In a perfectly symmetric distribution, which statement is true about quartile distances?

In a perfectly symmetric distribution, which statement is true about quartile distances?

• A. The distance from Q1 to Q2 equals the distance from Q2 to Q3.
• B. Q1 equals Q2 equals Q3.
• C. Q2 equals the mean.
• D. Q3 equals the maximum.

Answer: (A)

The main idea is that symmetry around the center makes the central portion of the distribution evenly spaced. In a perfectly symmetric distribution, the quartiles sit at equal distances from the median: Q1 is 25th percentile, Q2 is the median, and Q3 is the 75th percentile. Because the left and right halves mirror each other, the gap from Q1 up to Q2 is the same as the gap from Q2 up to Q3. This direct consequence of symmetry is why the statement about equal distances is true. For context, think of the data arranged from smallest to largest and divided into four equal-area parts around the center. The center acts as a mirror, so the distances on either side of the median match. The other statements either describe different quantities or rely on extra conditions. For example, Q3 equaling the maximum would only happen in a degenerate dataset where all values are the same as the maximum, which isn’t what symmetry implies. And while Q2 equaling the mean can occur in some symmetric distributions with a finite mean, that fact doesn’t address the quartile gaps, which is the focus of the question.

The main idea is that symmetry around the center makes the central portion of the distribution evenly spaced. In a perfectly symmetric distribution, the quartiles sit at equal distances from the median: Q1 is 25th percentile, Q2 is the median, and Q3 is the 75th percentile. Because the left and right halves mirror each other, the gap from Q1 up to Q2 is the same as the gap from Q2 up to Q3. This direct consequence of symmetry is why the statement about equal distances is true.

For context, think of the data arranged from smallest to largest and divided into four equal-area parts around the center. The center acts as a mirror, so the distances on either side of the median match.

The other statements either describe different quantities or rely on extra conditions. For example, Q3 equaling the maximum would only happen in a degenerate dataset where all values are the same as the maximum, which isn’t what symmetry implies. And while Q2 equaling the mean can occur in some symmetric distributions with a finite mean, that fact doesn’t address the quartile gaps, which is the focus of the question.