Paper 4: Fundamentals of Business Mathematics & Statistic

5.2 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Measures of Central Tendency and Measures of Dispersion

Some important result :

(i) ( )

n n n

∑j 1= xj+yj =∑ ∑j 1=xj+j 1= y 1

(ii)

n

j 1

A A A .... A nA (A is constant)

= [n times]

= + + + =

∑

(iii)

n

∑j 1= Axj=Ax 1 +Ax 2 +... Ax+ n

5.1.1. MEAN :

There are three types of mean :

(i) Arithmetic Mean (A.M.) (ii) Geometric Mean (G. M.) (iii) Harmonic Mean (H.M.)

Of these the Arithmetic mean is the most commonly used. In fact, if not specifically mentioned by mean we

shall always refer to arithmetic Mean (AM) and calculate accordingly.

1. Arithmetic Mean :
   (i) Simple Arithmetic mean : (Calculating mean from ungrouped data)

The simple arithmetic mean (x) of a given series of values, say, x 1 , x 2 ,........ x n is defined as the sum of these

values divided by their total number : thus

( )

n

x 1 x 2 .... xn j 1xj x

x xbar n =n n

+ + +

= = =

∑ ∑

Note. Often we do not write xj , x means summation over all the observations.

Example 1 : Find the arithmetic mean of 3,6,24 and 48.

Required A.M.=^3 + +6 24 48 81 4 + = 4 =20.25

(ii) Weighted Arithmetic Mean : (Calculating the mean from grouped data)

If the number x 1 , x 2 , ....... x n occur f 1 , f 2 .......f n times respectively (i.e. occur with frequencies f 1 , f 2 ........f n)

the arithmetic mean is

n

j=1 jj

n 1 1 2 2 n n

j=1j^12 n

x f f x + f x ..... + f x f x f x

x = = = f = N

f f + f + +f

(^)
L^
Where N = f is the total frequency, i.e., total number of cases. This mean x is called the weighted
Arithmetic mean, with weights f 1 , f 2 .......fn respectively.
In particular, when the weights (or frequencies) f 1 , f 2 ......f n are all equal. We get the simple Arithmetic
Mean.