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Question #40997

Difficulty: 8/10 | 🟢 Free

The Question

A grouped data distribution has three classes with class marks $x_1 = 1$ , $x_2 = 2$ , and $x_3 = 3$ , and corresponding frequencies $f_1, f_2, f_3$ . The total frequency is $\sum_{i=1}^3 f_i = 266$ . It is given that the harmonic mean of the distribution is exactly $2$ . If $V$ is the maximum possible variance of this distribution, what is the value of $\frac{133^2}{240} V$ ?

Options

A.
55
B.
66
C.
133
D.
266

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Calculation of Variance for Grouped Data|Harmonic Mean Constraints|Maximum/Minimum of Quadratic Expressions|Global Extrema on Intervals

Why This Question Is Tricky

Common Mistakes

This problem tests the ability to express the variance of a grouped distribution as a function of a single frequency variable, subject to a harmonic mean constraint.

The primary trap lies in the quadratic nature of the variance function.

Students often blindly differentiate and find the vertex of the parabola ( $f_1 = 133$ ), assuming it gives the maximum variance.

However, they must realize that the frequencies are constrained by $f_i \ge 0$ , which restricts the domain of $f_1$ to $[0, 66]$ .

Failing to apply these boundary constraints leads to an incorrect global maximum.

The true maximum occurs at the boundary $f_1 = 66$ , testing the concept of global extrema on closed intervals.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Calculation of Variance for Grouped Data
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free