place (equal to the number having least decimal places). 26.234 is rounded off to 3.223 is rounded off to 143.4 is retained as 2.2260 is rounded off to
26.23 3.22 143.4 2.23
Sum =
175.08
Finally rounded to one decimal place the sum = 175.1 Answer iv)
In multiplication or division you can retain in each factor one more significant figure than that of a factor having the least significant figures (that is significant figures contained in the least precise factor). After calculations the answer is rounded off to the number of significant figures contained in the least precise factor. For example, in the multiplication 7.0783 × 0.00305 × 6.602 the middle factor has got the least (=3) significant figures hence the values will be written as 7.078 × 0.00305 × 6.602 = 0.1425233 The answer rounded off to 3 significant figures is = 0.143
v)
If a calculation involves addition / subtraction and multiplication /division then the individual steps must be treated separately. As good practice one extra figure may be retained in the intermediate calculations and the final result is then rounded off by dropping the superfluous figures.
vi)
When a calculator or computer is used, insert all available digits in the calculation. The final result is rounded off as desired.
vii)
One extra figure may be included in the average.
viii)
In logarithm calculations as you understand a logarithm of a given number is composed of two parts (i) the characteristic which is a whole number and is indicative of the position of the decimal in the given number and hence is not a significant figure, (ii) the mantissa which is a decimal fraction and is the same regardless of the position of decimal in the given number. For example, to express properly the logarithm of 2.4 × 104, characteristic is 4 and mantissa is
37
Basic Aspects
0.3802 and the logarithm is 4.3802. The result is rounded to 4.38 since the given number (2.4 × 104) has only two significant figures. Example
Calculate the pH of a 4.0 × 10–3 M solution of hydrochloric acid. Solution
pH = – log [H+] = – log 4.0 × 10–3 The – 3 is the characteristic (from 10–3). The mantissa is 0.6010 (from the logarithm of 4.0). But the concentration is known only upto two significant figures, hence pH = – (– 3 + 0.6010) = 3 – 0.60 = 2.40 Answer.
SAQ 3 List the proper number of significant figures in the following numbers i) 0.162 ii) 10.06 iii) 200.0 iv) 0.0260 …………………………………………………………………………………………... …………………………………………………………………………………………... …………………………………………………………………………………………...
2.7
SUMMARY
In this unit you have learnt that the errors may be classified as determinate and indeterminate. The reasons for the determinate errors can be assigned and hence can be corrected for, whereas no definite reason can be assigned for indeterminate errors and they follow the rules of chance. The two : accuracy and precision are frequently used in evaluation of analytical data. The accuracy tells the nearness to the true value and the precision tells the reproducibility of replicate determinations. The smaller the determinate error in the determination, the more accurate is the result considered to be. A result may be precise even without being accurate. On the other hand a result cannot be accurate unless it is precise. Correct use of significant figures, which is dependent on the application of various computation rules, is of fundamental importance in the reporting of analytical results. The important computation rules for correct use of significant figures are discussed.
2.8
TERMINAL QUESTIONS
1.
Crystalline barium chloride was found to contain 14.70% water of crystallization as against the formula BaCl2.2H2O the true value of 14.75%. Estimate the absolute error, relative error and the relative accuracy.
2.
An analyst reported the following percentages of FeO in a sample: 16.65, 16.70, 16.68, 16.60, 16.58 and 16.63. The true value of FeO in the sample was reported to be 16.55%. Calculate the mean error and the relative error in parts per thousand.
3.
How many significant figures area there in the following: a) 0.062005
4.
c) 0.00625
d) 1.008
e) 6.111
Express the result of each of the following calculations to the proper number of significant figures. a)
38
b) 31.4
4.1374 + 2.81 + 0.603
5.
b)
4.1374 – 0.0603
c)
4.1374 – 2.81
d)
14.37 × 6.44
Evaluation of Analytical Data I
Analysis of a sample of brass gave the following percentages: 2.47 Sn, 7.60 Pb, 62.08 Cu, 0.027 Fe & 27.86 Zn. Calculate the total percentage determined. Apply the rules for correct use of significant figures.
6.
Express the result with the maximum number of significant figures. 40.36 × 0.0999 × 51.9961 346.6
7.
2.9
Express the result of each of the following calculations to the proper number of significant figures: a)
4.178 + 4.032 1.217
b)
14.276 × 0.00202 × 0.5503 2.321
(b)
(6.3194 − 4.1387) ( 204.2) 0.2148
ANSWERS
Self Assessment Questions 1.
E = 5.24 – 5.28 = - 0.04 g − 0.04 RE (ppt) = × 1000 = – 7.58 ppt 5.28
2.
The accuracy is nearness to true value and the precision tells the reproducibility of replicate determinations. A result may be precise even without being accurate. On the other hand a result cannot be accurate unless it is precise.
3.
i) 3
ii) 4
iii) 4
iv) 3
Terminal Questions 1.
Absolute error E = O – T = 14.70 – 14.75 = – 0.05% E − 0.05 Relative Error (% R.E.) = × 100 = × 100 = – 0.34% T 14.75 14.70 Relative accuracy = ×100 = 99.66% 14.75
2.
