Question

If there are two groups with 75 and 65 as harmonic mean and containing 15 and 13 observations then the combined harmonic mean....?

Answer 1

Hint: Here, we are given two groups with their harmonic means and their total observations too. So, we will use the definition of the harmonic mean and apply it to both the groups differently. After doing so, we will combine it to find the harmonic mean and solving this, we will get the final output.

Complete step-by-step answer:
Given that,
The first group with 75 as harmonic mean contains 15 observations and the second group with 65 as harmonic mean contains 13 observations.

So, the reciprocal of the given terms of harmonic mean are:
\[\dfrac{1}{{75}}\] and \[\dfrac{1}{{65}}\]

Here, first we will find the harmonic mean of both the two groups given differently.
As we know that, the harmonic mean is the reciprocal of the average of reciprocals, and so applying the definition we will have:

Next, the sum of the reciprocals of all the terms in first group is as below:
\[ = \dfrac{1}{{75}} \times 15\]
\[ = \dfrac{1}{5}\]
Then, the sum of the reciprocals of all the terms in second group is as below:
\[ = \dfrac{1}{{65}} \times 13\]
\[ = \dfrac{1}{5}\]

Second, we will combine both the harmonic mean of the two groups i.e. we will have a total of:
\[ = 15 + 13\]
\[ = 28\] observations.

Again we will use the definition of harmonic mean, we will get,
\[ = \dfrac{{28}}{{\dfrac{1}{5} + \dfrac{1}{5}}}\]
Taking LCM as 5, we will get,
\[ = \dfrac{{28}}{{\dfrac{{1 + 1}}{5}}}\]
On evaluating this, we will get,
\[ = \dfrac{{28}}{{\dfrac{2}{5}}}\]
Here, we have split the numerator and the denominator and use the divide sign, we will get,
\[ = 28 \div \dfrac{2}{5}\]
Replace the division sign into multiplication sign, and so the second terms numerator and denominator place will interchange, we will get,
\[ = 28 \times \dfrac{5}{2}\]
On simplifying this, we will get,
\[ = 14 \times 5\]
\[ = 70\]

Hence, the combined harmonic mean of both the groups is 70.

Note: As we know that, to represent the data or value in series, the measure of central tendency is used and they are mean, median, and mode. The definition of harmonic mean (HM) is the reciprocal of the average of the reciprocals of the data values. In short, it is used when there is a necessity to give greater weight to the smaller items. It is also applied in the case of times and average rates too.