The document discusses analytical chemistry methods and concepts related to errors, precision, accuracy, and statistical analysis of data. It defines types of errors, describes methods to minimize errors, and explains concepts like absolute and relative error, precision, accuracy, and statistical measures including mean, median, mode, standard deviation, and t-tests and F-tests. It also provides an example of calculating average deviation and standard deviation from a set of concentration data and discusses the normal distribution curve.

1. Dr. Santanu Chakravorty
CHEMISTRY-DSE: ANALYTICAL METHODS IN
CHEMISTRY
Dr. Santanu Chakravorty
Assistant Professor
Mathabhanga College
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2. Dr. Santanu Chakravorty
Syllabus
Qualitative and quantitative aspects of analysis:
Sampling, evaluation of analytical data, errors, accuracy and precision,
methods of their expression, normal law of distribution if indeterminate
errors, statistical test of data; F, Q and t test, rejection of data, and confidence
intervals.
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Errors
• Determinate (systematic) Errors: These are errors which can be avoided or whose
magnitude can be determined. These are definite or constant errors which include
instrumental, operational, personal, reagent, additive and proportional.
• Indeterminate (random) Errors: If a particular person carried out the analysis of a
substances in the same laboratory and under similar sets of conditions, most of
the time there is slight variation in the results obtains over which the analyst has
no controls. Such errors are called indeterminate errors.
Main Differences: Determinate errors are those errors which have definite values
whereas indeterminate errors are those which possess indefinite value and follow
law and invariably have fluctuating values.
•Determinate errors are of various kinds depending upon their origin viz. methodic,
operational, instrumental etc. Indeterminate errors arise from variation of
indeterminate errors or which are erratic in occurrence.
•The determinate errors can be reduced by controlling the origin of the error but
the analyst has no control over indeterminate errors.
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Minimization of Errors
Systematic (determinate) errors can be reduced by a no of methods:
• Calibration of apparatus and application of corrections: All instruments (weights,
flasks, burettes, pipettes etc) should be calibrated and the appropriate corrections
applied to the original measurements. In some cases where an error cannot be
eliminates it is possible to apply a correction for the effect that it produces. Thus
an impurity in a weighed precipitate may be determined and its weight deducted.
• Running a blank determination: This consists of in carrying out a separate
determination, the sample being omitted, under exactly the same experimental
conditions as are employed in the actual analysis of the sample. The object is to
find out the effect of the impurities introduced through the reagents and vessels,
or to determine the excess of standard solution necessary to establish the end-
point under the conditions met with in the titration of the unknown sample
 Indeterminate errors on the other hand can be minimized by repeating the
experimental work a no of times and then taking the average of the results
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Absolute and Relative Errors
• When a quantity is measured even with extreme care, it is found that the results
of successive determinations differ among themselves. The average value is
accepted as the most probable. This may not always be the true value.
• The absolute error of a determination is the difference between the observed (or
measured) value and the true value of the quantity measured. It is a measure of
the accuracy of the measurement.
• The relative error is the absolute error divided by the true value. It is usually
expressed in terms of percentage. The true or absolute values of a quantity can
not be compared with the most probable value.
• If x is the actual value of a quantity, x0 is the measured value of the quantity and
Δx is the absolute error, then the relative error can be measured using the below
formula.
• Relative error = (x0-x)/x = (Δx)/x
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Precision
Precision is defined as the degree of agreement between a numerical value and other
values of measurement. Precision doe not by any means imply accuracy it is used to
describe the reproducibility of results. It also implies nothing about their relation to the
true values. It is the close agreement of values among a group of experimental results
carried out by different persons on the same method or by different person on different
methods on the same measurement. The results may show a high degree of precision
or good agreement with one another, yet have poor accuracy or agreement with the
true value. The precision of a measurement is evaluated from data itself, by the
agreement among separate values. Various methods are used to describe the precision
of measurements. Precision is commonly evaluates by the application of statistical
methods to deviations among measurements, like average deviation, standard
deviation , range etc. Accuracy on the other hand is the agreement between the
observed value and most probable (true) values.
• Median: The median is the average or middle value of in a set. For an odd number of
values, the middle value is the median, and for an even no values, the average of the
two middle values is the median. The median is more likely to be closer to the true
value than the mean, because a single bad result may influence the mean more than
the median.