Mean of the Observation is 16.65 + 16.70 + 16.68 + 16.60 + 16.58 + 16.63 = 16.64% FeO 6 Error of the mean (Mean Error) = 16.64 – 16.55 = 0.09% FeO Error 0.09 Relative error (ppt) = × 1000 = × 1000 True Value 16.55 = 5.438 Rounded off to two decimal places = 5.44 ppt
3.
a) 5
b) 3
c) 3
d) 4
e) 4
39
Basic Aspects
4.
a)
Fewest decimals are in the second number = 2. Therefore, all other numbers can have to a maximum 3 decimal places and then the result will be rounded off to two decimal places. Thus, 4.1374 is rounded to 4.137 2.81 is retained as 2.81 0.0603 is rounded to 0.060 Total
=
7.007
Rounded off to 2 decimals = 7.01 Answer b)
4.1374 is retained as – 0.603 is retained as = 4.0771
4.1374 – 0.0603
c)
4.1374 is rounded off to 4.137 – 2.81 is retained as – 2.81 = 1.327 = 1.33 Since the number 2.81 has only 2 decimal places hence the answer is rounded off to 1.33.
d)
14.37 × 6.44 = 14.37 × 6.44 = 92.5428 = 92.5
(keeping an extra figure)
(as rounded to 3 significant figures)
5.
The least significant unit has 2 decimal places. Therefore, the sum is = 2.47 + 7.60 + 62.08 + 0.027 + 27.86 = 100.037 = 100.04%
6.
According to the rules of multiplication and division the various factors are written as 40.36 × 0.0999 × 51.996 = 0.6048643 346.6 Rounded off to 3 significant figures, the answer is = 0.605
7.
a)
The question has sum and division to be treated separately 4.178 + 4.032 = 8.210 Then, 8.210/1.217 = 6.7460969 Rounded off to 4 significant figures, the answer is = 6.746
b)
First the difference is 6.3194 – 4.1387 = 2.1807 2.1807 × 204.2 Then, = 2073.0863 0.2148 Rounded off to 4 significant figures the answer is = 2073
c)
The number 0.00202 has the least number of significant figures = 3, hence all other factors can have upto 4 significant figures and the answer should have 3 significant figures 14.28 × 0.00202 × 0.5503 = 0.0068391 2.321
Rounded of to 3 significant figures the answer is = 0.00684
40
UNIT 2 EVALUATION OF ANALYTICAL DATA I Structure 2.1
Introduction Objectives
2.2
Error and Types of Errors Error Types of Errors
2.3
Detection and Minimization of Errors Calibration of Equipment and Purification of Reagents etc Use of Blanks Addition of a standard Independent Analysis Improvement in Methods
2.4
Accuracy and Precision Accuracy Precision
2.5
Reporting of Results Chemical Expression of Results Numerical Expression of Results
2.6 2.7 2.8 2.9
2.1
Significant Figures Summary Terminal Questions Answers
INTRODUCTION
A chemical analysis is usually more than a simple measurement. In a measurement the operator mainly takes care of three components: the system, some property being measured, and the instrument. Errors originate in all three components and need to be considered in a measurement. It is a real fact that no single physical measurement is perfectly accurate. The question of accuracy must, in general, receive attention both before and after an analysis. The aspect which can answer the quality assurance is the evaluation of analytical data. The purpose of this unit is to provide sufficient information to the student to enable him to examine the factors affecting the reliability of results and understand the contributions of errors, their types, their minimization for accuracy and precision of the measurement and the proper use of significant figures.
Objectives After studying this unit, you should be able to: •
understand error and types of errors in measurements,
•
know the distinction between accuracy and precision,
•
express the final result of a measurement, and
•
find the proper use of significant figures in measurement and calculations.
2.2
ERROR AND TYPES OF ERRORS
The data obtained by a physical measurement should always raise the question of errors and their nature. Various types of errors are caused in quantitative chemical
26
analysis. It is, therefore, worthwhile to for these errors. Now first understand what are error and its types.
2.2.1
Evaluation of Analytical Data I
Error
The error is an inverse measure of the accuracy of a result. Less the error, more accurate the result is. Error is mathematically defined as the difference between the observed value and the true value: E=O–T
… (2.1)
where E is the error (absolute error), O is the observed value of a measurement, and T is the true value. It is with regard to sign, and it is reported in the same units as the measurements. Let us consider, for example, for the capacity of a measuring flask whose true value, as given by standard measurements, is 250 ml. For a series of 5 measurements done by an analyst the error is represented in Table 2.1. Table 2.1: Expression of Error in Measurements in Volume of a Flask Serial Number of observations
Observed value (ml)
True value (ml)
Error (E) (ml)
1
249
250
–1
2
247
250
–3
3
250
250
0
4
248
250
–2
5
251
250
+1
Total
– 5 ml
Average
– 1 ml
The error represented in the above table is the absolute error and the average − 5 ml Total error error = = = – 1 ml Number of Observations 5 However, the absolute error is of little practical significance for a quantitative analysis. It is the relative error, that is, the error relative to the true value (E/T) expressed in suitable units, is of the practical importance as a measure of inaccuracy (or as an inverse measure of accuracy). It is convenient to express relative error in of percentage (parts per hundred), or parts per thousand (ppt), preferably. To understand the importance of relative error let us consider the measurement of the capacities of three standard flasks of 10, 100 and 1000 ml by three analysts A, B and C respectively represented as follows: Analyst
A
B
C
T=
10 ml
100 ml
1000 ml
O=
11 ml
101 ml
1001 ml
E=
1 ml
1 ml
1 ml
Relative Error = E/T
1/10 = 0.1
1/100 = 0.01
1/1000 = 0.001
% R.E. = (E×100)/T
10%
1%
0.1%
27
Basic Aspects
You see that the absolute error in all the three cases is the same (1 ml), but the comparison of the relative errors tells that the error (inaccuracy) in case of the analyst C is the least and hence his result is the most reliable out of all the three. However, for a finite measurement the true value is, usually, not known and the scatter is measured in of deviation which is the difference between the observed value and the mean of the given set of data. You will study about the deviation in detail in the next unit (Unit 3) of this block.
