Calculate the Harmonic Mean of 1 , 1/3 , 1/6 and 1/9

Option b: 0.21. Solution: Let us calculate the Harmonic Mean ( HM ) of the observations: x1 = 1 , x2 = 1/3 , x3 = 1/6 , and x4 = 1/9 . 1. Formula for Harmonic Mean: HM = n/sum 1xi Here, the number of observations is n = 4 . 2. Reciprocals of the observations: 1/x1 = 1/1 = 1 1/x2 = 1/1/3 = 3 1/x3 = 1/1/6 = 6 1/x4 = 1/1/9 = 9 3. Sum of the reciprocals: sum 1/xi = 1 + 3 + 6 + 9 = 19 4. Harmonic Mean Calculation: HM = 4/19 ≈ 0.2105 Rounding to two decimal places, we get 0.21 . Hence, Option B is the correct answer.

Measures of Central Tendency and DispersionPYQ Sept 25Question 4174 of 473

All Questions

Calculate the Harmonic Mean of $\displaystyle 1$ , $\displaystyle \frac{1}{3}$ , $\displaystyle \frac{1}{6}$ and $\displaystyle \frac{1}{9}$

Options

A2.48

B0.21

C0.31

D0.25

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Correct Answer

✅ Option b — 0.21

All Options:

A2.48
B0.21
C0.31
D0.25

Detailed Solution & Explanation

Let us calculate the Harmonic Mean (

\displaystyle HM

) of the observations:

\displaystyle x_1 = 1

\displaystyle x_2 = \frac{1}{3}

\displaystyle x_3 = \frac{1}{6}

, and

\displaystyle x_4 = \frac{1}{9}

.
1. **Formula for Harmonic Mean**:

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

Here, the number of observations is

\displaystyle n = 4

.
2. **Reciprocals of the observations**:

\frac{1}{x_1} = \frac{1}{1} = 1

\frac{1}{x_2} = \frac{1}{1/3} = 3

\frac{1}{x_3} = \frac{1}{1/6} = 6

\frac{1}{x_4} = \frac{1}{1/9} = 9

3. **Sum of the reciprocals**:

\sum \frac{1}{x_i} = 1 + 3 + 6 + 9 = 19

4. **Harmonic Mean Calculation**:

HM = \frac{4}{19} \approx 0.2105

Rounding to two decimal places, we get

\displaystyle 0.21

.
Hence, **Option B** 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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Calculate the Harmonic Mean of 1\displaystyle 11, 13\displaystyle \frac{1}{3}31​, 16\displaystyle \frac{1}{6}61​ and 19\displaystyle \frac{1}{9}91​