Question

in distributions that are skewed to the left, what is the relationship of the mean, median, and mode?

a. \\(\text{mode} > \text{mean} > \text{median}\\)
b. \\(\text{mode} > \text{median} > \text{mean}\\)
c. \\(\text{mean} > \text{median} > \text{mode}\\)
d. \\(\text{mode} < \text{mean} < \text{median}\\)

Explanation:

⚡ Using what you learned: Describing Distributions

Step 1: Analyze Left-Skewed Distributions

In a left-skewed (negatively skewed) distribution, the tail of the distribution extends to the left toward the lower values.

• Mode: The peak of the distribution remains at the most frequent value, which is on the right side.
• Mean: The mean is highly sensitive to extreme values. The unusually small values in the left tail pull the mean down toward the left.
• Median: The median (the middle value) is more robust than the mean and sits between the mode and the mean.

Step 2: Determine the Inequality Relationship

Because the mean is pulled furthest to the left (lowest value) and the mode remains at the peak on the right (highest value), the order from smallest to largest is:

$$ \text{mean} < \text{median} < \text{mode} $$

Rewriting this from largest to smallest:

$$ \text{mode} > \text{median} > \text{mean} $$

Answer:

B. mode > median > mean