Missing Exam Grade A professor has recorded exam grades for 20 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 19 readable scores is 84, what is the value of the unreadable score?

Verified step by step guidance

1. Identify the total number of students, which is 20, and the overall mean score, which is 82.
2. Calculate the total sum of all 20 exam scores using the formula for the mean: \(\text{Total Sum} = \text{Mean} \times \text{Number of Students} = 82 \times 20\).
3. Note that the mean of the 19 readable scores is 84, so calculate the sum of these 19 scores: \(\text{Sum of 19 scores} = 84 \times 19\).
4. Set up an equation where the total sum of all 20 scores equals the sum of the 19 readable scores plus the unreadable score: \(\text{Total Sum} = \text{Sum of 19 scores} + \text{Unreadable Score}\).
5. Solve for the unreadable score by subtracting the sum of the 19 readable scores from the total sum: \(\text{Unreadable Score} = \text{Total Sum} - \text{Sum of 19 scores}\).

Key Concepts

Mean (Average)

The mean is the sum of all values divided by the number of values. It represents the central tendency of a data set. In this problem, the overall mean and the mean of a subset of scores are used to find the missing value.

Sum from Mean

Since the mean equals the total sum divided by the count, the total sum can be found by multiplying the mean by the number of observations. This allows calculation of the total sum of all scores and the sum of the known scores.

Solving for an Unknown Value

When one value is missing, it can be found by subtracting the sum of known values from the total sum. This approach uses algebraic manipulation to isolate and determine the unknown exam grade.