The mode is the value that appears most frequently in a data set, while the median is the middle value of a data set when it is arranged in order. The mean, also known as the average, is calculated by summing up all the values in a data set and dividing by the total number of values.

To understand why the difference between the mode and median is 2, let's consider an example. Suppose we have a data set of 5, 6, 8, 8, 9, and 10. In this case, the mode is 8 because it appears twice, while the median is also 8 because it is the middle value. The difference between the mode and median is 0 because they are the same value.

Now, let's consider a different example. Suppose we have a data set of 2, 3, 4, 4, 5, 5, 5, and 6. In this case, the mode is 5 because it appears three times, while the median is 4.5 because it is the average of the two middle values (4 and 5). The difference between the mode and median is 5 - 4.5 = 0.5.

From these examples, we can observe that the difference between the mode and median can vary. However, the question states that the difference is 2. Therefore, we need to find a scenario where the mode is 2 more than the median.

Let's consider a data set where the median is 3. In this case, the mode must be 3 + 2 = 5. To maintain this difference of 2, we can add other values to the data set, such as 5, 5, 5, and 5, so that the mode remains 5. The mean in this case would be (3 + 5 + 5 + 5 + 5 + 5) / 6 = 4.5.

Hence, we can conclude that if the difference between the mode and median is 2, the difference between the median and mean would be 4.