Business Statistics - Measures of Central Tendency (Formulas and Selected Problems and Solutions)

 

Business Statistics

Measures of Central Tendency

(Formulas and Selected Problems and Solutions)

 

Part A:

In Part A you will find Formulas of Mean (Arithmetic Mean, Geometric Mean and Harmonic Mean), Median and Mode for all types of data (i.e. for Simple Distribution, Simple Frequency Distribution and Grouped Frequency Distribution).

 

Part B:

In Part B you will find seven selected problems with solutions.

Part A

There are 3 measures of central tendency – Mean, Median and Mode. Again, Mean is of 3 types – Arithmetic Mean (A.M.), Geometric Mean (G.M.) and Harmonic Mean (H.M.)

 

Note: The words ‘mean’ and ‘average’ refer to Arithmetic Mean only.

 

Formulas of Mean (AM)

 

        A.         Direct Method:

For simple distribution, Mean = (∑x)/n

For frequency distribution, Mean = (∑fx)/ (∑f)

 

        B.         Short-cut Method:

For simple distribution,

Mean = A + (∑d)/n

[Here, A = Assumed Mean (as near as possible to the true mean); d = x – A]

 

For frequency distribution,

Mean = A + (∑fd)/ (∑f)

[Here, A = Assumed Mean (value of that observation or that mid-value which has the highest frequency); d = x – A]

 

        C.         Step-deviation Method:

For simple distribution,

Mean = A + [(∑d)/n]*c

[Here, A = Assumed Mean (as near as possible to the true mean); d = (x – A)/c; c = Common factor]

 

For frequency distribution,

Mean = A + [(∑fd)/ (∑f)]*c

[Here, A = Assumed Mean (value of that observation or that mid-value which has the highest frequency); d = (x – A)/c; c = Common factor or common width]

 

Important note:

Formula of mean for simple frequency distribution and grouped frequency distribution under all the above three methods are same. Only thing to be remembered is that in case of grouped frequency distribution x_i will be the mid-values of the class intervals.

 

Important Properties of Arithmetic Mean

 

1.       The total of a set of observations is equal to the product of their number and the A.M. Symbolically,

(i)          ∑x_i   = n*(Mean)

                   [n = Total number of observations]

(ii)        ∑f_ix_i = N*(Mean)     [N = ∑f_i]

2.       The sum of the deviations of a set of observations from their A.M. is always zero.

Symbolically,

(i)              ∑(x_i – Mean) = 0

[When Mean = (∑x_i)/n]

(ii)            ∑f_i(x_i – Mean) = 0   

[When Mean = (∑f_ix_i)/N]

3.       If two variables x and y are so related that

x = c + dy, where c and d are constants, then Mean of x = c + d(Mean of y)

4.       The sum of the squares of deviations of a set of observations has the smallest value, when deviations are taken from their A.M.

Symbolically,

(i)              ∑ (x_i – A) ^2 is minimum, when A = simple A.M.

(ii)            ∑f_i (x_i – A) ^2 is minimum, when A = weighted A.M.

Mean of Composite Group

For simple distribution,

Composite mean = [(∑x) + (∑y)]/ (n1 + n2)

= [(n1*Mean of x) + (n2*Mean of y)]/ (n1 + n2)

 

For frequency distribution,

Composite mean = [(∑fx) + (∑fy)]/ (N1 + N2)

= [(N1*Mean of x) + (N2*Mean of y)]/ (N1 + N2)

 

[Here,

x_i = Observations of Group- 1

y_i = Observations of Group- 2

n1 = Total number of observations of Group- 1

n2 = Total number of observations of Group- 2

N1 = Total frequency of Group- 1

N2 = Total frequency of Group- 2]

 

Formulas of Geometric Mean (GM)

 

For simple distribution,

G = (x1*x2*x3*..................*xn) ^ (1/n)

G = antilog [(1/n)*∑logx]

 

For frequency distribution,

G = [(x1^f1)*(x2^f2)*...........*(xn^fn)] ^ (1/∑f)

G = antilog [(1/∑f)*∑f (logx)]

 

Geometric Mean of Composite Group

G = [(G1^N1)*(G2^N2)] ^ (1/N)

 

[Here,

G1 = Geometric mean of Group- 1

G2 = Geometric mean of Group- 2

N1 = Total frequency of Group- 1

N2 = Total frequency of Group- 2

N = N1 + N2]

 

Formulas of Harmonic Mean (HM)

 

For simple distribution,

H = n/∑ (1/x)

 

For frequency distribution,

H = ∑f/∑ (f/x)

 

Harmonic Mean of Composite Group

H = [N1 + N2]/ [(N1/H1) + (N2/H2)]

 

[Here,

H1 = Harmonic mean of Group- 1

H2 = Harmonic mean of Group- 2

N1 = Total frequency of Group- 1

N2 = Total frequency of Group- 2

N = N1 + N2]

 

Relation between AM, GM and HM

GM = (AM * HM) ^ (1/2)

 

 

 

Formulas of Median

 

For simple distribution

When n is odd −

Step 1:

Arrange the data in ascending order.

