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Measures of Central Tendency
Central value of a set of observation is known as measure of central tendency or measures of
location. The measure of central tendency is broadly classified in to five categories, that are
- Arithmetic Mean, 2. Median, 3. Mode, 4. Geometric Mean and 5. Harmonic Mean
Arithmetic Mean
The arithmetic mean is the average of a group of numbers and is computed by summing all
numbers and dividing by the number of numbers.
The formulas for computing the mean of n observation and the mean for the data having
frequency are given as follow
Sample mean 𝑥̅ =
∑𝑥 𝑖
𝑛
𝑥 1
+𝑥 2
+⋯+𝑥 𝑛
𝑛
Mean for data having frequency distribution is given by
𝑖
𝑖
𝑖
Example:
a) Find the arithmetic mean of the following frequ8ency distribution
X: 1 2 3 4 5 6 7
f: 5 9 12 17 14 10 6
b) Calculate the arithmetic mean from the following data
x: 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60
f: 12 18 27 20 17 6
Solution
Properties of Arithmetic Mean:
I. Algebraic sum of the deviations of a set of values from their arithmetic mean is zero.
II. The sum of the squares of the deviations of a set of values is minimum when taken about
mean.
III. If 𝑥
𝑖
are the mean of k component series of size n i
respectively then the mean of component
series obtained on combining the component series is given by the formula
1
1
2
2
𝑘
𝑘
1
2
𝑘
Median :
Median is the middle most or central value of the observation made on a variable when the
values are arranged in ascending or descending order.
In case of discreate frequency distribution median is obtained by considering the cumulative
frequencies. The steps of calculating median are
i. Find N/
ii. See the cumulative frequency just greater than N/
iii. The corresponding value is the median
In case of grouped or continues frequency distribution the formula for the median is
Median = l + (h/f) *{(N/2) - C}
Where, l =lower limit, f = frequency, h = magnitude of the median class, c = cumulative frequency
preceding the median class
Example:
Obtain the median from the following frequency distribution
Wages (in RS.): 2000 - 3000 3000 - 4000 4000 - 5000 5000 - 6000 6000 - 7000
No. of Workers: 3 5 20 10 5
Measure of Dispersion
Literal meaning of dispersion is “scatteredness” of data. Dispersion is the measure of extent
to which individual items vary. This is divided into four categories, that are
I. Range:
The rage is the difference between two extreme values of an observation. It is given by
Range = H – L, where H is the largest and L is the lowest observation
II. Quartile deviation
Quartile or semi-inter quartile range Q is given as
Q = (Q
3
– Q
1
) / 2, Where Q 3
and Q 1
are 3rd and 1
st
quartiles respectively.
It is based on 50% of the data.
III. Mean Deviation
Mean deviation from the average A (usually mean, median or mode) is given
by
M.D. form average A =
1
𝑁
𝑖
𝑖
Science it is based on all observations it is a better measure of dispersion than range
and quartile deviation but it is useless for further mathematical treatment because it
artificially makes positive the sign of the deviation.
IV. Standard deviation
Standard deviation is the positive square root of the arithmetic mean of the
squares of the deviations of the given values from their arithmetic mean.
It is symbolised in the Greek letter σ and formulated as
σ = √
1
𝑁
Σ𝑓
𝑖
(𝑥
𝑖
− 𝑥̅ )
2
Square of standard deviation is known as variance. It is given by
σ
2
1
𝑁
𝑖
𝑖
2
Example:
Calculate the mean and standard deviation for the following table given the age distribution pf 542
members.
Age (in years): 20 - 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90
No. of members: 3 61 132 153 140 51 2
CORRELATION
Correlation implies mutual relationship between two or more variables. If two variables
deviate in the same direction, i.e., If the increase (or decrease) in the one result in a corresponding
increase (or decrease) in the other, correlation is said to be direct or positive. If they constantly deviate
in opposite direction, i.e., if the increase (or decrease) in the one result in a corresponding decrease
(or increase) in the other, corelation is called indirect or negative.
As a measure of intensity or degree of linear relationship between two variables, Karl Pearson
a British Biometrician, developed a formula called corelation coefficient which is given by
r(x,y) =
𝐶𝑜𝑣(𝑥,𝑦)
𝜎
𝑥
𝜎
𝑦
=
1
𝑛
Σ(𝑥
𝑖
−𝑥̅ )(𝑦
𝑖
−𝑦̅ )
√
1
𝑛
𝛴(𝑥
𝑖
−𝑥̅ )
2
√
1
𝑛
𝛴(𝑦
𝑖
−𝑦̅ )
2
=
1
𝑛
Σ𝑥
𝑖
𝑦
𝑖
−𝑥̅ 𝑦̅
√
1
𝑛
𝛴𝑥
𝑖
2
−𝑥̅
2
√
1
𝑛
𝛴𝑦
𝑖
2
−𝑦̅
2
Rank Correlation
In case of qualitative data rank correlation coefficient is used. It is also known as Spearman’s
rank correlation coefficient and it is denoted by r s
or ρ which is given by
6 Σ𝑑
𝑖
2
𝑛(𝑛
2
− 1 )
Where, Σ𝑑
𝑖
2
𝑖
𝑖
2
Example:
The ranks of same 16 students in mathematics and physics are follows. Two numbers within brackets
denote the ranks of the students in mathematics and physics :
Calculate the rank correlation coefficient for proficiencies of this group in mathematics and physics.
