How are quartiles typically determined in a data set?

• A. They are found by the overall mean and median.
• B. Q1 is the median of the lower half and Q3 is the median of the upper half.
• C. They are simply the minimum and maximum values.
• D. They are the 20th and 80th percentiles.

Answer: (B)

Quartiles are the points that divide a data set into four equal parts. To determine them, first order the data from smallest to largest, then split the data into a lower half and an upper half around the overall median. The first quartile is the median of the lower half, and the third quartile is the median of the upper half. This method places about 25% of observations at or below the first quartile and about 75% at or below the third quartile.

For example, with data 1, 2, 3, 4, 5, 6, 7, 8, the lower half is 1–4 and its median is (2 + 3) / 2 = 2.5, so Q1 = 2.5. The upper half is 5–8 and its median is (6 + 7) / 2 = 6.5, so Q3 = 6.5. If there’s an odd number of observations, the middle value is typically excluded from the halves, so Q1 and Q3 come from the medians of the lower and upper halves (e.g., for 1, 2, 3, 4, 5, Q1 = 1.5 and Q3 = 4.5). These correspond to the 25th and 75th percentiles under this approach.