Analytical Chemistry

1. STATISTICAL EVALUATION OF
ANALYTICAL DATA

2. ERRORS
• In quantitative analysis, the end result depends on the various data
measurements observed in an expt.
• The final result obtained might be wrong if there are any errors in the
data measurements taken.
• These errors arise due to the instruments or apparatus used or due to
the wrong observations made.
• The different kind of errors are

3. 1. Systematic or determinate errors
These are errors which can affect the results in a series of determinations
with a particular sample or with a particular observer or with a particular
method.
• Different types of determinate errors
• A) Personal errors
• This type of errors are due to the inability of the analyst to carry out the
measurements in a systematic way. It is not due to the inadequacy of the
method, procedure or the instruments used.
• Egs. 1. Errors in reading the burette
• 2. Errors in pipetting solutions
• 3. Mechanical loss of materials in the different steps of the expt like
pipetting, buretting, weighing ------

4. • 4. Insufficient heating of the crucibles with ppt or without ppt
• 5. Insufficient cooling of crucibles before weighing
• 6. Lack of care in handling very pure chemicals and materials
• 7. Ignition of ppts at incorrect temp
• 8. Allowing hygroscopic substances to absorb moisture before or
during weighing
• 9. Errors in every step of calculations

5. • B) Operational errors
• These errors are mostly physical in nature and occur when very good
analytical technique are not followed.
• Eg. Using an ordinary burette 0.01ml of a liquid can not be added, 0.1ml is
the least amount which can be added.
• C) Instrumental and reagent errors
• These errors arise due to many factors like
• 1. From faulty instruments like balances, potentiometer, conductometer,
colourimeter, -----
• 2. Use of improperly calibrated weights, glass wares and instruments
• 3. Use of impure reagents

6. • D) Methodical errors
• These errors originate from incorrect sampling and incompleteness
of a reaction.
• These errors are difficult to detect.
• Egs. Wrong standardisation of a pH meter, background absorption in
the atomic absorption spectroscopy, ----
• In the gravimetric analysis errors may arise due to co-precipitation,
decomposition or volatilisation of the ppt, weighing imperfectly
cooled substances, improper washing of the ppt, ----

7. • In titrimetric analysis errors may occur due to the failure of reactions
to completion, occurrence of induced or side reactions, reaction of
substances other than the constituent to be determined, difference
between the observed end point and the stochiometric equivalence
point -----
• In redox titrations errors may be due to redox potential, rate of
establishment of equilibrium, pH, presence of certain catalysts, non-
reversibility of redox indicators ---
• In precipitation titrations the main cause of error is the solubility of
the precipitate.

8. • The exes indicator used also can induce an error.
• Eg. In the ‘Mohr’s method’ an exes of K2Cr2O7 forms Ag2CrO4 before
the equivalence point has been reached. ( xxxxxxxxxx )
•

9. • E) Additive and proportional errors
• The absolute value of an additive error is independent of the
constituent present in the determination.
• Eg. Loss of weight of a crucible in which a ppt is ignited does not
depend on the weight of the ppt present.
• Proportional error is dependent of the constituent present and it
increases directly with the quantity of the constituent present.
• Eg. When Na2SO4 is coprecipitated with BaSO4 in an expt, its amount
increases with the amount of BaSO4 in direct ratio.

10. 2. Indeterminate or accidental errors
• These are errors due to
• 1. Inherent limitations of the instrument Eg. Using an ordinary
balance only two decimal positions of the weight can be noted.
• 2. Limitations in making the observations Eg. Lower meniscus level of
a liquid is not exactly adjusted to the mark.
• 3. Lack of care in making measurements Eg. Not stopping the
burette at the exact end point.
• These errors can be minimised by using high precision instruments
and by careful work. Random or indeterminate errors can not be
completely eliminated.

11. Minimisation of errors
• Many of these errors can be avoided or reduced by the following
methods
• 1. Calibration of the apparatus and application of corrections
• All measuring apparatus like burettes, pipettes, standard flasks,
measuring flasks, weights, ----- should be calibrated and appropriate
corrections applied to the original measurements.
• In the cases where an error can not be eliminated, it is possible to
apply a correction so that the error can be minimised.
• Eg. A reduction in the wt can be applied for an impurity present along
with a ppt in a gravimetric expt.

12. • 2. Running a blank determination
• This consists of carrying out a separate determination, except the
sample, under exactly the same exptl conditions as employed in the
actual analysis of the sample.
• The objective is to find out the effect of the impurities introduced
through the reagents, apparatus and vessels and to determine the
exes of standard solution necessary to establish the correct end point.
• A large blank correction is undesirable since the exact value then
becomes uncertain and the precision of the analysis may be reduced.

13. • 3. Performing analysis by independent methods
• In some cases the accuracy of a result can be established by carrying
out the analysis by an entirely different expt for the same sample.
• Eg. Iron metal in a sample can be determined gravimetrically by
precipitation as iron(III)hydroxide and then ignited to iron(III)oxide.
• This can be verified by doing a titrimetric method by reducing the
iron(III) to iron (II) state and then titrated against a standard solution
of K2Cr2O7 or cerium(IV)sulphate.

