Question

which of the following probability distribution types has a mean, median, and mode that are all equal?

constant
symmetric
positively skewed
negatively skewed

Explanation:

Analyze the properties of symmetric distributions

In a perfectly symmetric probability distribution, the left half of the distribution is a mirror image of the right half. Because of this perfect balance, the mean (arithmetic average) and the median (middle value) lie exactly at the center of symmetry. If the distribution is unimodal, the highest point (mode) also occurs at this exact center, making the mean, median, and mode all equal.

Analyze skewed distributions

In a positively skewed (skewed right) distribution, the long tail is on the right, which pulls the mean to the right of the median and mode (\(\text{mode} < \text{median} < \text{mean}\)). In a negatively skewed (skewed left) distribution, the long tail is on the left, pulling the mean to the left (\(\text{mean} < \text{median} < \text{mode}\)). Thus, skewed distributions do not have equal mean, median, and mode.

Evaluate the options

Comparing the characteristics, a symmetric distribution is the type where the mean, median, and mode are all equal.

Answer:

• constant
• symmetric (Correct answer)
• positively skewed
• negatively skewed