A dataset consists of 20 distinct numerical values arranged in increasing order. The lower… — GMAT

A dataset consists of 20 distinct numerical values arranged in increasing order. The lower quartile ( $Q_1$ ) of the dataset is equal to the average of the 5th and 6th values, and the upper quartile ( $Q_3$ ) is equal to the average of the 15th and 16th values. Given that the lower quartile $Q_1 = 42$ , the interquartile range ( $\text{IQR} = Q_3 - Q_1$ ) is 38, and the 15th value in the dataset is 74, what is the value of the 16th term?

Answer: 86

Answer

To find the 16th value, first determine the upper quartile

Q_3

using the interquartile range:

Q_3 = Q_1 + \text{IQR} = 42 + 38 = 80

. Since

Q_3

is the average of the 15th and 16th terms, set up the equation

\frac{74 + x_{16}}{2} = 80

. Multiplying by 2 gives

74 + x_{16} = 160

, so

x_{16} = 86

Step-by-Step Solution

Calculate the upper quartile (

Q_3

) using the given lower quartile (

Q_1

) and interquartile range (IQR).

Q_3 = Q_1 + \text{IQR} = 42 + 38 = 80

By definition, the interquartile range is the difference between the upper and lower quartiles (

\text{IQR} = Q_3 - Q_1

Relate the upper quartile (

Q_3

) to the 15th and 16th values of the ordered dataset.

Q_3 = \frac{x_{15} + x_{16}}{2} = 80

For an ordered dataset of 20 elements,

Q_3

is the median of the upper half of the data (the 11th through 20th terms), which equals the arithmetic mean of the 15th and 16th terms.

Substitute the known value of the 15th term (

x_{15} = 74

) into the equation and solve for the 16th term (

x_{16}

\frac{74 + x_{16}}{2} = 80 \implies 74 + x_{16} = 160 \implies x_{16} = 86

Multiplying both sides by 2 gives 160, and subtracting 74 yields 86.

Key Concept

Interquartile Range and Quartile Calculations

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