Question

Students very often ask their professors whether they will be "curving the grades." The traditional interpretation of "curving grades" required that the grades have a normal distribution, and that the grades will be assigned in these proportions: $$ \begin{array}{l|lllll} \text { Letter Grade } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{F} \\ \hline \text { Proportion of Students } & 10 \% & 20 \% & 40 \% & 20 \% & 10 \% \end{array} $$ a. If the average "C" grade is centered at the average grade for all students, and if we assume that the grades are normally distributed, how many standard deviations on either side of the mean will constitute the "C" grades? b. How many deviations on either side of the mean will be the cutoff points for the "B" and "D" grades?

Short answer

Answer: The "C" grades are within 0.524 standard deviations on either side of the mean. The cutoff points for the "B" and "D" grades are at 0.674 standard deviations on either side of the mean.

Step-by-step solution

Find the value of the "C" grades in terms of z-score

We're given that 40% of students will have "C" grades. As "C" grades are centered at the average grade, we need to find the z-score which corresponds to the proportion of students in the upper and lower tails of the normal distribution (since the "C" grades are between these tails). Since "C" grades represent 40% of the students, there will be (100% - 40%) / 2 = 30% of the students in each tail. We'll use a z-table or a calculator to find the z-score corresponding to the proportion of 0.3.

Find the z-score corresponding to the proportion for "C" grades

Using a z-table or calculator for the cumulative normal distribution, we find that the z-score corresponding to the proportion of 0.3 in the upper and lower tails is approximately 0.524. This means that the "C" grades are within 0.524 standard deviations on either side of the mean.

Find the value of the "B" and "D" grades in terms of z-score

We're given that 20% of students will have "B" and "D" grades. Since these grades are immediate neighbors to the "C" grades, we need to find the z-score corresponding to the proportion of students in the "B" and "D" grades combined. 20% of the students will have a "B" grade, and 20% will have a "D" grade, resulting in a total of 40%. Therefore, using the z-score that we calculated for the "C" grades (0.524), we can find the proportion of students that are beyond that z-score in only one tail (20% - as only one tail, either upper or lower, is considered for "B" and "D" grades).

Find the z-score corresponding to the proportion for "B" and "D" grades

Combining the proportion for the "B" or "D" grade (20%) with the proportion for a tail of the "C" grade (30%), we get a total proportion of 50%. Using a z-table or calculator for the cumulative normal distribution, we find that the z-score corresponding to the proportion of 0.5 in one tail is 0.674. This means that the "B" grades are between 0.524 and 0.674 standard deviations above the mean, and the "D" grades are between 0.524 and 0.674 standard deviations below the mean. To summarize, the "C" grades are within 0.524 standard deviations on either side of the mean, and the cutoff points for the "B" and "D" grades are at 0.674 standard deviations on either side of the mean.

Key concepts

Normal Distribution

When it comes to understanding how grades are allocated, knowledge of the normal distribution is key. This statistical concept helps us visualize how a set of data, like test scores, tends to cluster around a central value, usually the mean. In a perfectly normal distribution, also known as a bell curve, most observations fall near the mean, creating a symmetric shape around it.

This distribution is characterized by two parameters: the mean, which determines the center of the distribution, and the standard deviation, which measures the spread of the data around the mean. When we talk about grading on a curve, we refer to fitting the grades into this type of distribution. For instance, if 40% of students are to receive a 'C', this indicates that their scores lie around the midpoint of the distribution, clustering near the average grade.

Z-Score

A z-score is a crucial statistical measure used to describe the position of a score within a distribution. It represents the number of standard deviations an observation is from the mean. A z-score helps interpret individual scores by comparing them to the distribution as a whole.

In the context of grades, the z-score can identify exactly how far from the mean a specific cutoff point lies. For instance, if we're looking at the 'C' grade cutoff with 40% of students falling within this category, we calculate the z-score to find out how many standard deviations away from the mean this group ranges. Knowing the z-scores for different grades helps in assigning letter grades relative to how everyone in the class has performed.

Standard Deviation

Understanding the standard deviation is fundamental in interpreting how grades are dispersed. It measures the amount of variability or dispersion in a set of scores. A low standard deviation means most of the numbers are close to the average score, while a high standard deviation indicates the numbers are more spread out from the mean.

Grades typically follow a normal distribution where the standard deviation has a direct impact on the spread of the grades. If a professor decides to use a stricter cutoff for letter grades (meaning smaller intervals), this implies using a smaller portion of the standard deviation from the mean. Conversely, more lenient cutoffs would encompass a larger portion of the standard deviation, allowing for a wider range of scores within the same letter grade.

Cumulative Normal Distribution

The cumulative normal distribution plays a pivotal role in grading curves. This function gives us the probability that a random observation from a normally distributed dataset will be less than or equal to a certain value. In simpler terms, it tells us what percentage of scores fall below a specific cutoff point.

With grading, this cumulative aspect helps determine the z-scores for different grade thresholds. For example, in the exercise, we analyze what proportion of students fall into the 'C' category, and by extension, where the cutoff points are for 'B' and 'D' grades. By consulting the cumulative normal distribution, we can set precise points for each grade, thereby ensuring a fair and standardized grading process for students.