The harmonic mean of n observations, none of which is zero, is defined as the reciprocal of the arithmetic mean of their reciprocals.
Calculation of Harmonic Mean
(a) Individual series
If there are n observations X1, X2, ...... Xn, their harmonic mean is defined as
Example : A train travels 50 kms at a speed of 40 kms/hour, 60 kms at a speed of 50 kms/hour and 40 kms at a speed of 60 kms/hour. Calculate the weighted harmonic mean of the speed of the train taking distances travelled as weights. Verify that this harmonic mean represents an appropriate average of the speed of train.
Verification : Average speed = Total distance travelled/Total time taken We note that the numerator of Equation (1) gives the total distance travelled by train. Further, its denominator represents total time taken by the train in travelling 150 kms, Since 50/40 is time taken by the train in travelling 50 kms at a speed of 40 kms/hour.
Similarly 60/50 and 40/60 are time taken by the train in travelling 60 kms and 40 kms at the speeds of 50 kms./hour and 60 kms/hour respectively. Hence, weighted harmonic mean is most appropriate average in this case.
Example : Ram goes from his house to office on a cycle at a speed of 12 kms/hour and returns at a speed of 14 kms/hour. Find his average speed.Solution: Since the distances of travel at various speeds are equal, the average speed of Ram will be given by the simple harmonic mean of the given speeds.
Choice between Harmonic Mean and Arithmetic Mean
The harmonic mean, like arithmetic mean, is also used in averaging of rates like price per unit, kms per hour, work done per hour, etc., under certain conditions. To explain the method of choosing an appropriate average, consider the following illustration.
Let the price of a commodity be Rs 3, 4 and 5 per unit in three successive years. If we take A.M. of these prices, i.e., 3+4+5/3 = 4, then it will denote average price when equal quantities of the commodity are purchased in each year. To verify this, let us assume that 10 units of commodity are purchased in each year.
Total expenditure on the commodity in 3 years = 10*3 + 10*4 + 10*5.
which is arithmetic mean of the prices in three years.
Further, if we take harmonic mean of the given prices, i.e.
it will denote the average price when equal amounts of money are spent on the commodity in three years. To verify this let us assume that Rs 100 is spent in each year on the purchase of the commodity.
Next, we consider a situation where different quantities are purchased in the three years. Let us assume that 10, 15 and 20 units of the commodity are purchased at prices of Rs 3, 4 and 5 respectively.
which is weighted arithmetic mean of the prices taking respective quantities as weights. Further, if Rs 150, 200 and 250 are spent on the purchase of the commodity at prices of Rs 3, 4 and 5 respectively, then
purchased in respective situations. The above average price is equal to the weighted harmonic mean of prices taking money spent as weights.
Therefore, to decide about the type of average to be used in a given situation, the first step is to examine the rate to be averaged. It may be noted here that a rate represents a ratio, e.g., price = money/quantity, speed = distance/time , work done per hour = work done/time taken , etc.
We have seen above that arithmetic mean is appropriate average of prices (Money/quantity) when quantities, which appear in the denominator of the rate to be averaged, purchased in different situations are given. Similarly, harmonic mean will be appropriate when sums of money, that appear in the numerator of the rate to be averaged, spent in different situations are given.
To conclude, we can say that the average of a rate, defined by the ratio p/q, is given by the arithmetic mean of its values in different situations if the conditions are given in terms of q and by the harmonic mean if the conditions are given in terms of p. Further, if the conditions are same in different situations, use simple AM or HM and otherwise use weighted AM or HM.
Example : An individual purchases three qualities of pencils. The relevant data are given below:
Example : In a 400 metre athlete competition, a participant covers the distance as given below. Find his average speed.
Example : Peter travelled by a car for four days. He drove 10 hours each day. He drove first day at the rate of 45 kms/hour, second day at the rate of 40 kms/hour, third day at the rate of 38 kms/hour and fourth day at the rate of 37 kms/hour. What was his average speed.
Solution: Since the rate to be averaged is speed= (Distance/time) and the conditions are given in terms of time, therefore AM will be appropriate. Further, since Peter travelled for equal number of hours on each of the four days, simple AM will be calculated.
