Question

indira makes a box-and-whisker plot of her data. she finds that the distance from the minimum value to the first quartile is greater than the distance between the third quartile and the maximum value. which is most likely true?
the mean is greater than the median because the data is skewed to the right.
the mean is greater than the median because the data is skewed to the left.
the mean is less than the median because the data is skewed to the right.
the mean is less than the median because the data is skewed to the left.

Explanation:

1. Recall the concept of skewness in box - and - whisker plots:

• The distance from the minimum to the first quartile (\(Q_1 - \text{min}\)) and the distance from the third quartile to the maximum (\(\text{max}-Q_3\)) give an idea about the skewness of the data.
• If \(Q_1 - \text{min}>\text{max}-Q_3\), it means the left - hand tail (from min to \(Q_1\)) is longer than the right - hand tail (from \(Q_3\) to max), so the data is skewed to the left.

1. Recall the relationship between mean and median in skewed data:

• For left - skewed (negatively skewed) data, the mean is pulled towards the left - hand tail (towards the smaller values) and is less than the median. For right - skewed (positively skewed) data, the mean is pulled towards the right - hand tail (towards the larger values) and is greater than the median.
• Let's analyze each option:
• Option 1: If data is skewed to the right, \(\text{max}-Q_3>Q_1 - \text{min}\), and mean > median. But our data has \(Q_1 - \text{min}>\text{max}-Q_3\), so this is wrong.
• Option 2: For left - skewed data, mean < median, not mean > median. So this is wrong.
• Option 3: If data is skewed to the right, \(\text{max}-Q_3>Q_1 - \text{min}\), and mean > median. But our data has \(Q_1 - \text{min}>\text{max}-Q_3\), so this is wrong.
• Option 4: Since the data is skewed to the left (because \(Q_1 - \text{min}>\text{max}-Q_3\)) and for left - skewed data, the mean is less than the median, this option is correct.

Answer:

D. The mean is less than the median because the data is skewed to the left.