14. • 4. Testing any method by control analysis
• 5. Using internal standards
• 6. By a standard addition
• 7. By an isotopic dilution
• 8. By amplification methods

15. Expression of Error
• It is the numerical difference between a measured value and the true value
in an analytical determination. Egs --------
• The true value is also called accepted value.
• The error can be expressed as
• 1. Absolute error ( E ) :- It is the difference between the measured value
and the true value.
• E = Xi – Xt ; where Xi is the measured value and Xt is the true value
• 2. Relative error ( Er ) :- It is the ratio of the absolute error to the true
value. Er = (Xi – Xt ) / Xt
• Relative error is usually expressed as ‘per cent’ or ‘parts per thousand’
( ppt )
• Er = (Xi – Xt ) x 100 / Xt OR Er = (Xi – Xt ) x 1000 / Xt

16. • P-1. In two separate determinations, the concentration of Iron in a
given sample was found to be 20.17 ppm & 19.80 ppm. Given that
the std value is 20.00 ppm, calculate absolute error and relative error.
• A) Absolute error = Xi – Xt = 20.17 – 20.00 = 0.17 ppm
• Relative error = (Xi – Xt) x 1000 / Xt = 0.17 x 1000 / 20 = 8.50 ppt
• B) Absolute error = Xi – Xt = 19.80 – 20.00 = - 0.20 ppm
• Relative error = (Xi – Xt) x 1000 / Xt = - 0.20 x 1000 / 20 = -10.00
ppt

17. Precision and Accuracy of measurements in
quantitative methods
• These two terms refers to the two types of errors associated with
measurements.
• To ensure the correctness of the data, an analyst record
measurements in many trials.
• If the data obtained in the different trials are equal or closely agreeing
with each other, the measurement is said to be very ‘precise’
• ie ‘Precision’ means how closely the data obtained in different trials
agree with one another.

18. • When the data obtained by precise measurements agree with the
theoretical data, then it is said to be accurate.
• ie ‘Accuracy’ means the closeness of the measured average value
with the correct theoretical value.
• All precise data need not be accurate due to some systematic errors.
• But all accurate values are generally precise.
• Accuracy can be expressed in terms of absolute and relative error.

19. Mean, median and standard deviation
• The ‘mean’ is the numerical value obtained by dividing the sum of a
set of measurements by the number of individual results in the set.
• It is also known as ‘arithmetic mean’ or ‘average’
• Eg. Find the mean of the values 46.62, 46.47, 46.64 & 46.76
• Mean = (46.62 + 46.47+ 46.64 +46.76) / 4 = 46.62
• Median is a value about which all other values are equally distributed.
• The mean and median may not be the same.

20. Average deviation from the mean
• The precision of experimental data can be expressed conveniently in
terms of average deviation from the mean. It can be written as
• AD = ε (Xi – X ) / N ; i values between 1 to N ; where Xi is the
individual measurement value, X is the mean measurement value and
N is the total number of measurements.
• Average deviation is expressed relative to the magnitude of the
measured quantity. It is then termed as ‘relative average deviation
from the mean’
• Relative AD (%) = ε (Xi –X ) x 100 / X. N ; i values between 1 to N
• Relative AD (ppt) = ε (Xi –X ) x1000 /X. N ; i values between 1 to N

21. Standard deviation ( S )
• Also called root mean square deviation.
• It is obtained by the summation of the squares of the individual
deviations from the mean, dividing the sum by ( N -1 ) and then
taking the square root.
• S = [ ε (Xi – X )2 / N-1 ]½ ; i values between 1 to N
•

22. • P-2. In a titration using 10.0 ml of a solution, the titre volumes obtained in
six trials are 9.98, 9.99, 9.98, 9.95, 10.0 & 10.02 ml. Calculate the standard
deviation.
• Mean of the six measurements, X = ε Xi /N ; where i = 1 to 6
= 9.98 +9.99 + 9.98 + 9.95 + 10.0 + 10.02 / 6 = 9.99
• Trial no 1 2 3 4 5 6
• Xi 9.98 9.99 9.98 9.95 10.0 10.02
• Xi – X 0.01 0.00 0.01 0.04 0.01 0.03
• (Xi – X)2 0.0001 0.0000 0.0001 0.0016 0.0001 0.0009
• ჻ ε (Xi – X)2 = 0.0028
• ჻ S = √ ε (Xi – X )2 / N-1 = √ 0.0028/5 = 0.02367

23. Significant figures
• Significant figures in a numerical expression are defined as all those digits whose values
are known with certainty with one additional digit whose value is uncertain.
• These are digits in the data whose values are important for the data.
• The digit zero is significant figure except when it is the first fig in a number.
• Eg1. The mass of a substance is 0.0035 kg
• Here the zeros are not significant figures. So significant fig are only two.
• Eg2. The mass of a substance 2.03765 g
• Only the first four fig are meaningful. The last digit known with certainty is ‘7’. The digit
‘6’ is uncertain and indicates that the mass is more than 2.037 g. the last digit ‘5’ is
meaning less.
• Eg3. 2500 0.004 5000.0 0.02765 4.20 x 1010
sig fig 4 1 5 4 3