For the odd values 18,19, 20, 21, 22, the median = 20
For the even values 18, 19, 20, 21, 22, 23 the median = 20+21/2 = 20.5
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Continued……
• Arithmetic mean or average: The mean ( x̄ ) is the sum of the individual
measurements (xi) of some quantity divided by the number of measurements (n).
The mean is calculated by the formula:
where x̄ is the ith measured value of x.
• Mode: The mode is the frequency or no of occasion upon which a given value
appears in the data. For example, for a set of observations 5, 4, 6, 6, 4, 5, 6, 6, 3,
the mode is 6 and the frequency of this value is 4.
• Deviation: The difference between any results of an analysis and the mean of a
series of results is called the deviation from the mean. The deviation is positive if
the value is greater than the mean, and negative if it is less than the mean. It is
denoted by d.
• Average deviation: It is the average of the absolute value of deviations in a series
of determinations. Its relative magnitude is indicative of the precision of a series of
determinations. The smaller the average deviation, the greater is the precision.
The average deviation is represented by the given formula:
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Continued……
• Standard deviation: When there are four or more determinations, we often use
the term standard deviation instead of average deviation. The standard deviation
carries more statistical significance than the average deviation. The standard
deviation is represented as s and is given by:
• Range: The range is the arithmetical differences between the smallest and the
largest values of a series
• Variance: The square of the standard deviation is called variance. The coefficient
of variance is an accurate measure of the precision and is given by
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Frequency Distribution: Distribution of Experimental
Data
It is a method to represent large number of finite population of organized data after
proper correction of determinate errors, the remaining fluctuations in the data are
found to be random in nature. These data may be associated with undetectable random
errors that are attributable to uncontrollable variable in the experiment. This small
random errors in most of the cases cancel one another and their effect is minimum.
However, these errors may be unidirectional and thus they produce large positive or
negative net error. The sources of random errors may be due to the visual judgment,
temperature fluctuation, time variation. We cannot identify the contribution of any one
of these sources of error in the measurement so that their combined effect is
responsible for the scatter of data around the mean.
These unrecognized data are organized to form the frequency distribution data which
consist of range from the lowest to the highest value into a convenient number of
intervals or range and then count the number of values falling within each range. In this
process there involves some loss of information of the data but this is more convenient
to represent the data in condensed form. Usually the range is divided into equal
intervals and sometimes confusion is avoided by choosing range boundaries halfway
between possible observed data.
A plot of the frequency distribution data is known as frequency distribution curve or
histogram. This is also known as bar graph or frequency polygon curve in which the
percentage of measured values are plotted against the range values. This curve also
known as Gaussian Curve.
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The normal error curve Gaussian curve
It is quite evident that there may be the infinite number of measurements
comprising an universe of data and it is actually called infinite population to which
the normal error function is related. An analyst may take finite number of replicate
measurements which is called sample. The sample may be made in a random
fashion from a hypothetical infinite population, thus it is expected that sample is
representative one and fluctuations in its individual values may be considered to
be normally distributed so that all terms associated with the normal error function
may be employed in their analysis.
Continuous data resulting from infinite population fall within a range of values
which satisfy the normal of Gaussian distribution. This a bell-shaped curve which is
symmetrical about the mean. An analyst can differentiate between a sample mean
ẋ and population mean μ.
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Statistical Evaluation of Analytical data
The statistical evaluation of analytical data is based on experimental
measurements and the judgment of these experiment data is made trough sample
mean (ẋ), population mean (μ), population standard deviation (Ϭ) etc. , which are
already discussed. In the case of replicate data we have to consider statically the
following aspect:
1. confidence interval and confidence limit of replicate data
2. establishment of confidence interval of the mean of the experimental data
3. determination of probability whether an experimental mean is different from
accepted value.
4. establishment of a given probability level whether two experimental mean are
different.
5. to establish the probability level whether the precision of two sets of
measurements differ
6. to establish the detection limits for a measurement etc.
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Confidence limit, confidence level and
confidence interval
• A confidence interval (CI) is a type of interval estimate, computed from the
statistics of the observed data, that might contain the true value of unknown
population parameter.
• Confidence level is quantities the level of confidence that the deterministic
parameter is captured by the interval. More strictly speaking, the confidence level
represents the frequency of possible confidence intervals that contain the true
value of the unknown population parameter.
