GEOMETRIC
MEAN
Submitted to : Sir Umar
Work Distribution:
 Ghayoor Abbas
(Introduction)
 Awais Ghaffar
(Questionx..)
 Syed Saleh Haider (Geometric mean for ungroup data)
 Sohail Waqar
(Questionx of ungroup data)
 Sajawal Hussain
(Geometric mean for group data)
 M. Shoaib Malik
(Questionx of group data)
Definition :The geometric mean ‘G’ of ‘n’ positive values is
defined as the nth root of their product. Thus it is
obtained by multiplying together all the ‘n’ values and
then taking the nth root of the product.
G =[𝑥1 . 𝑥2 . 𝑥3 … … . . 𝑥𝑛 ]1/𝑛 =
n: number of observations
x: various values.
𝑛
𝑥1 . 𝑥2 . 𝑥3 … … . . 𝑥𝑛
The relation connecting arithmetic
geometric mean and harmonic mean
mean,
Arithmetic Mean ≥ Geometric Mean ≥ Harmonic
Mean
Uses of Geometric Mean : Find the average percentage in sales,
production etc.
 Find the index numbers since it shows the
relative change.
 When large weights are given to small items
and small weights are given to large items, the
best measure of central tendency is Geometric
Mean. That is, when there are extreme values,
the best measure of central tendency to be
used is Geometric Mean.
Properties of geometric
mean : The geometric mean is less than arithmetic
mean, i.e. G.M<A.M.
 The product of the items remains unchanged
if each item is replaced by the geometric
mean.
 The geometric mean of the ratio of
corresponding observations in two series is
equal to the ratios of their geometric means.
 The geometric mean of the products of
corresponding items in two series is equal to
the product of their geometric means.
Merits And Demerits Of
Geometric Mean :Merits :
 Geometric Mean is calculated
observations in the series.
based
on
all
 Geometric Mean is clearly defined.
 Geometric Mean is not affected by extreme values in
the series.
 Geometric Mean is amenable to further algebraic
treatment.
 Geometric Mean is useful in averaging ratios and
percentages.
Demerits :
 Geometric Mean is difficult to understand.
 We cannot compute geometric mean if there are both
positive and negative values occur in the series.
 We cannot compute geometric mean if one or more of
the values in the series is zero.
Question no 1:
Question no 2 :
Sequences : Arithmetic Sequence:
Is a pattern of numbers where any term (number in the sequence) is
determined by adding or subtracting the previous term by a constant
called the common difference.
Example: 2, 5, 8, 11, 14, 17 , 20 , 23 Common difference = 3
 Geometric Sequence:
Is a pattern of numbers where any term (number in the sequence) is
determined by multiplying the previous term by a common factor.
Example: 2, 6, 18, 54, 162, 486 , 1458 , 4374 Common difference = 3
Geometric Mean : Fact
Consecutive terms of a geometric sequence are proportional.
Example: Consider the geometric sequence with a common
factor 10.
4 , 40 , 400
4
40
cross-products are equal
=
40
400
(4)(400) = (40)(40)
1600 = 1600
Therefore …
To find the geometric mean between 7 and 28 ...
X , 28
7 , ___
label the missing term x
7
write a proportion
cross multiply
solve
X
X 2 = (7)(28)
X2 =
196
=
X
28
X 2 = 196
X = 14
Geometric Mean (Ungroup-data)
Formula :
G = Antilog (
𝑙𝑜𝑔𝑥
𝑛
)
where :
Log x = log of x
Σ logx = sum of (log x) values
n = Total number of values
Geometric Mean (Ungroup-data)
Example : Find the geometric mean of the
following values: 4, 6, 10, 15, 100.
Step 1 : Find the log of the Given values
X
Log x
4
0.6024
6
0.7781
10
1.0000
15
1.1760
100
2.0000
Geometric Mean (Ungroup-data)
Step 2 : Add the Values
∑logx = 0.6024 + 0.7781 + 1.0000 + 1.1760 + 2.0000
= 5.5505
Step 3 : Calculate the total numbers and there are 5
numbers, So n=5.
Step 4 : Divide the values = ∑logx/n = 5.5505/5
= 1.1101
Step 5 : Taking Antilog = Antilog (1.1101) = 12.8854
Answer : Geometric Mean = 12.8854
Question no 1
Question no 2
Geometric Mean (Group-data)
Formula :
G = Antilog (
𝑓𝑙𝑜𝑔𝑥
𝑓
)
where :
Log x = log of given values
(f.Logx) = Multiplying values of logx with f
Σ (f.logx) = Sum of (f.logx) values
Σf = sum of f (frequency)
Geometric Mean (Group-data)
Example : Find the geometric mean of the following
data:
Marks: 0-10, 10-20, 20-30, 30-40, 40-50
No of Students (f) : 4, 8, 10, 6, 7
Step 1 : Find X through distribution of marks
Class
X=(starting limit + ending limit)/2
0-10
(0+10)/2 = 5
10-20
(10+20)/2 = 15
20-30
(20+30)/2 = 25
30-40
(30+40)/2 = 35
40-50
(40+50)/2 = 45
Geometric Mean (Group-data)
Step 2 : Find the log of x
x
Log x
5
0.6990
15
1.1761
25
1.3969
35
1.5441
45
1.6532
Geometric Mean (Group-data)
Step 3 : Multiply logx values with f
f
Log x
(f.logx)
4
0.6990
2.7960
8
1.1761
9.4088
10
1.3969
13.9790
6
1.5441
9.2646
7
1.6532
11.5724
Step 4 : Finding Σ (f.logx) by Adding the values of (f.logx)
2.7960+9.4088+13.9790+9.2646+11.5754= 47.0208
Geometric Mean (Group-data)
Step 5 : Find Σf By adding the values of f
4 + 8 +10 + 6 + 7 = 35
Step 6 : Find (
𝑓𝑙𝑜𝑔𝑥
𝑓
)
47.0208/35 = 1.3435
Step 7 : Taking Antilog (
𝑓𝑙𝑜𝑔𝑥
𝑓
)
Antilog (1.3435) = 22.06
Answer : Geometric Mean = 22.06
Geometric Mean (Discrete series)
Formula :
G = Antilog (
𝑓𝑙𝑜𝑔𝑥
𝑁
)
Geometric Mean (Discrete series)
Example 1: Calculate the geometric mean of
the following data:
X= 12 13 14 15 16 17.
F= 5 4 4 3 2 1.
X
F
Log x
F x Logx
12
5
1.0792
5.3960
13
4
1.1139
4.4556
14
4
1.1461
4.5844
15
3
1.1761
3.5283
16
2
1.2041
2.4092
17
1
1.2304
1.2304
Total
19
21.6029
Geometric Mean (Discrete series)
f∑logx=21.6029 and N = 19
= Antilog of 21.6029/19
= Antilog of 1.1371
= 13.71
Answer: Geometric Mean = 13.71
Question no 1
Find the geometric mean of the following data:
Marks: 60-80, 80-100, 100-120, 120-140, 140160, 160-180, 180-200.
No of Students (f) : 5, 14, 17, 10, 1, 0
Question no 2
Calculate the geometric mean of the following
data:
X= 1,2,3,4,5,6,7.
F= 7,6,5,4,3,2,1.
Any