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 + n2/1H1 + 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: sumall 1/xi = n1/H1 + n2/H2 Total observations = n1 + n2 Step 3: Combined HM. H = n1 + n2/1H1 + 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.

Measures of Central Tendency and DispersionMCQPYQ 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.

About This Chapter: Measures of Central Tendency and Dispersion

Paper

Paper 3: Quantitative Aptitude

Weightage

12-15 Marks

Key Topics

Mean, Median, Mode, Range, Mean Deviation, Standard Deviation

The core foundation of Statistics. This chapter covers Mean (Arithmetic, Geometric, Harmonic), Median, Mode, and their properties. It also explores measures of spread like Range, Mean Deviation, Standard Deviation, and Quartile Deviation.

View Official ICAI Syllabus

Exam Strategy Tip

Do not just memorize formulas; ICAI loves asking about the mathematical properties (e.g., 'sum of deviations from the AM is always zero'). You can usually eliminate 2 options just by knowing the properties.

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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