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Measures of Central Tendency and Dispersion

Harmonic Mean and Geometric Mean

You all are well aware with finding squares, cubes, and other powers of a base.  Also, you know about square roots, cube roots, and nth roots of a number. Moreover, you are aware of the reciprocal of a number and its importance. What are all these in statistics?  How are they helpful? They form the basis of the geometric mean and harmonic mean in Statistics. Let us get started to learn more about the geometric and harmonic mean.

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Geometric and Harmonic Mean

The geometric mean (G.M.) and the harmonic mean (H.M.) forms an important measure of the central tendency of data. They tell us about the central value of the data about which all the set of values of data lies. Suppose we have a huge data set and we want to know about the central tendency of this data set.

We have so many ways by which we can do so. But what if the data sets are fluctuating or we need to add or remove some of the data value? Calculating the average value or the central value will be a tiresome and troublesome task. So, we use geometric and harmonic means as our rescuer.

Browse more Topics under Measures Of Central Tendency And Dispersion

Geometric Mean

A geometric mean is a mean or average which shows the central tendency of a set of numbers by using the product of their values. For a set of n observations, a geometric mean is the nth root of their product. The geometric mean G.M., for a set of numbers x1, x2, … , xn is given as

G.M. = (x1. x2 … xn)1⁄n

or, G. M. = (π i = 1n xi) 1⁄n = n√( x1, x2, … , xn).

The geometric mean of two numbers, say x, and y is the square root of their product x×y. For three numbers, it will be the cube root of their products i.e., (x y z) 1⁄3.

Relation Between Geometric Mean and Logarithms

In order to make our calculation easy and less time consuming we use the concept of logarithms in the calculation of geometric means.

Since, G.M. = (x1. x2 … xn) 1⁄n

Taking log on both sides, we have

log G.M. = 1⁄n (log ((x1. x2 … xn))

or, log G.M. = 1⁄n (log x1 + log x2 + … + log xn)

or, log G.M. = (1⁄n) ∑ i= 1n log xi

or, G.M. = Antilog(1⁄n (∑ i= 1n log xi)).

Geometric Mean of Frequency Distribution

For a grouped frequency distribution, the geometric mean G.M. is

G.M. = (x1 f1. x2 f2 …  xn fn) 1⁄N , where N = ∑ i= 1n fi

Taking logarithms on both sides, we get

log G.M. = 1⁄N (f1 log x1 + f2 log x2 + … + fn log xn) = 1⁄N [∑ i= 1n fi log x].

Properties of Geometric Means

  • The logarithm of geometric mean is the arithmetic mean of the logarithms of given values
  • If all the observations assumed by a variable are constants, say K >0, then the G.M. of the observation is also K
  • The geometric mean of the ratio of two variables is the ratio of the geometric means of the two variables
  • The geometric mean of the product of two variables is the product of their geometric means

Geometric Mean of a Combined Group

Suppose G1, and G2 are the geometric means of two series of sizes n1, and n2 respectively. The geometric mean G, of the combined groups, is:

log G = (n1 log G1 + n2 log G2) ⁄ (n1 + n2)

or, G = antilog [(log G1 + n2 log G2) ⁄ (n1 + n2)]

In general for ni geometric means, i = 1 to k, we have

G = antilog [(log G1 + n2 log G2 + … + nk log Gk) ⁄ (n1 + n2 + … +nk)]

Advantages of Geometric Mean

  • A geometric mean is based upon all the observations
  • It is rigidly defined
  • The fluctuations of the observations do not affect the geometric mean
  • It gives more weight to small items

Disadvantages of Geometric Mean

  • A geometric mean is not easily understandable by a non-mathematical person
  • If any of the observations is zero, the geometric mean becomes zero
  • If any of the observation is negative, the geometric mean becomes imaginary

Harmonic Mean

A simple way to define a harmonic mean is to call it the reciprocal of the arithmetic mean of the reciprocals of the observations. The most important criteria for it is that none of the observations should be zero.

A harmonic mean is used in averaging of ratios. The most common examples of ratios are that of speed and time, cost and unit of material, work and time etc. The harmonic mean (H.M.) of n observations is

H.M. = 1÷ (1⁄n ∑ i= 1n (1⁄xi) )

In the case of frequency distribution, a harmonic mean is given by

H.M. = 1÷ [1⁄N (∑ i= 1n (fi ⁄ xi)], where N = ∑ i= 1n fi

Properties of Harmonic Mean

  • If all the observation taken by a variable are constants, say k, then the harmonic mean of the observations is also k
  • The harmonic mean has the least value when compared to the geometric mean and the arithmetic mean

harmonic mean

Advantages of Harmonic Mean

  • A harmonic mean is rigidly defined
  • It is based upon all the observations
  • The fluctuations of the observations do not affect the harmonic mean
  • More weight is given to smaller items

Disadvantages of Harmonic Mean

  • Not easily understandable
  • Difficult to compute

Important Note

Suppose a person cover a certain distance d at a speed of x. He returns back to the starting point with a speed of y. In this case, the average speed of the person is calculated by the harmonic mean.

Average speed = Total distance covered / Total time taken = 2d (d⁄x + d⁄y).

In other words, if an equal distance is covered with speeds S1, S2, … , Sn, then

Average speed = n ÷ ∑ (1⁄S).

If different distances D1, D2, … , Dn, is covered with different speeds S1, S2, … , Sn, the average speed is

Average Speed = [∑ i= 1n  Di] ⁄ [∑ i= 1n (Di ⁄ Si)]

Solved Example for You

Problem: Calculate the geometric and harmonic mean of the given data

x 2 4 5 8
f 3 3 2 2

Solution: Geometric mean = G.M. = (x1 f1. x2 f2 …  xn fn) 1⁄N

Here, N = 3 + 3 + 2 + 2 = 10

G.M. = (2 3 × 4 3 × 5 2 × 8 2)1/10

or, G.M. =(8×64×25×64)1/10 = (819200)1/10

H.M. = 1 ÷ [1 ⁄ N ∑ i= 1n (fi ⁄ xi) ] = 1÷[1⁄10 × (3⁄2 + ¾ + 2⁄5 + 2⁄8)] = 100⁄29.

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