1. Coefficient of Variation 2. Modal Value 3. Rank Correlation Coefficient

1. Name – Chandresh Madhyan Subject – Business Statistics
Coefficient of Variation
Coefficient of Variation (CV) is used in probability theory and statistics.
It refers to normalized measure of dispersion of a frequency
distribution or probability distribution.
It is defined as the ratio of the standard deviation to the arithmetic
mean. It was introduced by Karl Pearson and is expressed as percentage.
Coefficient of Variation = Standard Deviation/Mean
1.) It is widely used to compare the variability or homogeneity or
consistency or stability of two or more sets of data.
2.) The data which has higher value of coefficient of variation (CV) is
more dispersed and less uniform.
3.) Coefficient of variation is computed only for data measured on ratio scale
and takes non-negative values.
4.) As compared to Standard deviation, CV is better as it is independent
of unit of measurement and can be used to compare different sets of
data with different units.
In the given data, the number of workers are mentioned against the range of
daily wages.
1.) As the data depicts, maximum number of the workers earn in the
range of 50-55.
2.) There are less number of workers who fall under the lowest daily
wages and highest daily wages.
3.) In the investing world, the coefficient of variation helps to determine
how much volatility (risk) one is assuming in comparison to the
amount of return you can expect from your investment. In simple
language, the lower the ratio of standard deviation to mean, the better
your risk-return tradeoff.
4.)The arithmetic mean value of Rs. 54 which depicts that on an
average a worker earns Rs. 54 as wage.
5.)The standard deviation value of Rs. 11.02 shows the amount of
variation or dispersion from the average value. A low value in this case

2. indicates that the data points are tend to be very close to the
mean. A high value indicates that data points are spread out over a large
range of values.
6.) In this case the CV value is 20.41% which means the variability of
wages in terms of number of workers is in the range of 20 odd
percent. If the expected return in the denominator of the calculation is
negative or zero, the ratio will not make sense.
Below is the calculation of coefficient of variation. The CV is 20.41%.
Class
Intervals
(Daily
Wages)
No. of workers (f) Mid-Values (X) u=X-52.5 fu fu2
30-35 17 32.5 -20.00 -340.00 6800.00
35-40 27 37.5 -15.00 -405.00 6075.00
40-45 42 42.5 -10.00 -420.00 4200.00
45-50 61 47.5 -5.00 -305.00 1525.00
50-55 72 52.5 0.00 0.00 0.00
55-60 65 57.5 5.00 325.00 1625.00
60-65 47 62.5 10.00 470.00 4700.00
65-70 34 67.5 15.00 510.00 7650.00
70-75 22 72.5 20.00 440.00 8800.00
75-80 13 77.5 25.00 325.00 8125.00
Total 400 600 49500
X (dash) Working 52.5+(600/400)
X (dash) - Arithmetic
mean
54
Sigma Working 123.75 2.25
Sigma(square) 121.5
Sigma (Standard
Deviation)
11.02270384
Answer (Coefficient
of Variation)
20.41241452

3. Modal Value
Modal mean is the arithmetic average of modes in a data set. It refers
to the most frequently occurring result in a data set.
1.) The data provided is mapping between marks interval and number of
students in each interval. Since this is a grouped frequency
distribution, we need to first calculate the modal class.
2.) Modal class is the class in which mode of the distribution lies. If the
distribution is regular (as in given data), we can determine modal
class by inspection (i.e. 50-70) else we need to follow method of
grouping.
3.) After determining the class, we need to find the location of mode in
class. This will depend on upon the frequencies of the classes
immediately proceeding and following it.
4.) Since the frequency of class before and after modal class (50-70) is
different, the mode will not lie at middle of modal class.
5.) The position of mode will be to the right of middle point as the
frequency of class after modal class is more (i.e. 12) than the
frequency of class before it (i.e. 10). Further the mode is determined
by using the below formulae. The Modal value is 63.33
MO = Lm + (Delta1/Delta2) * h where
Lm = lower limit of modal class
fm = frequency of modal class and h = width
Delta1 = fm – f1 Delta2 = fm – f2
Marks No. of
Students
10-30 4 Modal Class =
50-70
30-50 10 Lm 50
50-70 14 Delta 1 4
70-90 12 Delta 2 2
90-110 8 h 20
110-130 6
Mo 63.3333

4. Rank Correlation Coefficient
Spearman’s Rank Correlation coefficient is a non-parametric measure of
statistical dependence between two variables (X,Y).
It refers to measure of the extent to which, one variable increases and
the other variable tends to increase, without requiring that increase
to be represented by linear relationship. It is crude method of computing
the relationship between two variables which can be described using a
monotonic function.
It is appropriate for both continuous and discrete variables. The
formula for calculation of Rank correlation for a sample size n and scores
(Xi, Yi) when converted to ranks xi, yi is -
Where di = xi - yi
1.) If there are no repeated values in data, a perfect Spearman rank
correlation coefficient (rho) of +1 or −1 occurs, which means each
of the variables is a perfect monotone function of the other.
2.) It is used when quantitative measurements of the characteristics
are not possible. Example – Drawing competition
3.) In the given data, marks of 8 students are plotted for Computer
Graphics (X) and Probability & Statistics (Y). The Spearman’s rank
correlation is calculated by using the above formulae.
4.) Since there are repeated ranks, mean is assigned to those ranks. After
doing calculations as per above formulae, rank correlation value comes to
0.030.
5.) This low value shows that the correlation between CG and P&S marks
is very low. Below is the working of Rank Correlation Coefficient.

5. Calculation of Spearman’s Rank Correlation Coefficient
CG(Xi) PS(Yi) rank xi rank yi di= yi - xi di
2
15 40 2 6 4.00 16.00
20 30 3.5 4 0.50 0.25
28 50 5 7 2.00 4.00
12 30 1 4 3.00 9.00
40 20 6 2 -4.00 16.00
60 10 7 1 -6.00 36.00
20 30 3.5 4 0.50 0.25
80 60 8 8 0.00 0.00
Total 81.50
n 8.00
Edi
2
81.50
n(n2
-1) 504.00
Numerator (6*Edi
2
) 489.00
Denominator 504.00
Answer (Rank
Correlation)
1 -(Numerator/
Denominator)
0.030