For certain data, the mode is $ 24.6 $ and mean is $ 20.1 $. W... | Mathematics MCQ

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For certain data, the mode is $24.6$ and mean is $20.1$ . What is the median of that data, using empirical formula?

22.2

23.5

24.1

21.6

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option D

21.6

Quick Summary:

Rearrange the empirical formula for median: $\text{Median} = \frac{\text{Mode} + 2 \times \text{Mean}}{3}$ and substitute the given values.

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📐MAMath SolutionDirect Formula

📋 Given

Mode = 24.6, Mean = 20.1

🔢 Formula Used

Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}Mode=3×Median−2×Mean

⚡ Exam Hall Shortcut / Speed Trick

Rearrange the empirical formula for median: $\text{Median} = \frac{\text{Mode} + 2 \times \text{Mean}}{3}$ and substitute the given values.

⚠️ Common Student Trap / Pitfall

Confusing the coefficients of Median (3) and Mean (2) in the empirical relationship between mean, median, and mode.

🔢 Step-by-Step Solution

State Empirical Formula

The relationship between mean, median, and mode for a moderately skewed distribution is given by the empirical formula:

Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}Mode=3×Median−2×Mean

Substitute Given Values

Substitute Mode = $24.6$ and Mean = $20.1$ into the formula:

24.6=3×Median−2×20.124.6 = 3 \times \text{Median} - 2 \times 20.124.6=3×Median−2×20.1

Calculate the Median

Simplify and solve for Median:

24.6=3×Median−40.224.6 = 3 \times \text{Median} - 40.224.6=3×Median−40.2

Final Answer Extraction

Add $40.2$ to both sides and divide by $3$ :

3×Median=24.6+40.2=64.8  ⟹  Median=64.83=21.63 \times \text{Median} = 24.6 + 40.2 = 64.8 \implies \text{Median} = \frac{64.8}{3} = 21.63×Median=24.6+40.2=64.8⟹Median=364.8​=21.6

✅

D is correct because using the empirical formula gives a median of 21.6.

Core Concepts Used

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Empirical Relationship between Mean, Median, and Mode Measures of Central Tendency Skewed Distributions

💡 EXAM TIP

For symmetrical distributions, mean = median = mode. For moderately asymmetrical distributions, always use the empirical formula $\text{Mode} = 3Median - 2Mean$ .

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