Question

High school students in an International Baccalaureate (IB) program are placed in accelerated or advanced courses and must take IB examinations in each of six subject areas at the end of their junior or senior year. Students are scored on a scale of \(1-7,\) with \(1-2\) being poor, 3 mediocre, 4 average, and \(5-7\) excellent. During its first year of operation at John W. North High School in Riverside, California, 17 juniors attempted the IB economics exam, with these results: $$ \begin{array}{cc} \text { Exam Grade } & \text { Number of Students } \\ \hline 7 & 1 \\ 6 & 4 \\ 5 & 4 \\ 4 & 4 \\ 3 & 4 \end{array} $$ Calculate the mean and standard deviation for these scores.

Short answer

Answer: The mean is approximately 4.65 and the standard deviation is approximately 1.04.

Step-by-step solution

Calculate the Total Score and Number of Students

First, we will calculate the total score by multiplying the exam grade by the number of students receiving that grade and then summing up the results. We will also find the total number of students who took the exam. Total Score: \((7*1) + (6*4) + (5*4) + (4*4) + (3*4) = 7 + 24 + 20 + 16 + 12 = 79\) Total Number of Students: \(1 + 4 + 4 + 4 + 4 = 17\)

Calculate the Mean

We will now find the mean by dividing the total score by the total number of students: Mean = \(\frac{Total Score}{Total Number of Students} = \frac{79}{17} \approx 4.65\)

Calculate the Squared Differences from the Mean

Next, we will calculate the squared differences of each score from the mean: \((7 - 4.65)^2 = 5.5225, (6 - 4.65)^2 = 1.8225, (5 - 4.65)^2 = 0.1225, (4 - 4.65)^2 = 0.4225,\) and \((3 - 4.65)^2 = 2.7225\)

Calculate the Average of the Squared Differences

We will now calculate the average of the squared differences by multiplying each squared difference by the number of students receiving that grade, summing the results, and dividing by the total number of students: Average of squared differences = \(\frac{(5.5225*1) + (1.8225*4) + (0.1225*4) + (0.4225*4) + (2.7225*4)}{17} \approx 1.080882\)

Calculate the Standard Deviation

Finally, we will calculate the standard deviation by taking the square root of the average of the squared differences: Standard Deviation = \(\sqrt{1.080882} \approx 1.04\) Thus, the mean for these scores is approximately \(4.65\), and the standard deviation is approximately \(1.04\).

Key concepts

Mean Calculation

Calculating the mean is one of the fundamental concepts in statistics. It represents the average value of a dataset. To find the mean, you sum up all the values and divide by the total number of values. In the context of the International Baccalaureate program exam scores, the mean gives us a picture of how well the group performed collectively.
In our exercise, students scored between 3 and 7. By multiplying the score by the number of students achieving each score, and summing them, we find a total score of 79. Then, we divide this sum by the total number of students, 17, to get the mean:

\[ \text{Mean} = \frac{79}{17} \approx 4.65 \]

This mean of approximately 4.65 indicates that, as a group, the students performed slightly above average in their economics exam.

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range.
In the IB program exam scenario, calculating the standard deviation helps us understand how students' scores deviate from the mean of 4.65. To find it, we calculate the squared differences between each score and the mean, multiply each squared difference by the number of students for each score, and then find the average of these squared differences.

\[ \text{Average of squared differences} = \frac{(5.5225 \times 1) + (1.8225 \times 4) + (0.1225 \times 4) + (0.4225 \times 4) + (2.7225 \times 4)}{17} \approx 1.080882 \]

Finally, taking the square root of this average gives us the standard deviation:

\[ \text{Standard Deviation} = \sqrt{1.080882} \approx 1.04 \]

This result shows a moderate dispersion of exam scores around the mean.

Probability Distribution

Understanding probability distributions is crucial in statistics. It describes how the probabilities are distributed over the values of a random variable. In this case, the probability distribution of students' scores can guide predictions for future exam outcomes.
The exam score distribution reflects the chances of students receiving a particular score. This distribution can be viewed from the rates of occurrence in our dataset:

• Score of 7: 1 student
• Score of 6: 4 students
• Score of 5: 4 students
• Score of 4: 4 students
• Score of 3: 4 students

The highest frequency is within scores 3 to 4, suggesting that most students are scoring at or just above average.

International Baccalaureate Program

The International Baccalaureate (IB) program is recognized globally for its rigorous education standards for students, particularly in high schools. It aims to develop knowledgeable and disciplined students who can think critically and manage complex problems in all their six-subject areas.
Scores in the IB program exams reflect the students' grasp and understanding of the subject knowledge. Scores range from 1 to 7, with 4 being the average. In our example, a mean score of approximately 4.65 reflects that students in the economics course performed relatively well. This outcome supports the idea that IB students, enrolled in advanced or accelerated courses, tend to exhibit high performance levels as they are better prepared for academic challenges.