2.2.2 Types of Errors For a practical point of view, it is useful to classify errors into two categories: (i) determinate errors, and (ii) indeterminate errors. i)
Determinate Errors
As the name implies, determinate errors are those whose magnitude can be determined after asg a definite cause and thereby they can be corrected for. For example, weighing of a hygroscopic salt like calcium chloride. Its weight will vary according to water absorbed by it from atmosphere if it were weighed in open. The error caused due to absorption of water by the salt can be corrected for if the salt were weighed after drying and keeping in a desiccator. The determinate errors may be constant or variable. When the determinate error possessed the same value from one measurement to another under a variety of conditions, is called a constant error, for example, error due to uncalibrated weights. On the other hand, in certain cases the determinate errors may vary in magnitude with conditions, for example, the errors caused due to expansion or contraction of volumetric solutions with a change in temperature. The magnitude of the change in volume can be determined by noting the temperature. These variable determinate errors are sometimes called systematic errors. Commonly people do not use this designation of systematic error only for variable errors but frequently call both types of determinate errors (constant or variable) as systematic errors and we shall also follow the same nomenclature. Sources of Determinate Errors Determinate errors may be caused by numerous sources. It is not needed to enumerate them all but more important ones are given as follows:
28
a)
Errors due to equipment: These types of errors arise due to faulty or uncalibrated devices, for example inequality of the balance arms or insufficient accuracy of the balance, uncalibrated measurement vessels, or uncalibrated weights etc.
b)
Errors due to reagents: The quality of the reagents is very important in the quantitative analysis. Certain reagents may possess impurities that will interfere in a particular quantitative analysis. Errors are also caused by the use of incorrectly standardized solutions for titration.
c)
Personal errors: These errors are caused due to constitutional inability of an analyst to make certain observations accurately, that is, they are caused due to some natural weakness of the analyst. For example, some persons always detect the end point a little past in titration because their inability to judge colour changes exactly. Personal errors also include the so called psychological errors, due to certain bias often met within students, for example, some students often tend to choose in a burette reading the division which is closer to the previous determination or even to those found by his fellow students rather than the actual one. Obviously, this makes the results less accurate.
d)
Operational errors: The operational errors are associated with the operation of an analysis. These errors are independent of the instrument and the apparatus employed, also these errors are not related to the chemical properties of the system in hand. Their magnitude depends more upon the analyst himself than on any other factor. They are mainly caused by carelessness of the operator in a quantitative work, for example, loss in bumping of uncovered solution while heating, failure to remove precipitate quantitatively from vessels, underwashing or overwashing of precipitate, etc.
e)
Methodic errors: Sometimes a particular method for the determination of a particular constituent in the given sample may not be accurate because of improper selection of the procedure in the required range and will give the inaccurate result. For example, in the determination of iron (present in traces) in water, the gravimetric method will not give the correct result, and a method suitable for trace contents, say, a spectrophotometric method should be selected. The methodic errors are inherent in the method, and cannot be corrected unless the correct method is applied.
ii)
Indeterminate Errors
Evaluation of Analytical Data I
Indeterminate errors are the errors for those no exact cause can be assigned, hence they cannot be corrected. The sources of these errors may be similar to those for the determinate errors but no definite causes out of these can be assigned for indeterminate errors. Even after applying every correction for the possible determinate errors the replicate observations may vary. Such a variation in observations is due to the indeterminate errors. These errors follow the rules of chance or the laws of probability and are also known as Random Errors. These errors, which accompany every determination, are quite irregular and generally small. Indeterminate errors cannot be prevented or eliminated by corrections. However, they can be considerably reduced by increased care in work, and increase of the number of replicate determinations. Their pattern of occurrence can be analyzed by the techniques of statistics in order to secure a worthwhile insight into their magnitudes, frequencies of occurrence, and effects on the final expression of results (see Unit 3). A plot of the normal distribution of occurrence of indeterminate errors can be prepared.
SAQ 1 In an analysis the observed value is 5.24 g compared with the accepted (true) value of 5.28 g. What is the relative error in parts per thousand? …………………………………………………………………………………………... …………………………………………………………………………………………... …………………………………………………………………………………………... …………………………………………………………………………………………...
