Consider the distribution of exam scores (graded from 0 to 100) for 86 students: 38 students received an A, 28 received a B, and 20 received a C. Complete the following parts: a. How many peaks would you expect in the distribution?

In statistics, the distribution of data points is often examined to determine its shape, which can help in understanding the underlying patterns and trends. One way the shape of a distribution is described is by its number of peaks, which are referred to as "modes." A distribution can be unimodal (one peak), bimodal (two peaks), trimodal (three peaks), or multimodal (more than three peaks).

Given the distribution of exam scores for the 86 students, you mentioned that there are three distinct groups of students based on their grades (38 got an A, 28 got a B, and 20 got a C). Assuming that each grade category (A, B, and C) results in a concentration of scores around certain values within the 0 to 100 range, this would suggest different peaks for each group.

Therefore, based on the given information, you would expect the distribution to potentially have three peaks, making it trimodal. One peak would correspond to the scores where students received an A, another peak would correspond to the scores where students received a B, and the third peak would correspond to the scores where students received a C.

However, it is important to note that seeing these actual peaks would depend on the specific score distribution within each grade category. If the individual scores within each of the A, B, or C categories are also spread out or concentrated around one or two score ranges, the actual observed peaks may vary. As such, without more specific score data, we are simply making an assumption based on the counts of each grade.

Extra: The mode of a distribution is the score or value that appears most frequently. In a perfectly shaped bell curve, or normal distribution, there is only one peak, and it is also the mean and median of the distribution. However, not all distributions follow this pattern.

A multimodal distribution often indicates that the data may have multiple underlying groups or clusters, each cluster having its own peak. For instance, in a classroom with a trimodal distribution of exam scores, it might be that the three groups reflect different levels of understanding of the material (e.g., high, medium, low) or study habits among students.

It's also worth noting that the presence of a peak in a distribution does not always align with the formal definition of mode in terms of frequency of individual scores. This is because in grouped data, like grades categories, what you're observing is not individual scores, but rather the concentration of scores around a central value for each group. Each peak represents a range of scores where a number of students lie, rather than a specific score that occurs most frequently.