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The mode and median of a data is 19.5 and 85, respectively. What is the mean of the data? (Use empirical formula.)
119.1
118
117.8
118.6
117.8
Use the relationship 2├ЧMean=3├ЧMedianтИТMode. Calculate 3├Ч85=255, subtract 19.5 to get 235.5, then divide by 2 to find the mean.
Mode (Mo) = 19.5, Median (Me) = 85. We need to find the Mean (X).
Mode=3├ЧMedianтИТ2├ЧMean
Use the relationship 2├ЧMean=3├ЧMedianтИТMode. Calculate 3├Ч85=255, subtract 19.5 to get 235.5, then divide by 2 to find the mean.
Students often confuse the order of the empirical formula, frequently writing Mean=3├ЧMedianтИТ2├ЧMode, which leads to an incorrect result.
State the empirical relationship
The relationship between Mean, Median, and Mode for a moderately skewed distribution is given by the empirical formula.
Mode=3├ЧMedianтИТ2├ЧMean
Substitute the given values
Substitute the values Mode=19.5 and Median=85 into the formula.
19.5=3(85)тИТ2(Mean)
Simplify and solve for Mean
Perform the multiplication and algebraic rearrangement: 19.5=255тИТ2├ЧMean. This implies 2├ЧMean=255тИТ19.5=235.5.
Mean=2235.5тАЛ=117.75
Final Result Verification
The calculated mean is 117.75, which rounds to 117.8 as per the provided options.
MeanтЙИ117.8
C is correct because the application of the empirical formula Mode=3├ЧMedianтИТ2├ЧMean yields 117.75, which aligns with option C (117.8).
This empirical formula is a standard relationship in descriptive statistics; always remember that the mode is generally 'pulled' towards the mean in skewed datasets, making this relation robust for exam questions.
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