If Arithmetic Mean and Median of a Series are 10 and 15 respectively, what will be the value of mode ?

Question

Accepted Answer

The correct option is 2:
The correct option is 2: 25.
The calculation is based on the empirical relationship between the Mean, Median, and Mode for a moderately skewed distribution, a core concept in statistics. The formula states that Mode is approximately equal to three times the Median minus two times the Mean (Mode $\approx 3 \times \text{Median} - 2 \times \text{Mean}$). This approximation is widely used when dealing with asymmetrical distributions, as it reflects the distance between the measures of central tendency in such series. By substituting the given values into the formula, where the Arithmetic Mean is 10 and the Median is 15, the calculation is $3 \times 15 - 2 \times 10 = 45 - 20$, which results in an estimated Mode of 25. This value precisely matches Option 2, confirming it as the correct answer derived from statistical principles.

The incorrect option is 1:
The value 22.5 is the result of an incorrect application or a basic mathematical error when using the empirical formula. It is crucial to remember the exact weightings in the relationship: the Median is tripled and the Mean is doubled before the subtraction. Any confusion in these coefficients—for example, using incorrect multipliers—will lead to an answer like 22.5. In a moderately skewed series, the Mean, Median, and Mode maintain a fixed approximate distance from each other, which the standard formula accurately captures. The value 22.5 does not respect this statistical proportionality based on the given Mean and Median values.

The incorrect option is 3:
The value 25.82 is numerically close to the correct answer but does not arise from the straightforward application of the standard empirical formula. This small deviation indicates that it might be the result of a computational error or a flawed rounding in a more complex, multi-step calculation that is unnecessary for this problem. When encountering a straightforward question about the relationship between the three central tendencies, one must rely on the widely accepted empirical relationship. Therefore, 25.82, being outside the result of the standard formula, must be dismissed as an incorrect estimate.

The incorrect option is 4:
The value 22.82 is an incorrect result and is lower than the value yielded by the standard formula. This discrepancy confirms that the underlying calculation that produced this figure was erroneous. It is important for students to correctly recall and apply the formula Mode $\approx 3 \times \text{Median} - 2 \times \text{Mean}$ to avoid selecting a statistically inaccurate answer like 22.82. The purpose of this formula is to provide the best possible estimate, and any figure other than 25 suggests a failure to use the established relationship properly.