Unit I
Data Analysis
Science is based on quantitative data. They are generally derived from experimental
measurements and may have some error. So it is important to study the error to get accurate
value. Thus, here we discuss some methods used by analytical chemists in assessing the
significance of experimental results.
Error
The term error is used to deal the numerical difference between a measured value and
the true value. The true value of any quantity is something we never know, although we
generally accept a value as being true when it is believed that the uncertainty in the value is
less than the uncertainty in something else with which it is being compared. The percentage
composition of a standard sample is certified by the National Institute of Standards and
Technology (NIST), which may be regarded as standard or correct value.
Thus, error may be defined as the differences between the standard values and the
results obtained by the new method are treated as error.
While discussing the term error we should learn more about their different types. Generally,
error can be described by determinate and indeterminate error. Determinate errors are
generally unidirectional with respect to the correct or true value. However, indeterminate
errors lead to both high and low results with equal probability.
Determinate errors have been classified as methodic, operative and instrumental
accordance with their origin.
Determinate errors can also be classified as constant and proportional.
Examples of sources of determinate errors are incorrectly calibrated instruments such as pH
meter, balance, burette, etc.
Indeterminate errors cannot be attributed to any known cause, but they invariably
attend measurements made by human beings. These errors cannot be corrected and hence are
the ultimate limitation of the measurement.
Accuracy and Precision
A result can be treated as accurate when the value agrees closely with the true value
of a measured quantity. A comparison is usually made on the basis of an inverse measure of
the accuracy, i.e. the error. The absolute error is the difference between the experimental
value and the true value. Suppose an analyst obtained a value of 24.24% copper in a sample
which actually contains 24.14%, the absolute error is
24.24 -24.14 = 0.10%
It can be measured in percentage or in parts per thousand. Relative error is the absolute error
divided by the true value. It is measured in percentage or parts per thousand Here the relative
error is
(0.10/24.14) x 100 = 0.41%
Or, (0.10/24.14)x 1000 = 4.14 ppt
Precision is defined as the concordance of a series of measurement of the same
quantity. It implies nothing about their relation to the true value. The term precise is
commonly stated in terms of the standard deviation, average deviation or range.
The Normal Error Curve: Gaussian Distribution Curve
When a large number of replicate readings, at least 50 numbers are taken of a
continuous variable, the results attained will usually be distributed about the mean in a
roughly symmetrical manner. The mathematical model that best satisfies such a distribution
of random error is called the normal Gaussian distribution.
Significance: Gaussian distribution curve is a bell-shaped that is symmetrical about the mean
as shown in figure 1.
This curve satisfies the equation
1/(2) e- (x - )2/22
Where, = standard deviation
= mean of total population
In Gaussian distribution about 68% of all values will fall within one standard deviation on
either side of the mean, 95% will fall within two standard deviations and 99.7% within three
standard deviations.
Fig. 1: Normal or Gaussian Distribution curve
Statistical Treatment of Finite Samples
The central tendency of a group of results is simply that value about which the individual
results tend to “cluster”. For an infinite population, it is , the mean of a such sample.
Mean
The mean of a finite number of measurements, x1, x2, x3, x4,……..,xn is often
designated 𝑥̅ to distinguish it from . Of course, 𝑥̅ approaches as a limit when the measured
value approaches infinity.
𝑥̅ = (x1 + x2 + x3 + x4 + ……..+ xn)n = ∑𝑖=𝑛
𝑖=1 𝑥𝑖 /𝑛
It may be shown that mean of n results is √𝑛 times as reliable as any of the individual results.
The mean of 4 results is twice as reliable as 1 result in measuring central tendency. The mean
of 9 results is three times as reliable, the mean of 25 results, five times as reliable etc.
Median
The median of an odd number of results is simply the middle value when the results
are listed in order. However the median of an even number of results is the average of the tow
middle ones. In a symmetrical distribution, the mean and the median are identical. The
median is a less efficient measure of central tendency than is the mean.
Range
It is the difference between the largest and smallest values. Like the median, the range
is sometimes useful in small statistics, but generally speaking it is an inefficient measure of
variability.
Average deviation
The average deviation from the mean is often given in scientific papers as a measure
of variability, although strictly it is not very significant from a statistical point of view. For a
large group of data which is normally distributed, the average deviation approaches 0.8 to
calculate the average or mean deviation, one simply finds the differences between individual
results and the mean, regardless of sign, adds these individual results, and by divides by the
number of results
Average deviation, 𝑑̅ = ∑𝑖=𝑛
𝑖=1 ⋮ 𝑥𝑖 − 𝑥̅ ⋮/n
Relative standard deviation
Often the average deviation is expressed relative to the magnitude of the measured
quantity, for example, as a percentage
Relative standard deviation (%), 𝑑̅ /𝑥̅ x 100 ={(∑𝑖=𝑛
𝑖=1 ⋮ 𝑥𝑖 − 𝑥̅ ⋮/n)/ 𝑥̅ }x100
in parts per thousand
Relative standard deviation (ppt), 𝑑̅ /𝑥̅ x 1000 ={(∑𝑖=𝑛
𝑖=1 ⋮ 𝑥𝑖 − 𝑥̅ ⋮/n)/ 𝑥̅ }x1000
Standard deviation
The standard deviation is much more meaningful statistically than is average
deviation. The symbol s is used for the standard deviation of a finite number of values. The
standard deviation, which may be thought as a root mean square deviation of values from
their average.