Solution:
Example:
Ten competitors in a musical competition were ranked by three judges A, B and C in the following
order:
Ranks of A: 1 6 5 10 3 2 4 9 7 8
Ranks of B: 3 5 8 4 7 10 2 1 6 9
Ranks of C: 6 4 9 8 1 2 3 10 5 7
Using the rank correlation coefficient method, discuss which pair of judges has the nearest approach
to commonages in music:
REGRESSION
Regression analysis is a mathematical measure of the average relationship between two or
more variables in terms of the original units of the data. In regression analysis there two types of
variables. The variable whose value is influenced or is to be predicted is called dependent variable,
also known as regressed or explained variable. The variable which influences the value or which is
used for prediction is called independent variable, also known as regressed or explained variable.
Regression equation:
Regression equation is an algebraic method. It is an algebraic expression of regression line X on Y and
Yon X. it can be solve with help of method of least square.
The regression equation X on Y is given by
Example:
Determine the equation of straight line which best fits the data.
X: 10 12 13 16 17 20 25
Y: 10 22 24 27 29 33 37
Min Z = 3x 1 + 2x 2
Subject to the constraints
5x 1
+ x 2
2x 1
+ 2x 2
x 1
+ 4x 2
x 1
, x 2
Graphical Solution
The major steps in solution of a linear programming problem by graphical method are
- Identify the problem
- Set up the mathematical formulation of the problem,
- Plot a graph representing all the constraints of the problem and identify the feasible region.
The feasible region is the intersection of all regions represented by the constraints of the
problem and is restricted to the first quadrant.
- The feasible region obtain in step 3 may be bounded or unbounded. Compute the coordinates
of all the corner points of the feasible region.
- Find out the value of the objective function at each corner point determined in step 4.
- Select the corner point that optimizes the value of the objective function. It gives the optimum
feasible solution.
Example:
Solve the following LPP by graphical method
Minimize Z = 20 x 1
Subject to the Constraints
x 1
3x 1
4x 1 + 3x 2 ≥ 60
x 1
, x 2
Two-Phase Method
Two- phase method is another method to solve LPP involving artificial variables. Here solution
is obtained in two-phase. The steps are
Phase – 1
- Assign a cost - 1 to each artificial variables and get a new objective function Z
= - A
1
- A
2
where A i
’ s are artificial variables.
- Write down the auxiliary LPP in LPP in which the new objective function is to be
maximized, subject to the given set of constraints.
- Solve the auxiliary LPP by simplex method until either of the following three cases arise
i. Max Z
<0 and at least one artificial variable appears in the optimum basis at
positive level.
ii. Max Z
= 0 and at least one artificial variable appears in the optimum basis at zero
level.
iii. Max Z
= 0 and no artificial variable appears in the optimum.
Phase- 2
Consider the optimum basic feasible solution of phase 1 as a starting basic feasible solution
for the original LPP. Assign actual coefficients to the variables in the objective function and a value
zero to the artificial variables that appear at zero value in the final simplex table of phase 1.
Example:
`Use two-phase simplex method to maximize z = 5x 1
Subject to the constraints:
2x 1
6x 1
+5x 2
8x 1
+6x 3
x 1
, x 2
, x 3
Big- M (Penalty) Method:
In this method a very high penalty M is assigned to the artificial variable in the objective
function. The iterative procedure for the algorithm is given follows:
- Write the given LPP into its standard form and check whether there exists a starting basic
feasible solution.
a) If there is ready starting BFS then move to step 3.
b) If there does not exist a ready BFS the move to step 2.
- Add artificial variable to the left side of each equation that has no obvious starting basic
variables. Assign a very high penalty M to these variables in the objective function.
- Apply simplex method to the modified LPP. following case may arise at the last iteration:
a) At least one artificial variable is present in the basis with zero values. In such a case the
current optimum basic feasible solution is degenerate.
b) At least one artificial variable is present in the basis with a positive value. In such a case,
the given LPP dose not posses an optimum basic feasible solution the given problem is
said to have a pseudo-optimum BFS.
Example:
Use penalty (or Big – M) method to
Maximize z = 6x 1
Subject to the constraints:
2x 1
3x 1
X
1
x 1
, x 2
