Question

Mean and median The figure displays two density curves that model different

distributions of quantitative data. Identify the location of the mean and median by letter for each graph. Justify your answers.

Short answer

Part (a) Mean is located at $A$and the median is located at $A$

Part (b) Mean is located at $A$ and the median is also located at $B$

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Concept

Neither of these transformations changes the shape of the distribution.

Part (a) Step 3: Explanation

The graph on the left of $A$ is the inverse of the graph on the right. The distribution is thus symmetric.

Both the median and the mean should be at the center of the distribution for the distribution to be symmetric (at the point $A$).

As a result, A stands for both the mean and the median.

Part (b) Step 1: Given information

Part (b) Step 2: Explanation

The distribution's apex is to the right, and the distribution is skewed to the left, with a tail of more unusual values to the left in the graph.

Because the median is resistant while the mean is not, the mean is more influenced by the left-skewed distribution's extremely high values.

Because both the mean and the median are influenced by unusually high values in the left-skewed distribution, one measure impacts more than the other, neither the median nor the mean is predicted to reside at the distribution's peak. The mean is lower than the median, and neither is at its highest point.

As a result, $A$ denotes the mean and $B$ denotes the median.