 

Step 2:

Median = Value of [(n +1)/2]^th observation.

 

When n is even –

Step 1:

Arrange the data in ascending order.

 

Step 2:

Median = ½ of [Value of (n/2)^th observation +

              Value of ((n + 2)/2)^th observation]

 

For simple frequency distribution

Step 1:

Construct a cumulative frequency distribution table (“less than” type)

 

Step 2:

Median = Value of observation corresponding to cumulative frequency [(N + 1)/2]

 

For grouped frequency distribution

Step 1:

If the class intervals are given in ‘limits’, convert the class limits into class boundaries

 

Step 2:

Construct a cumulative frequency distribution table (“less than” type)

 

Step 3:

Median = L1 + [(L2 – L1)/f]*(m – c)

 

[Here,

L1 = Lower boundary of the median class

L2 = Upper boundary of the median class

f = Frequency of the median class

m = N/2

N = Total frequency

c = Cumulative frequency of the class interval immediately preceding the median class

Median class = The class interval in which the (N/2)^th observation lies

 

 

 

 

Formulas of Mode

 

For simple distribution

Mode is the value of that observation which occurs for maximum number of times in the given data.

 

For simple frequency distribution

Mode is the value of that observation which corresponds to the largest frequency.

 

For grouped frequency distribution

Step 1:

If the class intervals are given in ‘limits’, convert the class limits into class boundaries

 

Step 2:

Mode = L1 + [(f_m – f₁)/(2f_m – f₁ – f₂)]*c

 

[Here,

L1 = Lower boundary of the modal class

f_m = Frequency of the modal class

f₁ = Frequency of the class interval immediately preceding the modal class

f₂ = Frequency of the class interval immediately following the modal class

c = Width of the modal class

Modal class = The class interval having the largest frequency

 

Relation between Mean, Median and Mode (Empirical Relation)

 

Mean – Mode = 3(Mean – Median)

Or, Mode = 3Median – 2Mean

Part B

Statistics

Measures of Central Tendency

Selected Problems and Solutions

 

Problem: 1

The following table gives the life time in hours of 400 tubes of a certain make. Find the mean life time of the tubes.

Life time (hrs)	No. of tubes	Life time (hrs)	No. of tubes
Less than 300	0	Less than 800	265
Less than 400	20	Less than 900	324
Less than 500	60	Less than 1,000	374
Less than 600	116	Less than 1,100	392
Less than 700	194	Less than 1,200	400

 

Solution: 1

Problem: 2

Calculate mean of the following grouped frequency distribution:

Class Interval	Frequency	Class Interval	Frequency
25 – 50	21	100 – 125	89
50 – 75	47	125 – 150	55
75 – 100	67	150 – 175	21

 

 

Solution: 2

Problem: 3

Find out the mode of the following series:

Size (x)	Frequency (f)	Size (x)	Frequency (f)
5	48	13	52
6	52	14	41
7	56	15	57
8	60	16	63
9	63	17	52
10	57	18	48
11	55	19	40
12	50

 

Solution: 3

Problem: 4

From the data in the following table calculate the average marks of the M.Com examinees in Statistics at a class test:

Marks	No. of Examinees
30 – 39	2
40 – 49	3
50 – 59	11
60 – 69	20
70 – 79	32
80 – 89	25
90 – 99	7

 

 

Solution: 4

Problem: 5

The following table gives the rise in price of 300 commodities between two dates. Calculate the mean rise in price:

Rise in price (%)	No. of Commodities
0 – 5	12
5 – 10	30
10 – 15	51
15 – 25	84
25 – 35	66
35 – 45	35
45 – 60	15
60 – 80	7

 

Solution: 5

Problem: 6

The table below gives the numbers of candidates (f) obtaining marks (x) or higher in a certain examination (all marks are given in whole numbers):

Marks (x) or more	Number of candidates (f)
10	140
20	133
30	118
40	100
50	75
60	45
70	25
80	9
90	2
100	0

 

Calculate the mean and the median marks obtained by the candidates.

 

Solution: 6

Problem: 7

Calculate the value of (i) mean, (ii) median, and (iii) the two quartiles from the following grouped frequency distribution of income earned by the persons of a society.

Income (Rs in ’000)	No. of persons
0 – 1	13
1 – 2	90
2 – 3	81
3 – 5	117
5 – 10	66
10 – 25	27
25 – 50	6
50 – 100	2
100 – 1,000	2

 

Solution: 7