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The mode and median of a data is 26.7 and 71, respectively. What is the mean of the data? (Use empirical formula.)
95.1
93.2
94.6
94.3
93.2
Use the approximation: Mean=23├ЧMedianтИТModeтАЛ. Since 3├Ч71=213, subtract 26.7 to get 186.3, then halve it to get 93.15, which rounds to 93.2.
Mode (Z) = 26.7, Median (M) = 71.
Mode=3├ЧMedianтИТ2├ЧMean
Use the approximation: Mean=23├ЧMedianтИТModeтАЛ. Since 3├Ч71=213, subtract 26.7 to get 186.3, then halve it to get 93.15, which rounds to 93.2.
Students often swap the coefficients in the formula, mistakenly writing Mean=3├ЧModeтИТ2├ЧMedian instead of the correct empirical relation.
Identify the Empirical Relationship
The relationship between Mean, Median, and Mode for a moderately skewed distribution is given by the empirical formula.
Z=3MтИТ2X
Substitute the Given Values
Insert the values of Mode (Z=26.7) and Median (M=71) into the formula.
26.7=3(71)тИТ2X
Calculate the Mean
Perform the arithmetic: 3├Ч71=213. Then, isolate the Mean term: 2X=213тИТ26.7=186.3. Finally, divide by 2.
X=2186.3тАЛ=93.15
B is correct because the calculated mean is 93.15, which rounds to the provided option value 93.2.
This empirical formula is frequently used in Statistical Quality Control and competitive aptitude sections to relate distribution parameters when direct data points are unavailable.
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