Question

An incoming freshman took her college's placement exams in French and mathematics. In French, she scored 82 and in math 86 . The overall results on the French exam had a mean of 72 and a standard deviation of 8 , while the mean math score was 68 , with a standard deviation of \(12 .\) On which exam did she do better compared with the other freshmen?

Short answer

She did better in math based on the z-scores.

Step-by-step solution

Understand the Problem

We need to compare the student's performance on two exams: French and Math. To determine which exam she did better on compared to others, we'll use the concept of z-scores, which measure how many standard deviations a score is from the mean of the distribution.

Calculate Z-Score for French

To calculate the z-score for the French exam, use the formula: \( z = \frac{X - \mu}{\sigma} \), where \( X \) is the student's score, \( \mu \) is the mean score, and \( \sigma \) is the standard deviation. Substituting in the given values for French: \( X = 82 \), \( \mu = 72 \), and \( \sigma = 8 \), we get: \[ z = \frac{82 - 72}{8} = 1.25 \].

Calculate Z-Score for Math

Repeat the process to calculate the z-score for the Math exam using the same formula. For Math: \( X = 86 \), \( \mu = 68 \), and \( \sigma = 12 \). Therefore, we have: \[ z = \frac{86 - 68}{12} = 1.5 \].

Compare Z-Scores

The z-score tells us how far each score is from the mean in terms of standard deviations. A higher z-score indicates a better performance relative to the mean of that exam. For French, the z-score is 1.25, and for Math, the z-score is 1.5. Thus, the student did better on the Math exam compared to her peers, as the z-score is higher.

Key concepts

Understanding Placement Exams

Placement exams are essential in determining where an incoming student stands academically in specific subjects.
These exams place students in the appropriate level of coursework, ensuring they are neither under-challenged nor overwhelmed.
They typically assess knowledge that should have been gained before entering college and help schools decide optimal class levels for each student.

• Placement exams can cover a variety of subjects such as languages, mathematics, and sciences.
• They serve as a diagnostic tool for educators to identify strengths and areas for improvement in students.
• The outcome directs students to their starting point in a college curriculum.

Understanding how placement exams relate to overall academic performance is key for students.
While absolute scores are important, comparing scores to the performance of others is also crucial.

What is Standard Deviation?

Standard deviation is a measure of the amount of variation or dispersion in a set of values.
In the context of placement exams, it helps us understand the spread of students' scores and how any individual score compares to the group's scores.
A smaller standard deviation means scores are more closely clustered around the mean. A larger one shows that scores are spread out over a wider range of values.

• Standard deviation provides an insight into the predictability and consistency of student performances.
• It's used in conjunction with the mean to calculate z-scores, which shows the relative performance of individual scores in a distribution.
• This measure is critical in educational settings, as it identifies the range within which most scores fall.

By understanding standard deviation, students and educators better interpret exam results and potential impacts on course placements.

Grasping Mean Score

The mean score is the average score of a set of numbers, calculated by adding all scores and dividing by the number of scores.
In placement exams, the mean score provides a central benchmark to which individual scores can be compared.

• The mean helps establish what is considered a 'typical' score in a group.
• It acts as a reference point for calculating z-scores, offering a way to assess an individual's performance relative to peers.
• If a student scores above the mean, their performance is considered better than average.

Understanding the mean helps students view their position within a larger group context.
It is an essential component in assessing academic readiness and in planning educational paths.