Harmonic Mean Lecture 07

 Harmonic Mean 

Lecture 07

Introduction

The harmonic mean is a measure of central tendency and very useful in averaging rates and ratios types of data, such as price (Rs./kg), speed (km/hour), and productivity (output/manhour). The harmonic mean is applicable in the following situations:.

i. When ratio is expressed as x per y, if x is given, use hormonic, and if y is given, then use arithmatic mean. 

ii. The data have a few points that are much higher than the rest.

Definition

The reciprocal of the arithmatic mean of the reciprocal of values.

Let X1, X2,..., Xn be the values of a data. The harmonic mean is obtained as:

When the data is expressed in the frequency distribution, then the harmonic mean can be obtained as:

Example 3.18: The distance from residential area to market is 40 km. A driver drove to a residential area at a speed of 50 km per hour and returned to the market at a speed of 80 km per hour. What is the average speed for the whole journey?

Solution: As the data is expressed as per (50/per hour) and x is given, then we use harmonic mean.

Example 3.19: A bus traveled 40 km in four segments at a speed of 100 km per hour for the first 10 km, 110 km per hour for the second 10 km, 90 km per hour for the third 10 km, and 120 km per hour for the fourth 10 km. What is the average speed of the bus?

Solution:

Number	Distance	Speed (Km per hour)	Time (hour)
1	10	100	0.100
2	10	110	0.091
3	10	90	0.111
4	10	120	0.083
Total	40		0.385

The harmonic mean formula is:

Example 3.20: Find the harmonic mean for the data given below:

10, 16, 20, 25, 28, 32, 46.

Solution: The harmonic mean for a discrete set of data is given below:

Example 3.21: Find the harmonic mean for the following frequency distribution.

Absentees	2	3	4	5	6	7	8	9
Number of students	15	10	16	20	21	15	9	7

Solution:

Example 3.22: Find the hamonic mean for the following age distribution.

Age	10 - 20	20 - 30	30 – 40	40 – 50	50- 60	60 - 70
No. of person	50	80	145	95	60	55

Solution:

Merits and Demerits of Harmonic Mean

Merits of harmonic mean

i. It is rigorously defined by proper mathematical formula.

ii. It is based on all the observations of the data.

iii. It is suitable average in case series having some observation very large.

iv. It is a suitable average for rates and ratios.

Demerits of harmonic mean

i. It is not easy to understand and interpret.

ii. It gives too much weight to smaller observations of the data.

iii. It cannot be computed if any value in the data is zero.

iv. It cannot represent distribution.

Relationship between Mean, Geometric Mean and Harmonic Mean

i. If all the observations are the same, the arithmetic mean, geometric mean, and harmonic mean are equal.

ii. If all the observations are not the same, then mean will be greater than geometric mean and geometric mean will be greater than harmonic mean.

iii. If there are two values, then the square of the geometric mean is equal to the product of arithmetic and harmonic means.

Example 3.23: The arithmetic and geometric means of two values are 25 and 23.50, respectively. Find the harmonic mean.

Solution: The mean and G.M. of two values are given. Now to find the H.M., we use the following relation:

Example 3.24: Compare the arithmetic mean, geometric mean, and harmonic mean of the following data.

10, 16, 20, 25, 32.

Solution:

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