∴ Average speed = 45+40+38+37/4 = 40 kms/hour
Example : In a certain factory, a unit of work is completed by A in 4 minutes, by B in 5 minutes, by C in 6 minutes, by D in 10 minutes and by E in 12 minutes. What is their average rate of working? What is the average number of units of work completed per minute? At this rate, how many units of work each of them, on the average, will complete in a six hour day? Also find the total units of work completed.
Solution: Here the rate to be averaged is time taken to complete a unit of work, i.e., time/units of work done . Since we have to determine the average with reference to a (six hours) day, therefore, HM of the rates will give us appropriate average.
Thus, the average rate of working =
The average number of units of work completed per minute = 1/6.25 = 0.16. The average number of units of work completed by each person = 0.16 *360 = 57.6. Total units of work completed by all the five persons = 57.6 * 5 = 288.0.
Example : A scooterist purchased petrol at the rate of Rs 14, 15.50 and 16 per litre during three successive years. Calculate the average price of petrol (i) if he purchased 150, 160 and 170 litres of petrol in the respective years and (ii) if he spent Rs 2,200, 2,500 and 2,600 in the three years.
Solution: The rate to be averaged is expressed as Money/litre (i) Since the condition is given in terms of different litres of petrol in three years, therefore, weighted AM will be appropriate
Merits and Demerits of Harmonic Mean
Merits
It is rigidly defined average and its value is always definite.Its value is based on all observation in a given series.It is capable of further algebraic treatment.It is not affected by sampling fluctuations.In problems relating to time and rates, it gives better results as compared to other averages. Harmonic mean gives the best result when distance covered are the same, but speed of coverage varies.
Demerits
It is not easily understood and hence its application is ignored.It is not easy to calculate as it involves reciprocal values (The use of calculators can help to remove this difficulty).It gives undue weights to small items and ignores bigger items. This restricts its use in the analysis of economic data.In case of zero or negative values, it cannot be computed.
Relationship among AM, GM and HM
If all the observations of a variable are same, all the three measures of central tendency coincide, i.e., AM = GM = HM. Otherwise, we have AM > GM > HM.
Example : Show that for any two positive numbers a and b, AM ³ GM ³ HM.
Solution: The three averages of a and b are:
Exercise with Hints
A train runs 25 miles at a speed of 30 m.p.h., another 50 miles at a speed of 40 m.p.h., then due to repairs of the track, 6 miles at a speed of 10 m.p.h. What should be the speed of the train to cover additional distance of 24 miles so that the average speed of the whole run of 105 miles is 35 m.p.h?
Hint: Let x be the speed to cover a distance of 24 miles,
Hint: Find simple HM in (ii) and weighted AM in (iii).
Typist A can type a letter in five minutes, B in ten minutes and C in fifteen minutes. What is the average number of letters typed per hour per typist?
Hint: Since we are given conditions in terms of per hour, therefore, simple HM of speed will give the average time taken to type one letter. From this we can obtain the average number of letters typed in one hour by each typist.
Hint: First find the prices per dozen in three situations and since equal money is spent,
HM is the appropriate average.
A country accumulates Rs 100 crores of capital stock at the rate of Rs 10 crores/year, another Rs 100 crores at the rate of Rs 20 crores/year and Rs 100 crores at the rate of Rs 25 crores/year. What is the average rate of accumulation?
Hint: Since Rs 100 crores, each, is accumulated at the rates of Rs 10, 20 and 25 crores/year, simple HM of these rates would be most appropriate.
A motor car covered a distance of 50 miles 4 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.Hint: Use HM.The interest paid on each of the three different sums of money yielding 10%, 12% and 15% simple interest p.a. is the same. What is the average yield percent on the sum invested?Hint: Use HM
Quadratic Mean
Quadratic mean is the square root of the arithmetic mean of squares of observations. If X1, X2 ...... Xn are n observations, their quadratic mean is given by
Moving Average
This is a special type of average used to eliminate periodic fluctuations from the time series data.
Progressive Average
A progressive average is a cumulative average which is computed by taking all the available figures in each succeeding years. The average for different periods is obtained as shown below:
This average is often used in the early years of a business.
Composite Average
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