Question

which of the following describes the probability distribution below?

probability distribution

chart showing probability distribution with x-values 1, 2, 3, 4, 5 and corresponding probabilities

• the median is greater than the mean, and the majority of the data points are to the left of the mean.
• the median is greater than the mean, and the majority of the data points are to the right of the mean.
• the mean is greater than the median, and the majority of the data points are to the left of the mean.
• the mean is greater than the median, and the majority of the data points are to the right of the mean.

Explanation:

Identify the distribution shape

Using the Skewed Left knowledge point
The probability distribution has a long tail extending to the left (lower values \(x = 1, 2, 3\)) with the peak concentrated on the right at \(x = 4\). This shape is classified as left-skewed (skewed left).

Compare the mean and median

Using the Mean vs Median in Skewed Data knowledge point
For a left-skewed distribution, the mean is pulled down toward the long left tail by the lower values, making the mean less than the median:

$$ \text{mean} < \text{median} \quad \text{or} \quad \text{median} > \text{mean} $$

Determine the majority of data points relative to the mean

Using the Mean vs Median in Skewed Data knowledge point
The peak of the distribution is at \(x = 4\), which has a probability of approximately \(0.45\). Since the mean is pulled to the left of the peak (typically between 3 and 4), the majority of the probability mass (data points) lies to the right of the mean (at \(x = 4\)).

Match with the correct option

Using the Mean vs Median in Skewed Data knowledge point
The correct statement is: "The median is greater than the mean, and the majority of the data points are to the right of the mean."

Answer:

• The median is greater than the mean, and the majority of the data points are to the left of the mean.
• The median is greater than the mean, and the majority of the data points are to the right of the mean. (Correct answer)
• The mean is greater than the median, and the majority of the data points are to the left of the mean.
• The mean is greater than the median, and the majority of the data points are to the right of the mean.