Standard deviation, s = √[ ∑𝑖=𝑛
𝑖=1 ⋮ 𝑥𝑖 − 𝑥̅ ⋮ / (n-1)
2
If n is large (50 or more), then it is immaterial whether the term in the denominator is n-1 or
n. When the standard deviation is expressed as a percentage of the mean, it is called the
coefficient of variance,
=(s/𝑥̅ ) x 100
Variance
It is the square of standard deviation, which is designated as s2. The variance is
fundamentally more important in statistics than is s itself.
Variance, s2 = (Standard deviation)2
Exercises
Ex. 1: An iron core gives the following results during the Fe estimation as the value 7.08,
7.21, 7.12, 7.09, 7.16, 7.14, 7.07, 7.14, 7.18, 7.11. Calculate the mean, the standard deviation
and coefficient of variance for the values.
Solution:
The mean, 𝑥̅ = (7.08 + 7.21 + 7.12 + 7.09 + 7.16 + 7.14 + 7.07 + 7.14 + 7.18 + 7.11)/10
= 71.98/10 =7.13
Standard deviation, s = √[ ∑𝑖=𝑛
𝑖=1 ⋮ 𝑥𝑖 − 𝑥̅ ⋮ / (n-1)
2
= √(7.08-7.13)2 + (7.21-7.13)2 + …………..(7.18-7.13)2 + (7.11-7.13)2/9
= √181.98 x 10-4/9
= 4.49 x 10-2 = 0.0449
Coefficient of variance, = (s/𝑥̅ ) x 100
= {0.045/7.13} x 100
= 0.63 %
Ex. 2: The normality of a solution is determined by four analysts. The results being 0.2041,
0.2049, 0.2039 and 0.2043. Calculate mean, median, range, average deviation, relative
average deviation, standard deviation and coefficient of variance.
Solution:
The mean, 𝑥̅ = (0.2041 + 0.2049 + 0.2039 + 0.2043)/4
= 0.2043
Median, M = (0.2041 +0.2043)/2
= 0.2042
Range, R = 0.2043 -0.2039
= 0.0010
Average deviation, 𝑑̅ = ∑𝑖=𝑛
𝑖=1 ⋮ 𝑥𝑖 − 𝑥̅ ⋮/n
= (0.0002) + (0.0006) + (0.0004) + (0.0000)/4
= 0.0003
Relative standard deviation (ppt), 𝑑̅ /𝑥̅ x 1000 = {(∑𝑖=𝑛
𝑖=1 ⋮ 𝑥𝑖 − 𝑥̅ ⋮/n)/ 𝑥̅ }x1000
= (0.0003/0.2043) x 1000
= 1.5 ppt
Standard deviation, s = √[ ∑𝑖=𝑛
𝑖=1 ⋮ 𝑥𝑖 − 𝑥̅ ⋮ / (n-1)
2
= 0.0004
Coefficient of variance, = (s/𝑥̅ ) x 100
= {0.0004/.2043} x 100
= 0.2 %
Analysis of results
Analysis of results can be made by two different ways. They are:
(i) Comparison of results
(ii) Reliability of results
Comparison of sets of values with either true value or with another set of values gives us
the trick to determine whether the sets of values or the analytical procedure is accurate of
precise.
There are two common methods
(a) Student’s t -test
(b) Variance ration test or F –test
t –test: Student’s t –test is used for small samples. It can also be used to test the
difference between the mean of two sets of data’s ( 𝑥̅ 1 and 𝑥̅ 2 ). The purpose of the test is
to compare the mean of samples with some standard value and to express some level of
confidence in the significance of comparison.
The t –test is obtained as
t = (𝑥̅ + ) √𝑁/s
Where, 𝑥̅ is the mean value
is true value
N is number of determination
s is standard value
These calculated values is then compared with the sets of values obtained for different
probabilities and degree of freedom from the given table
(degree of freedom: It may be defined as the number of individual observations that could be
allowed to vary under the condition that 𝑥̅ and s, once determined, be held constant.)
Confidence interval of the mean
By rearranging the above equation, we obtain the confidence interval of the mean or
confidence limits
= 𝑥̅ t.s/ √𝑁
We can use this equation to estimate the probability that the population mean, , lies within a
certain region centred at 𝑥̅ , the experimental mean of our measurements.
Exercises
Ex.3: If 𝑥̅ of twelve determination is 8.37 and a true value, =7.91, say whether or not these
results is significant if the standard deviation, s is 0.17.
Solution.
The t value is
t = [(𝑥̅ - )√𝑁]/s
= [(8.37 – 7.91)√12]/0.17
= 9.37
20% 10% 5% 2% 1%
Probability (α) 0.20 0.10 0.05 0.02 0.01
Given t value 1.363 1.796 2.20 2.72 3.71
The calculated value for t is 9.37. Comparison of this value with true value, it implies that the
calculated t value is highly significant. The t value tells us that the probability of obtaining
the difference of 0.46 is less than 1 in 100.
Ex. 4: For the estimation of iron from a sample following results are obtained:
8.43, 8.41, 8.4, 8.32, 8.34, 8.24
Find out the mean standard deviation coefficient of variance and say whether or not these
results are significant if the true value is 8.37
F-test: This test is used to compare the precision of two sets of datas. It is calculated as
F = SA2/ SB2
Where SA is standard deviation for one method
SB is standard deviation of another method
Generally, SASB
The value obtained for ‘F’ is then check for its significance against values in the F table. If
the calculated F value can relate with the lower probability then the two sets of data’s is
highly significant.
The F test may be used to determine the validity of the sample t test described here, but it
may also be interest in its own right to determine whether two analytical procedures yield
significantly different precision.
Source: Quantitative Analysis, 6th edn, R.A. Day et al.