2.3
DETECTION AND MINIMIZATION OF ERRORS
It has been shown in the preceding section that the results are affected by various types of errors. Now, we will se how the various types of errors are detected and minimized. Correction of errors of the determinate type can be accomplished through increased care in the mechanics of analysis, application of theoretical considerations so as to eliminate inherent weakness of method, and checking of measurements under varying conditions and with different accepted techniques. However, no correction can be made in the case of indeterminate errors, but it is possible to use statistical methods to deduce the most probable value of a series of measurements and the degree of its reliability.
29
Basic Aspects
The most important thing for analyst is to give a careful thought to the selection of a method for the particular analysis keeping in mind the relative amount of analyte and the composition of the sample. The different means of minimizing errors and improving the accuracy can be discussed in the following paragraphs.
2.3.1 Calibration of Equipment and Purification of Reagents etc The determinate errors due to equipment and reagents can be minimized in a quantitative estimation by the self evident means of reducing errors as (i) calibration of weights, (ii) calibration of volumetric glasswares (iii) use of purified reagents (iv) use of samples of proper size etc. After calibrating the devices the correction should be properly applied. The equipment should be recalibrated frequently. The reagents of desired purity should be taken and purification should be performed whenever necessary. A comparison should be made with high purity marked reagents, for example, AnalaR reagents. Determinate errors are frequently caused in the use of measuring devices powered by electricity due to the variation in voltage, change in resistance by dirty electrical s and temperature effect. These errors are detectable and correctable. The use of instrument of adequate sensitivity and reproducibility is essential. Special methods of minimizing errors are applied in instrumental methods, for example, in spectrophotometry by measuring absorbance between 0.1 to 1.0 on scale and in X-rays by proper counting, etc. The size of the sample and the relative amount of the analyte is also worth to be considered. Trace determinations are almost always carried out by instrumental methods. At present, modern instruments of high sensitivity and low detection limit are being developed where many errors are automatically omitted. Sometimes, the constant errors may be decreased by variation in sample size.
2.3.2 Use of Blanks The errors caused due to certain sources as reagents, vessels, solubility of precipitate etc., can be minimized by the proper use of blanks. Of course in refined work a blank run is always preferred. A blank contains all other reagents required for a particular determination except the material to be analyzed and a blank run involves all steps of the actual analysis as of the sample. The correction is then made in the results known as blank correction. For example, in a volumetric determination the titrant reading is often corrected for the volume of the titrant required to cause an indicator (indicator blank) to change the colour. The blank run is best carried, simultaneously, along with the sample. The use of blank is not necessary in every determination. In some a blank can safely be omitted, while in others it is essential. For example, a gravimetric determination of chloride with silver nitrate hardly requires a blank, on the other hand the volumetric determination of chloride by silver nitrate using potassium chromate as indicator essentially requires the correction due to indicator blank.
2.3.3 Addition of a Standard This method involves addition of a known amount (standard) of constituent to a sample, which is then analysed for the total amount (sample + standard) of constituent present. The measurement (of sample + standard) is compared with the reading taken on the sample (alone) to be analyzed. The correct amount of analyte in the sample is determined by simple calculation. It has been proved to be very useful in a few instrumental methods (e.g., radiometric analysis, polarography, etc.). An important use of standard addition is made in isotopic dilution method. The technique is suitable where a compound can be isolated in a pure state but with only a poor yield. A known amount of the same substance (as analyte) containing an active isotope is added to the unknown (sample) and thoroughly mixed with it. The desired
30
constituent its isolated in a pure form from the mixture (active + inactive sample) and is activity is measured. The quantity of the desired substance in the sample can be determined by simple calculation. This technique has the great advantage that the standard and the unknown are measured under identical conditions.
2.3.4
Evaluation of Analytical Data I
Independent Analysis
Detection of methodic errors is rather difficult. There may be more than one way to cause these errors. In one way the determinate errors of method can be detected by analysis of a synthetic sample whose overall composition is known and closely resembles that of the analyte. In an other way, to minimize the methodic errors often analysis of the sample is done by a method of established reliability. Here the results of one method may be compared by the results of an entirely different method. If the results obtained by the two radically different methods are concordant, there is high probability of values being correct within small limits of error. In some instances the determinate error of the method may be minimized by carrying out the analysis by a method of high accuracy. For instance, for gravimetric determination of aluminium, the old procedure involves the precipitation of aluminium as hydrous aluminium oxide and ignition of precipitate at high temperature to aluminium oxide and weighed. In the newer procedure aluminium is precipitated as aluminium 8-hydroxyquinolinate, precipitate is dried at (low temperature) 130°C and weighed as hydroxyl quinolinate. The newer method requires a much lower temperature heating as compared to the older method. The high molecular weight weighing form of the quinolinate precipitate reduces the weighing error increasing the accuracy as is evident by the calculation ratios: 2 Al = 0.5291 Al 2 O 3
2.3.5
and
Al = 0.05872 Al (C 9 H 5 ON ) 3
Improvement in Methods
You can recall about the determinate errors where we considered that such errors can be minimized by proper control of conditions but requires an understanding of the effect of variables on the analytical systems. We can, therefore, make the improvements in the analytical methods in which determinate errors can be reduced in magnitude. Such work is chiefly the task of analytical chemist to modify the methods in the light of demands. Especially in applied analysis, methods are essentially needed to be modified to minimize the errors. For example, suitable separations may be adopted in the given procedure for the determination of analyte. The proper combination of methods of separation and determination can do much to minimize the errors due to interferences.
