If the mean of a data is $ 65 $ and its mode is $ 23 $, then t... | Mathematics MCQ

MathematicsPrevious Year Paper 2024

If the mean of a data is $65$ and its mode is $23$ , then the median using empirical formula is:

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option B

Quick Summary:

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

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📐MAMath SolutionDirect Formula

📋 Given

Mean = 65, Mode = 23

🔢 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 as $\text{Median} = \frac{\text{Mode} + 2 \times \text{Mean}}{3}$ and directly substitute the given values.

⚠️ Common Student Trap / Pitfall

Confusing the coefficient of Mean and Median in the empirical relationship between mean, median, and mode.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Step 1: Write the empirical relationship

The empirical relationship connecting mean, median, and mode in statistics is given by: $\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}$

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

Step 2: Express Median in terms of Mode and Mean

Rearranging the formula to solve for Median: $3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean}\text{Median} = \frac{\text{Mode} + 2 \times \text{Mean}}{3}$

Median=Mode+2×Mean3\text{Median} = \frac{\text{Mode} + 2 \times \text{Mean}}{3}Median=3Mode+2×Mean​

Step 3: Substitute the given values

Substitute $\text{Mean} = 65$ and $\text{Mode} = 23$ into the equation: $\text{Median} = \frac{23 + 2 \times 65}{3}$

Median=23+1303\text{Median} = \frac{23 + 130}{3}Median=323+130​

Step 4: Calculate the final value

$\text{Median} = \frac{153}{3} = 51$

Median=51\text{Median} = 51Median=51

✅

B is correct because using the empirical formula gives a median of 51.

Core Concepts Used

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Empirical relationship between mean, median, and mode Measures of central tendency

💡 EXAM TIP

Questions involving skewed distributions frequently test this precise empirical relationship among mean, median, and mode.

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