Question

From the prices of shares X and Y below, find out which is more stable in value: X 35 54 52 53 56 58 52 50 51 49 Y 108 107 105 105 106 107 104 103 104 101

Answer 1

The given data is:

X	35	54	52	53	56	58	52	50	51	49
Y	108	107	105	105	106	107	104	103	104	101

First calculate the values for share X. The prices of the share X are 34, 54, 52, 53, 56, 58, 52, 50, 51, and 49. The formula to calculate the mean is the ratio of sum of observations to the number of observations.

x ¯ = 1 n ∑ i=1 n x i (1)

The sum of observations is,

∑ i=1 n x i =34+54+52+53+56+58+52+50+51+49 =510

Since the total number of observation is 10, thus substitute 10 for n and 510 for ∑ i=1 n x i in equation (1).

x ¯ = 1 10 ( 510 ) =51

Therefore, the mean of the given data is 51.

The formula to calculate the variance is,

σ 2 = 1 n ∑ i=1 n ( x i − x ¯ ) 2 (2)

Substitute 51 for x ¯ and 34, 54, 52, 53, 56, 58, 52, 50, 51, 49 for x i respectively in x i − x ¯ and make a table to evaluate the variance.

x i	x i − x ¯	( x i − x ¯ ) 2
34	−16	256
54	3	9
52	1	1
53	2	4
56	5	25
58	7	49
52	1	1
50	−1	1
51	0	0
49	−2	4
∑ i=1 n x i =510		∑ i=1 n ( x i − x ¯ ) 2 =350

Substitute 10 for n and 350 for ∑ i=1 n ( x i − x ¯ ) 2 in equation (2).

σ 2 = 350 10 =35

Thus, the variance of the given data is 35.

The formula to calculate the standard deviation is,

S.D.= σ 2 (3)

Substitute 35 for σ 2 in equation (3).

S.D.= 35 =5.91

The formula to calculate the coefficient of variation is,

C.V.= σ x ¯ ×100(4)

Substitute 5.91 for σ and 51 for x ¯ in equation (4).

C.V.= 5.91 51 ×100 =11.58

Thus, the coefficient of variation is 11.58.

Now, calculate the values for share Y. The prices of the share Y are 108, 107, 105, 105, 106, 107, 104, 103, 104, 101. The formula to calculate the mean is the ratio of sum of observations to the number of observations.

y ¯ = 1 n ∑ i=1 n y i (5)

The sum of observations is,

∑ i=1 n y i =108+107+105+105+106+107+104+103+104+101 =1050

Since the total number of observation is 10, thus substitute 10 for n and 1050 for ∑ i=1 n y i in equation (5).

y ¯ = 1 10 ( 1050 ) =105

Therefore, the mean of the given data is 105.

The formula to calculate the variance is,

σ 2 = 1 n ∑ i=1 n ( y i − y ¯ ) 2 (6)

Substitute 51 for y ¯ and 108, 107, 105, 105, 106, 107, 104, 103, 104, 101 for y i respectively in y i − y ¯ and make a table to evaluate the variance.

y i	y i − y ¯	( y i − y ¯ ) 2
108	3	9
107	2	4
105	0	0
105	0	0
106	1	1
107	2	4
104	−1	1
103	−2	4
104	−1	1
101	−4	16
∑ i=1 n y i =1050		∑ i=1 n ( y i − y ¯ ) 2 =40

Substitute 10 for n and 40 for ∑ i=1 n ( y i − y ¯ ) 2 in equation (6).

σ 2 = 40 10 =4

Thus, the variance of the given data is 35.

The formula to calculate the standard deviation is,

S.D.= σ 2 (7)

Substitute 4 for σ 2 in equation (7).

S.D.= 4 =2

The formula to calculate the coefficient of variation is,

C.V.= σ y ¯ ×100(8)

Substitute 2 for σ and 105 for y ¯ in equation (8).

C.V.= 2 105 ×100 =1.9

Thus, the coefficient of variation is 1.9.

This shows that coefficient of variable for X shares are greater than Y shares.

Thus, the prices of shares Y are more stable than the prices of shares X.