The geometric mean is a type of average that indicates the central tendency of a set of numbers using their product, as opposed to the arithmetic mean which uses their sum, and it is calculated by taking the nth root of the product of the numbers. The geometric mean is more appropriate than the arithmetic mean for describing proportional growth and ratios, and it has various applications in fields like optics, signal processing, geometry, and finance.

1. In Mathematics, the geometric
mean is a type of mean or average,
which indicates the central
tendency or typical value of a set of
numbers by using the product of
their values(as opposed to the
arithmetic mean which uses their
sum).

2. Formula:
G.M.=
For example,
GM of two numbers 4 and 9 is
GM of three numbers 1, 4 and 128 is
𝑋1 ∗ 𝑋2 ∗ 𝑋3 ∗ …..∗ 𝑋4
𝑛
=(X1*X2*X3*…….Xn)1/n
4 ∗ 9 = 36 = 6
1∗4∗128
3
= 512 = 8

4. 1)Individual Observations
Geometric mean of a set of n observations
X1,X2,X3…..,Xn is given by
GM=(X1*X2*X3*….*Xn)1/n
Taking logarithms of both sides, we obtain
1/nlog(X1*X2*X3*….*Xn) =
1/n(logX1+logX2+logX3+….+logXn)=
1/n∑logX

5. We see that G.M. of a set of observations is the
antilog of the arithmetic mean of their
logarithms.
Example: Find the geometric mean of
2,4,8,12,16 and 24.

6. Diameter (mm) (X) No. of screws (f) log X f log X
13o 3
2.1139 6.3417
135 4
2.1303 8.5212
140 6
2.1461 12.8766
145 6
2.1614 12.9684
143 3
2.1553 6.4659
N = ∑ f = 31
∑ f log X = 66.7239

8. In case of continuous frequency
distribution, G.M. is given by
G.M. = AL [1/Nf∑logX]
Where X1,X2,X3…..,Xn are the class marks
(or mid values) of a set of grouped data with
corresponding class frequencies f1, f2,f3,….fn

9. Example: Find the geometric mean for the
following distribution:
Marks No. of Students
0-10 10
10-20
9
20-30 25
30-40
30
40-50

10. Marks Mid-value (X) No. of Students log X f log X
0-10 5 10
0.6990 6.99
10-20 15 9
1.1761 10.5849
20-30 25 25
1.3979 34.9475
30-40 35 30
1.5441 46.323
40-50 45 16
N = ∑ f = 100 ∑ f log X
= 142.7006

12. Proportional Growth
The geometric mean is more appropriate than the
arithmetic mean for describing proportional
growth, both exponential growth (constant
proportional growth) and varying growth; in
business the geometric mean of growth rates is
known as the compound annual growth rate
(CAGR).

13.  In Social Sciences
The geometric mean has been used relatively
rare in computing social statistics, but starting
from 2010 the United Nations Human
Development Index did switch to this mode of
calculation, on the grounds that it better
reflected the non-substitutable nature of the
statistics being compiled and compared.

14. Aspect Ratio
The geometric mean has been used in choosing a
compromise aspect ratio in film and video: given
two aspect ratios, the geometric mean of them
provides a compromise between them, distorting or
cropping both in some sense equally.

15. Anti-reflective coatings
In optical coatings, where reflection needs to be
minimized between two media of refractive
indices n0 and n2, the optimum refractive
index n1 of the anti-reflective coating is given by the
geometric mean.

16.  Spectral Flatness
In signal processing, spectral flatness, a measure
of how flat or spiky a spectrum is, is defined as the
ratio of the geometric mean of the power spectrum
to its arithmetic mean.

17. Geometry
In the case of a right triangle, its altitude is the
length of a line extending perpendicularly from the
hypotenuse to its 90° vertex. Imagining that this
line splits the hypotenuse into two segments, the
geometric mean of these segment lengths is the
length of the altitude.

18. Financial
The geometric mean has from time to time been
used to calculate financial indices (the averaging is
over the components of the index). For example in
the past the FT 30 index used a geometric mean. It
is also used in the recently introduced "RPIJ"
measure of inflation in the United Kingdom and
elsewhere in the European Union.

19. Submitted by:
Sumit Kumar 130326
Sujeet Kumar 130357
Nitin Kumar 130528
Neeraj 130532