• To calculate the confidence limits for a measurement variable, multiply the
standard error of the mean times the appropriate t-value. The t-value is
determined by the probability and the degree of freedom (n-1)
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t and F test
• A t-test is a form of the statistical hypothesis test, based on Student’s t-statistic
and t-distribution to find out the p-value (probability) which can be used to accept
or reject the null hypothesis.
• T-test analyses if the means of two data sets are greatly different from each other,
i.e. whether the population mean is equal to or different from the standard mean.
It can also be used to ascertain whether the regression line has a slope different
from zero. The test relies on a number of assumptions, which are:
• The population is infinite and normal.
• Population variance is unknown and estimated from the sample.
• The mean is known.
• Sample observations are random and independent.
• The sample size is small.
• Mean and standard deviation of the two samples are used to make comparison
between them, such that:
x̄1 = Mean of the first dataset, x̄2 = Mean of the second dataset
S1 = Standard deviation of the first dataset, S2 = Standard deviation of the second dataset, n1 = Size
of first dataset, n2 = Size of second dataset
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t and F test
• F-test is described as a type of hypothesis test, that is based on Snedecor f-
distribution, under the null hypothesis. The test is performed when it is not known
whether the two populations have the same variance.
• F-test can also be used to check if the data conforms to a regression model, which
is acquired through least square analysis. When there is multiple linear regression
analysis, it examines the overall validity of the model or determines whether any
of the independent variables is having a linear relationship with the dependent
variable. A number of predictions can be made through, the comparison of the
two datasets. The expression of the f-test value is in the ratio of variances of the
two observations, which is shown below:
Where, S2 = variance
• The assumptions on which f-test relies are:
• The population is normally distributed.
• Samples have been drawn randomly.
• Observations are independent.
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Key Differences Between T-test and F-test
• The difference between t-test and f-test can be drawn clearly on the following
grounds:
• A univariate hypothesis test that is applied when the standard deviation is not
known and the sample size is small is t-test. On the other hand, a statistical test,
which determines the equality of the variances of the two normal datasets, is
known as f-test.
• The t-test is based on T-statistic follows Student t-distribution, under the null
hypothesis. Conversely, the basis of the f-test is F-statistic follows Snedecor f-
distribution, under the null hypothesis.
• The t-test is used to compare the means of two populations. In contrast, f-test is
used to compare two population variances.
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Q test
In a set of replicate measurements of large number of data if one of the result may
vary excessively from the average value i.e. the value is either too high or two low
from the average, then a decision must be made whether we shall retain or reject
it. Several statistical test are performed for this. However there is no universal rule
can be applied to settle the question of retention or rejection of that particular
data. A simplified test which is widely used is called Q-test (quotient test)
described by Dean and Dixon. It involve the following steps:
(a) The results of measurements are arranged in decreasing order.
(b) The range of the results is calculated
(c) The difference between the questionable or suspected results xq and its
nearest neighbor xn is divided by the spread or range R of the entire set to obtain a
quantity Qexp as:
(d) The Qexp is then compared with the critical value of the rejection ratio Qcrit
found at appropriate confidence level
(e) If Qexp> Qcrit, the questionable or the suspected result can be discarded at the
appropriate confidence level
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Mathematical Problem
• Four sample of a silver alloy were analyzed and were found to have the following
percentages of silver Sample 1(Ag% 16.37), Sample 2 (Ag% 16.29), Sample 3 (Ag%
16.39), sample 4 (Ag% 16.35), calculate the average deviation and standard
deviation.
Average value ẋ = 16.37+ 16.29 + 16.39 +16.35/4 = 16.35%
The deviation d1 = 16.37-16.35 = 0.04
The deviation d2 = 16.29-16.35 = -0.06
The deviation d3 = 16.39-16.35 = 0.04
The deviation d4 = 16.35-16.35 = 0.00
Average deviation = d1+d2+d3+d4/4 = 0.02+0.06+0.04+0.00/4 = 0.03
Standard deviation
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= 0.0435
• The normality of a solution determined by four separate titrations results being
0.2041, 0.2049, 0.2039 and 0.2043. Calculate mean, median, range, average,
deviation, relative average deviation, standard deviation and the coefficient of
variances (do it yourself)