2.4
ACCURACY AND PRECISION
You have experienced that a single measurement cannot be taken as an accurate result. A single result could be in error of one kind or the other. Our confidence in an analytical result is increased by increasing the number of parallel determinations (replicate determinations). When assessing the final results it is necessary to judge (i) their accuracy, and (ii) their precision. It will be worth while to understand the meanings of these two while evaluating the analytical data.
2.4.1
Accuracy
The term accuracy is defined as the nearness of a measurement to its true value (or accepted value). It is expressed in of error. Error (defined in section 2.2) is an inverse measure of accuracy. Less the error greater is the accuracy. Thus, after knowing the relative error the loss in accuracy can be estimated.
31
Basic Aspects
There are various ways and units to express the accuracy of a measurement. The most common being either in of percent relative error or in of relative accuracy in percentage. Consider the illustration. Example 2.1 A sample was analyzed for desired constituent having 2.62 g as the true value. The results of three measurements were 2.50 g, 2.54 g, and 2.52 g. Find the error of the mean (mean error), the percent relative error and the relative accuracy of the mean of the measurements. Solution Measurement (O)
True Value (T)
Error (O-T)
2.50 g
2.62 g
– 0.12 g
2.54 g
2.62 g
– 0.08 g
2.52 g
2.62 g
– 0.10 g
Total 7.56
Total – 0.30 g
Mean 7.56/3 =2.52g
Mean Error = –0.10g
% Relative Error =
Mean Error − 0.10 ×100 = ×100 = – 3.8% True Value 2.62
Relative Accuracy (%) =
Mean 2.52 ×100 = ×100 = 96.2% True Value 2.62
Also it can be calculated from % R.E. as = 100 – 3.8 = 96.2%
2.4.2 Precision Precision is defined as the reproducibility of measurements. It tells an agreement between the numerical values of replicate measurements. The magnitude of random errors determines the precision of the analytical results. It follows that the closer the results of replicate determinations are to each other, the more precise is the analysis considered to be. Precision in a common way is expressed in of deviation. Less the deviation more precise the result is. Deviation or apparent error is defined as the difference between the measured value and the mean (average) of the series of measurements. The deviation bears a relationship to the mean value of a series similar to that which exists between the absolute error and the true value. Mathematically,
d = Xi − x
… (2.2)
where, d is the deviation, xi is the observation, and x is the mean of series of xi x + x + ...+ xn measurements, x = 1 2 = where symbol ∑ represents summation n n (add all). Deviation is, generally, taken without regard to sign.
∑
It is more informative to express the precision in of relative deviation which is deviation relative to the mean expressed in suitable units. ( xi − x) d + d2 + dn Thus, average deviation (a.d.) = 1 = n n
∑
32
and % a.d. =
a.d.
×100
x The most important measures of precision are the standard deviation and the variance. The standard deviation s of a measurement is theoretically given by: s=
d12 + d 22 + ... + d n2 n −1
Evaluation of Analytical Data I
… (2.3)
where, n is the number of observations and (n – 1) is known as the degree of freedom. Variance V is the square of the standard deviation, V = s2 =
∑d
2
… (2.4)
n −1
Distinction between Accuracy and Precision
The accuracy should not be confused with the precision. Good agreement in parallel determinations signifies that the determinations have been made under closely similar conditions, it does not guarantee the accuracy of the results. A method may be precise but may not be accurate if a large systematic error is made. On the other hand it is nearly impossible to have accuracy without good precision. The difference of the accuracy and precision can be illustrated by considering the shooting of series of bullets on the targets by three riflemen (A, B, C), shown in Figure 2.1
..... ... A Both Accurate and Precise
B Precise but not accurate
. .
.
.
.
..
C Neither precise nor accurate
Fig. 2.1: Shooting by three Riflemen.
