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If the mean of a distribution is 12.5 and its median is 13.5, then the mode of the distribution is __________ (using the empirical relation Mode = 3 Median - 2 Mean).
14.5
15.5
16.5
13.5
15.5
Use the deviation logic: Mode - Mean = 3(Median - Mean). Here, Median - Mean = 1, so Mode - Mean = 3(1) = 3, implying Mode = 12.5 + 3 = 15.5.
Mean = 12.5 and Median = 13.5
Mode=3├ЧMedianтИТ2├ЧMean
Use the deviation logic: Mode - Mean = 3(Median - Mean). Here, Median - Mean = 1, so Mode - Mean = 3(1) = 3, implying Mode = 12.5 + 3 = 15.5.
Many students accidentally swap the coefficients, calculating 2 times the Median minus 3 times the Mean, leading to incorrect results.
State the Empirical Relation
To find the mode of a symmetrical or moderately skewed distribution, use the Pearson's empirical formula relating the three measures of central tendency.
Mode=3├ЧMedianтИТ2├ЧMean
Substitute the Given Values
Substitute the given values for mean (12.5) and median (13.5) into the formula.
Mode=3(13.5)тИТ2(12.5)
Perform Calculation
Calculate the products: 3├Ч13.5=40.5 and 2├Ч12.5=25. Subtract the values to find the final mode.
Mode=40.5тИТ25=15.5
B is correct because the calculated value using the empirical formula is 40.5тИТ25=15.5.
This empirical relationship is valid only for moderately skewed distributions; it is a standard tool in statistics questions in exams like JEE and SSC CGL.