Harmonic Mean - Measures of Central Tendency, Business Mathematics & Statistics - Business Mathematics and Statistics - B Com

HARMONIC MEAN (H. M.) :

Definition. The Harmonic Mean (H) for n observations, x₁ , x₂ ,…….x_n is the total number divided by the sum of the reciprocals * of the numbers.

Again,  (i.e. reciprocal of H. M = A. M. of reciprocals of the numbers).

* For   , i.e. a reciprocal of b. And for 1/ a = b, b is reciprocal of a. Reciprocal of 2 is. 1/2 .

Example 17 : Find the H. M. of 3, 6, 12 and 15.

Example 18 : Find the H.M. of 1, 1/2 , 1/3 ,……. 1/n

[Note. The denominator is in A..P. use  

Example 19 : A motor car covered distance of 50 miles four times. The first time at 50 m. p. h, the second at 20 m. p. h., the third at 40 m. p. h, and the fourth at 25 m.p.h Calculate the average speed and explain the choice of the average.

Average Speed (H.M) = 

For the statement x units per hour, when the different values of x (i.e. distances) are given, to find average, use H.M. If again hours (i.e., time of journey) are given, to find average, we are to use A.M. In the above example, miles (distances) are given, so we have used H.M. Weighted H.M. The formula to be used is as follows :

Example 20 : (a) A person travelled 20 k.m. at 5 k.m.p.h. and again 24 k.m. at 4 k.m.p.; to find average speed. (b) A person travelled 20 hours at 5 k.m.p.h. and again 24 hours at 4.m.p.h.; to find average speed. (a) We are to apply H.M. (weight) in this case, since, distances are given.

Average speed (H.M.)  

Average speed (A.M.)  

Example 21 : Find the harmonic mean of the following numbers : 

Example 22 : An aeroplane flies around a square and sides of which measure 100 kms. Each. The aeroplane cover at a speed of 10 Kms per hour the first side, at200 kms per hour the second side, at 300 kms per hour the third side and at 400 kms per hour the fourth side. Use the correct mean to find the average speed round the square.
Here H.M. is the appropriate mean.
Let the required average speed be H kms per hours

then 

ADVANTANGES OF HARMONIC MEAN :
(i) Like A.M. and G. M. it is also based on all observations.
(ii) Capable of further algebraic treatment.
(iii) It is extremely useful while averaging certain types of rates and rations.

DISADVANTAGES OF HARMONIC MEAN :
(i) It is not readily understood nor can it be calculated with ease.
(ii) It is usually a value which may not be a member of the given set of numbers.
(iii) It cannot be calculated when there are both negative and positive values in a series or one of more values in zero.

It is useful in averaging speed, if the distance travelled is equal. When it is used to give target weight to smallest item, this average is used.