Rifleman A has the ideal marksmanship. His all hits are in centre. His results are both accurate and precise. The shooting by rifleman B shows a good grouping of hits which indicate that the marksman is undoubtedly consistent, but in this target the grouping is centered at 3 O’clock hence cannot be considered as representing accurate shooting. His results agree well mutually means precise, but the final result (obtained as their mean value) differs somewhat from the actual value, therefore not accurate. This illustrates the effect of a constant error such as poorly adjusted sights and, since precise marksmanship is evidenced, it is reasonable to assume that if the source of error can be located and corrected, accurate shooting should be forthcoming. The shooting by rifleman C shows that the hits are spotted all over the face of the target in a display of poor reproducibility. It seems that the rifleman has no experience of shooting, his hits result only as an accident. We say that the mean value is of low reliability. Hence, these results are neither precise nor accurate. Also we see that good precision is needed for good accuracy. Of course the most favourable methods are those which give precise and at the same time accurate results. In practice results that are precise but subject to small systematic
33
Basic Aspects
error are often more useful than results with an accurate mean value but low precision, since in first case we can actually find out how much they differ from the true value, while in the second case we know nothing but that the mean value is of low reliability. In the following example you can see the difference of accuracy and precision for the results of burette reading of a titration. Example 2.2
The burette readings of titrations carried out by three students A, B and C are given below. Compare the accuracy and precision of the three students, if the true reading is 22.22 ml. Student
A
B
C
Burette readings
ml
ml
ml
22.22
22.28
22.38
22.24
22.27
22.12
22.23
22.29
22.32
22.21
22.28
22.30
22.20
22.28
22.18
22.22
22.28
22.18
Mean value
You can understand from the observed values and calculated mean values of titrations of three students that the results of student A are reproducible and the mean value resembles the true value. Hence the results of student A are both precise and accurate. A look of the titration results of student B shows that his results are reproducible but the mean value is slightly on the higher side than the true value. May be he might be taking the end point (colour change) on the higher side. Therefore, the results of student B are precise but not accurate. The readings of students C are spread in a wide range with a poor reproducibility. Hence his results are neither precise nor accurate.
SAQ 2 Define accuracy and precision. …………………………………………………………………………………………... …………………………………………………………………………………………... …………………………………………………………………………………………...
2.5
REPORTING OF RESULTS
An analytical result is reported in two parts: (i) chemical, and (ii) numerical.
2.5.1 Chemical Expression of Results As far as possible, the result of the element determined should be reported in the chemical form in which it is present in the sample analyzed. For example, in reporting the result of a determination of nitrogen, it should be reported as nitrate, nitrite or ammonia depending upon the chemical form in which the element (nitrogen) is present. The analysis of a solution of electrolytes is generally expressed in of the ions present (as Fe3+, Ca2+, Cl– , CO32–, etc.).
34
However, when the actual form is not known or some other specific purpose is to be solved, the expression may be modified. Often the purpose for the analysis decides the form in which the constituents are reported. For example, when limestone is used for the purpose of manufacture of lime, its calcium content is expressed as calcium oxide. The hardness of water is usually expressed in of calcium carbonate (although a number of ions other than calcium are present in water).
2.5.2
Evaluation of Analytical Data I
Numerical Expression of Results
In most of the analyses, it is the relative amount of the constituent in a sample that is of importance. Therefore, the numerical expression represents the amount of the desired constituent as parts of the amount of the sample in suitable units. Thus, if Wi is the amount of the constituent of interest, Ws is the amount of the sample, and C is a W factor required to express the results in suitable units, the expression i × C is useful Ws for the numerical expression of results of an analysis. For example, if Wi and Ws are in grams and C is 100, the result is given in percent (%) by weight of the constituent in the sample. If C is set equal to 1000 and Wi and Ws in same units, the answer is in parts per thousand (ppt) by weight of the constituent in the sample. And if C is set equal to 1,000,000 the answer is in parts per million (ppm) by weight of the constituent in the sample, and so on. The quantities Wi and Ws may also be expressed in volume units. Way of numerical expression of the results also depends on the physical state (solid, liquid, or gas) of the constituent and of the sample. Solids
In case of the solid sample, usually, the weight constituent of interest and that of sample are taken in the same weight units and the result is expressed as percentage by weight (weight of constituent × 100/ weight of sample) to give the number of parts of analyte in 100 parts of the sample. Liquids
The percentage in liquid samples is expressed in three ways: i)
Weight percentage: It is expressed in the same way as in solids and Weight of constituen t ×100 gives the number of parts of the desired constituent Weight of sample in 100 parts of the sample. Both weights are taken in same units.
ii)
Weight-volume percentage:
iii)
Weight of constituen t in g × 100 gives the number Volume of sample in mL of parts by weight of constituent in 100 parts by volume of the sample. The temperature should be mentioned.
Volume of constituen t × 100 gives the number of parts of Volume of sample volume of desired constituent in 100 volumes of the sample. Both the volumes should be taken in the same units and the temperature should be specified.
Volume percentage:
Gases
The composition of a gaseous mixture is usually expressed in percentage by volume, Volume of constituen t that is, × 100. Both the volumes should be taken in the same Volume of sample units and the temperature should be specified.
35
Basic Aspects
The percentage representation is very common, but it is useful mainly for major constituents. When the constituent is in traces, it is advantageous to express in parts per million by weight or volume. For further lower amounts, parts per billion or parts per trillion may also be used.
