Business Statistics - II B.Com.

1. Geometric Mean
Dr. RMKV
Asst. Prof

2. Geometric Mean
 The geometric mean of a series containing n observations is the nth
root of the product of the values. If x1,x2…, xn are observations
 That is G.M =
𝑛
𝑥1 ∗ 𝑥2 ∗ 𝑥3 … .∗ 𝑥𝑛
 Eg. Find GM from the following data 2, 4, 6, 8
 GM =
4
2 ∗ 4 ∗ 6 ∗ 8 =
4
384 = 4.4267=4.43

3. Geometric Mean
 In many problem, numerous values are given to calculation
 At that time, To simplify the calculation with help of logarithm
 How do simplify..
Arithmetic Calculation Logarithm Calculation
X +
÷ -
^ X
÷

4. Proof
 X – 3, 4, 5, 6, 7 GM =
5
3𝑥4𝑥5𝑥6𝑥7
 =
5
2520 = 4.7894
 Logarithm
 GM =AL(
log 3 +log 4+log 5+log 6 + log 7
5
)
 = AL(
3.4014
5
) =AL(0.68028)
 4.7894
X Log X
3 0.4771
4 0.6021
5 0.6990
6 0.7781
7 0.8451
log 𝑥 = 3.4014

5. Formula
 Individual observation
 G.M. = Antilog(
∑ log X
𝑵
)
 Discrete Series
 G.M. = Antilog(
∑f log X
𝑵
)
 Continuous series
 G.M. = Antilog(
∑f log 𝒎
𝑵
)

6. INDIVIDUAL OBSERVATION: Illustration
and Solution
 Daily income of 10 families of a particular place
is given below. Find out Geometric Mean
 X = 85, 70, 15, 75, 500, 8, 45, 250, 40, 36
 Solution
 GM = AL(
∑ log X
𝑵
)
 = AL(
𝟏𝟕.𝟔𝟑𝟕𝟐
𝟏𝟎
)
 AL(1.76372) = 58.04
 Geometric Mean of Daily Income Rs. 58.04
X Log x
85 1.9294
70 1.8451
15 1.1761
75 1.8751
500 2.6990
8 0.9031
45 1.6532
250 2.3979
40 1.6020
36 1.5563
log 𝑥 = 17.6372

7. INDIVIDUAL OBSERVATION: Illustration
and Solution
 Find out Geometric Mean from the following
data
 X = 125, 1462, 38, 7, 0.22, 0.08, 12.75, 0.5
 Solution
 GM = AL(
∑ log X
𝑵
)
 = AL(
𝟔.𝟕𝟑𝟔𝟕
𝟖
)
 AL(0.8421) = 6.952
 Geometric Mean = 6.95 or 7
X Log x
125 2.0969
1462 3.1649
38 1.5798
7 0.8451
0.22 -0.6576
0.08 -1.0969
12.75 1.1055
0.5 -0.3010
log 𝑥 = 6.7367

8. Discrete Series: Illustration and Solution
 Calculate the Geometric Mean Income of families
 Solution
 Let us take monthly incomes as X and No. of families as f
 G.M. = Antilog(
∑f log X
𝑵
)
Monthly
Income
5000 400 2000 3750 3000 750 600 300
No. of
families
2 100 50 4 6 8 6 10

9. x f Log x F log x
5000 2 3.6990 7.3980
400 100 2.6021 260.2100
2000 50 3.3010 165.0500
3750 4 3.5740 14.2960
3000 6 3.4771 20.8630
750 8 2.8751 23.0008
600 6 2.7782 16.6692
300 10 2.4771 24.7710
N = 186 ∑f log X = 532.2580
 G.M. = Antilog(
∑f log X
𝑵
)
 = AL(
𝟓𝟑𝟐.𝟐𝟓𝟖𝟎
𝟏𝟖𝟔
)
 = AL(2.8616)
 = 727.11
 Geometric Mean of family monthly
income Rs. 727.11

10. Discrete Series: Illustration and Solution
 The following table gives the diameter of screws obtained in a sample inquiry. Calculate the
mean diameter using GM
 Solution
 Let us take Diameter as X and No. of screws as f
 G.M. = Antilog(
∑f log X
𝑵
)
Diameter(in
mm)
130 135 140 145 150 155 160 165 170
No. of
screws
3 4 6 6 3 5 2 1 1

11. x f Log x F log x
130 3 2.1139 6.3417
135 4 2.1303 8.5212
140 6 2.1461 12.8766
145 6 2.1614 12.9684
150 3 2.1761 6.5283
155 5 2.1903 10.9515
160 2 2.2041 4.4082
165 1 2.2175 2.2175
170 1 2.2304 2.2304
N = 31 ∑f log X = 67.0438
 G.M. = Antilog(
∑f log X
𝑵
)
 = AL(
𝟔𝟕.𝟎𝟒𝟑𝟖
𝟑𝟏
)
 = AL(2.1627)
 = 145.446
 Geometric Mean diameter of screws
145.45

12. Continuous Series: Illustration and Solution
 Calculate Geometric Mean from the following data
 Let us take mark as X and No. of students as f
 G.M. = Antilog(
∑f log 𝒎
𝑵
)
Marks 4-8 8-12 12-16 16-20 20-24 24-28 28-32 32-36 36-40
No. of
Students
6 10 18 30 15 12 10 6 2

13. x f m Log m F log m
4-8 6 (4+8)/2
= 6
0.7782 4.6692
8-12 10 (8+12)/
2= 10
1.0000 10.0000
12-16 18 14 1.1461 20.6298
16-20 30 18 1.2553 37.6590
20-24 15 22 1.3424 20.1360
24-28 12 26 1.4150 16.9800
28-32 10 30 1.4771 14.7710
32-36 6 34 1.5315 9.1890
36-40 2 38 1.5798 3.1596
N = 109 ∑f log m =
137.1936
 G.M. = Antilog(
∑f log m
𝑵
)
 = AL(
𝟏𝟑𝟕.𝟏𝟗𝟑𝟔
𝟏𝟎𝟗
)
 = AL(1.2587)
 = 18.14
 Geometric Mean of Mark = 18.14

14. Continuous Series: Illustration and Solution
 Find the geometric mean for the data given below
 G.M. = Antilog(
∑f log 𝒎
𝑵
)
x 0-10 10-20 20-30 30-40 40-50
f 5 15 30 8 2

15. x f m Log m F log m
0-10 5 5 0.6990 3.4950
10-20 15 15 1.1761 17.6415
20-30 30 25 1.3979 41.9370
30-40 8 35 1.5441 12.3528
40-50 2 45 1.6532 3.3064
N = 60 ∑f logm =78.7327
 G.M. = Antilog(
∑f log m
𝑵
)
 = AL(
𝟕𝟖.𝟕𝟑𝟐𝟕
𝟔𝟎
)
 = AL(1.3122)
 = 20.52
 Geometric Mean = 20.52