If there are two groups with n1 and n2 observations and H1 and H2 are respective harmonic means, then the harmonic mean of combined observation is

Option c: n1 + n2/frac{n1{H1} + n2/H2}. Solution: Step 1: Recall the formula for Harmonic Mean of a group. If a group has n observations with HM = H, then: sum 1/xi = n/H Step 2: For the combined group. Total sum of reciprocals: sum{text{all}} 1/xi = n1/H1 + n2/H2 Total observations = n1 + n2 Step 3: Combined HM. H = n1 + n2/frac{n1{H1} + n2/H2} This matches Option C. Note: The given correct option is A, but mathematically the correct formula for combined HM is Option C. Hence, Option C is the correct answer.

Central Tendency & DispersionPYQ Jan. 21Question 3003 of 473

All Questions

If there are two groups with $\displaystyle n_1$ and $\displaystyle n_2$ observations and $\displaystyle H_1$ and $\displaystyle H_2$ are respective harmonic means, then the harmonic mean of combined observation is

Options

\displaystyle \frac{n_1 H_1 + n_2 H_2}{n_1 + n_2}

\displaystyle \frac{n_1 H_1 + n_2 H_2}{H_1 + H_2}

\displaystyle \frac{n_1 + n_2}{\frac{n_1}{H_1} + \frac{n_2}{H_2}}

\displaystyle \frac{(n_1 + n_2)H_1 H_2}{n_1 H_1 + n_2 H_2}

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Correct Answer

✅ Option c — $\displaystyle \frac{n_1 + n_2}{\frac{n_1}{H_1} + \frac{n_2}{H_2}}$

All Options:

A $\displaystyle \frac{n_1 H_1 + n_2 H_2}{n_1 + n_2}$
B $\displaystyle \frac{n_1 H_1 + n_2 H_2}{H_1 + H_2}$
C $\displaystyle \frac{n_1 + n_2}{\frac{n_1}{H_1} + \frac{n_2}{H_2}}$
D $\displaystyle \frac{(n_1 + n_2)H_1 H_2}{n_1 H_1 + n_2 H_2}$

Detailed Solution & Explanation

**Step 1: Recall the formula for Harmonic Mean of a group.** If a group has

\displaystyle n

observations with HM =

\displaystyle H

, then:

\sum \frac{1}{x_i} = \frac{n}{H}

**Step 2: For the combined group.** Total sum of reciprocals:

\sum_{\text{all}} \frac{1}{x_i} = \frac{n_1}{H_1} + \frac{n_2}{H_2}

Total observations =

\displaystyle n_1 + n_2

**Step 3: Combined HM.**

H = \frac{n_1 + n_2}{\frac{n_1}{H_1} + \frac{n_2}{H_2}}

This matches **Option C**. Note: The given correct option is A, but mathematically the correct formula for combined HM is Option C. Hence, **Option C** is the correct answer.

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If there are two groups with n1\displaystyle n_1n1​ and n2\displaystyle n_2n2​ observations and H1\displaystyle H_1H1​ and H2\displaystyle H_2H2​ are respective harmonic means, then the harmonic mean of combined observation is