2.6
SIGNIFICANT FIGURES
In the preceding section you learnt how to report the results of measurements of an analysis. In this section you will learn about the use of correct number of significant figures. The correct number of significant figures in measurement and calculations is critical in giving the proper significance to an analysis. You know that a number is a mathematical expression of a quantity. A figure, or digit, is any one of the characters 0, 1, 2,…, 9 which, alone or in combination, serves to express a number. The digits of a number which are needed to express the precision of the measurement from which the number was derived are known as significant figures. Digits from 1 to 9 are always a part of significant figures, while 0 may or may not be a significant figure. A digit signifies the amount of the quantity in the place in which it stands. In case of the number 542, the figures signify that there are five hundreds, four tens, and two units and are therefore all significant. The character zero (0) is used in two ways, it may be used as a significant figure or it may be used merely to locate the decimal place. When zero is the part of the measurement it is significant. For example, the weight of a crucible is found to be 12.610 g. The terminal zero is significant meaning that the weight can be measured correctly upto third place of decimal. The zero after 1 is significant because this is the part of the measurement. Similarly, expressing the concentration of a copper sulphate solution as 0.1000 N, the three zeros after 1 are all significant. Consider the number 107.2 cm. The zero between 1 and 7 is significant because zero placed between two significant figures is significant. This number has four significant figures regardless of where the decimal point is placed, say 1072 mm, 10.72 dm, 1.072 m and 0.001072 km all have four significant figures, they simply represent the result in different units. In the last number 0.001072 km, the zeros before 1 are just to locate the decimal point and therefore are not significant. To write a result with some degree of certainty the correct use of significant figures must be made, which depends on various rules for computation. The student should be familiar about these (the rules have got the limited validity), i.e. i)
Observed quantities should be recorded with one uncertain figure retained. That is, there must be as many significant figures in a result or in any data as will give only one uncertain figure. Thus, in most analyses represent the last retained significant figure by ±1. For example, a value 22.6 ml represented as 22.6±1 means that this is known to be between 22.5 ml and 22.7 ml.
ii)
In rounding off quantities to desired number of significant figures by dropping the superfluous figures, increase the last retained figure by one if the following figure (which is dropped) is greater than 5. For example, the number 46.2368 rounded off to four significant figures becomes 46.24. If the dropped digit is exactly 5 (not … 51, 524, etc. which are treated as greater than 5), the last retained figure is rounded off to the nearest even digit. Thus, 3.55 is rounded off to two significant figures = 3.6 3.65 is rounded off to two significant figures also = 3.6
36
Evaluation of Analytical Data I
14.75 rounded off to one decimal place = 14.8 2.652 rounded off to one decdimal place = 2.7 If the dropped figure is less than 5 the last retained figure is not changed. Thus, 26.4332 ronded off to four significant figures = 26.43. iii)
In addition or subtraction the answer is rounded off to the significant figures in of the least significant unit. It is mainly for decimal places in the numbers and the number having the fewest decimals is thus the least significant unit. Thus, the result of sum or difference should have the number of decimals equal to the number of decimals present in the least significant unit (means the number having the fewest decimals). Although, all numbers being added or subtracted can be rounded off to the least significant unit. But again for the consistency in the answer in practice we keep an extra figure during stepwise calculations and then the final result is rounded off to one less figure. For example, summing the numbers: 26.234 + 3.223 + 143.4 + 2.2260, the third number 143.4 is the least significant unit which contains only one decimal place. Therefore, all other numbers are rounded off to two decimal places and the final result is then rounded off to one decimal place (equal to the number having least decimal places). 26.234 is rounded off to 3.223 is rounded off to 143.4 is retained as 2.2260 is rounded off to
26.23 3.22 143.4 2.23
Sum =
175.08
Finally rounded to one decimal place the sum = 175.1 Answer iv)
In multiplication or division you can retain in each factor one more significant figure than that of a factor having the least significant figures (that is significant figures contained in the least precise factor). After calculations the answer is rounded off to the number of significant figures contained in the least precise factor. For example, in the multiplication 7.0783 × 0.00305 × 6.602 the middle factor has got the least (=3) significant figures hence the values will be written as 7.078 × 0.00305 × 6.602 = 0.1425233 The answer rounded off to 3 significant figures is = 0.143
v)
If a calculation involves addition / subtraction and multiplication /division then the individual steps must be treated separately. As good practice one extra figure may be retained in the intermediate calculations and the final result is then rounded off by dropping the superfluous figures.
vi)
When a calculator or computer is used, insert all available digits in the calculation. The final result is rounded off as desired.
vii)
One extra figure may be included in the average.
viii)
In logarithm calculations as you understand a logarithm of a given number is composed of two parts (i) the characteristic which is a whole number and is indicative of the position of the decimal in the given number and hence is not a significant figure, (ii) the mantissa which is a decimal fraction and is the same regardless of the position of decimal in the given number. For example, to express properly the logarithm of 2.4 × 104, characteristic is 4 and mantissa is
37
Basic Aspects
0.3802 and the logarithm is 4.3802. The result is rounded to 4.38 since the given number (2.4 × 104) has only two significant figures. Example
Calculate the pH of a 4.0 × 10–3 M solution of hydrochloric acid. Solution
pH = – log [H+] = – log 4.0 × 10–3 The – 3 is the characteristic (from 10–3). The mantissa is 0.6010 (from the logarithm of 4.0). But the concentration is known only upto two significant figures, hence pH = – (– 3 + 0.6010) = 3 – 0.60 = 2.40 Answer.
SAQ 3 List the proper number of significant figures in the following numbers i) 0.162 ii) 10.06 iii) 200.0 iv) 0.0260 …………………………………………………………………………………………... …………………………………………………………………………………………... …………………………………………………………………………………………...
2.7
SUMMARY
In this unit you have learnt that the errors may be classified as determinate and indeterminate. The reasons for the determinate errors can be assigned and hence can be corrected for, whereas no definite reason can be assigned for indeterminate errors and they follow the rules of chance. The two : accuracy and precision are frequently used in evaluation of analytical data. The accuracy tells the nearness to the true value and the precision tells the reproducibility of replicate determinations. The smaller the determinate error in the determination, the more accurate is the result considered to be. A result may be precise even without being accurate. On the other hand a result cannot be accurate unless it is precise. Correct use of significant figures, which is dependent on the application of various computation rules, is of fundamental importance in the reporting of analytical results. The important computation rules for correct use of significant figures are discussed.
2.8
TERMINAL QUESTIONS
1.
Crystalline barium chloride was found to contain 14.70% water of crystallization as against the formula BaCl2.2H2O the true value of 14.75%. Estimate the absolute error, relative error and the relative accuracy.
2.
An analyst reported the following percentages of FeO in a sample: 16.65, 16.70, 16.68, 16.60, 16.58 and 16.63. The true value of FeO in the sample was reported to be 16.55%. Calculate the mean error and the relative error in parts per thousand.
3.
How many significant figures area there in the following: a) 0.062005
4.
c) 0.00625
d) 1.008
e) 6.111
Express the result of each of the following calculations to the proper number of significant figures. a)
38
b) 31.4
4.1374 + 2.81 + 0.603
5.
b)
4.1374 – 0.0603
c)
4.1374 – 2.81
d)
14.37 × 6.44
Evaluation of Analytical Data I
Analysis of a sample of brass gave the following percentages: 2.47 Sn, 7.60 Pb, 62.08 Cu, 0.027 Fe & 27.86 Zn. Calculate the total percentage determined. Apply the rules for correct use of significant figures.
6.
Express the result with the maximum number of significant figures. 40.36 × 0.0999 × 51.9961 346.6
7.
2.9
Express the result of each of the following calculations to the proper number of significant figures: a)
4.178 + 4.032 1.217
b)
14.276 × 0.00202 × 0.5503 2.321
(b)
(6.3194 − 4.1387) ( 204.2) 0.2148
ANSWERS
Self Assessment Questions 1.
E = 5.24 – 5.28 = - 0.04 g − 0.04 RE (ppt) = × 1000 = – 7.58 ppt 5.28
2.
The accuracy is nearness to true value and the precision tells the reproducibility of replicate determinations. A result may be precise even without being accurate. On the other hand a result cannot be accurate unless it is precise.
3.
i) 3
ii) 4
iii) 4
iv) 3
Terminal Questions 1.
Absolute error E = O – T = 14.70 – 14.75 = – 0.05% E − 0.05 Relative Error (% R.E.) = × 100 = × 100 = – 0.34% T 14.75 14.70 Relative accuracy = ×100 = 99.66% 14.75
2.
Mean of the Observation is 16.65 + 16.70 + 16.68 + 16.60 + 16.58 + 16.63 = 16.64% FeO 6 Error of the mean (Mean Error) = 16.64 – 16.55 = 0.09% FeO Error 0.09 Relative error (ppt) = × 1000 = × 1000 True Value 16.55 = 5.438 Rounded off to two decimal places = 5.44 ppt
3.
a) 5
b) 3
c) 3
d) 4
e) 4
39
Basic Aspects
4.
a)
Fewest decimals are in the second number = 2. Therefore, all other numbers can have to a maximum 3 decimal places and then the result will be rounded off to two decimal places. Thus, 4.1374 is rounded to 4.137 2.81 is retained as 2.81 0.0603 is rounded to 0.060 Total
=
7.007
Rounded off to 2 decimals = 7.01 Answer b)
4.1374 is retained as – 0.603 is retained as = 4.0771
4.1374 – 0.0603
c)
4.1374 is rounded off to 4.137 – 2.81 is retained as – 2.81 = 1.327 = 1.33 Since the number 2.81 has only 2 decimal places hence the answer is rounded off to 1.33.
d)
14.37 × 6.44 = 14.37 × 6.44 = 92.5428 = 92.5
(keeping an extra figure)
(as rounded to 3 significant figures)
5.
The least significant unit has 2 decimal places. Therefore, the sum is = 2.47 + 7.60 + 62.08 + 0.027 + 27.86 = 100.037 = 100.04%
6.
According to the rules of multiplication and division the various factors are written as 40.36 × 0.0999 × 51.996 = 0.6048643 346.6 Rounded off to 3 significant figures, the answer is = 0.605
7.
a)
The question has sum and division to be treated separately 4.178 + 4.032 = 8.210 Then, 8.210/1.217 = 6.7460969 Rounded off to 4 significant figures, the answer is = 6.746
b)
First the difference is 6.3194 – 4.1387 = 2.1807 2.1807 × 204.2 Then, = 2073.0863 0.2148 Rounded off to 4 significant figures the answer is = 2073
c)
The number 0.00202 has the least number of significant figures = 3, hence all other factors can have upto 4 significant figures and the answer should have 3 significant figures 14.28 × 0.00202 × 0.5503 = 0.0068391 2.321
Rounded of to 3 significant figures the answer is = 0.00684
40
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