Question

The International Baccalaureate (IB) program is an accelerated academic program offered at a growing number of high schools throughout the country. Students enrolled in this 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 \(\mathrm{W}\). 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 4.647, and the standard deviation is approximately 1.217.

Step-by-step solution

Calculate the total sum of the scores and the number of students.

We can use the given table to calculate the total sum of the scores and the number of students. Note that we have 17 students in total. Total sum of scores = (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

To find the mean, we divide the total sum of the scores by the total number of students: Mean = Total sum of scores / Total number of students Mean = 79 / 17 = 4.647

Calculate the differences between scores and the mean

Now, we need to find the difference between each student's score and the mean. We'll use the table to calculate the sum of squared differences. Sum of squared differences = (7-4.647)^2 * 1 + (6-4.647)^2 * 4 + (5-4.647)^2 * 4 + (4-4.647)^2 * 4 + (3-4.647)^2 * 4 = 5.499 + 7.233 + 0.499 + 1.767 + 10.183 = 25.181

Calculate the average of the squared differences

Divide the sum of squared differences by the total number of students to find the average: Average of squared differences = Sum of squared differences / Total number of students Average of squared differences = 25.181 / 17 = 1.481

Calculate the standard deviation

Finally, take the square root of the average of squared differences to find the standard deviation: Standard deviation = √(Average of squared differences) Standard deviation = √(1.481) ≈ 1.217 Hence, the mean and standard deviation for these scores are approximately 4.647 and 1.217, respectively.

Key concepts

Mean Calculation

The mean, often referred to as the average, is a measure that represents the central or typical value in a set of data. In the context of the IB economics exam scores at John W. North High School, the mean is computed by adding up all of the exam scores and then dividing by the number of students who took the exam.

To ensure clarity in calculating the mean, let's consider an everyday scenario: imagine you and four friends have some apples. You have 2 apples, and each friend has 3, 5, 7, and 4 apples respectively. To find out the average number of apples each person has, you'd add up all the apples: 2 + 3 + 5 + 7 + 4 = 21 apples. Then divide by the number of people, which is 5. So, on average, each person has 21 divided by 5, which is 4.2 apples.

Now, applying this principle to the IB economics exam statistics, we sum the products of each exam grade and the corresponding number of students who scored that grade. Finally, we divide this total by the overall number of students. This calculation provides the mean score of the group, giving us an insight into the overall performance level of the students on the exam.

Standard Deviation Calculation

Standard deviation is a statistic that measures the dispersion or spread of a set of data points relative to its mean. A lower standard deviation indicates that the scores are close to the mean, whereas a higher standard deviation signifies that the scores are spread out over a larger range of values.

Imagine if a friend of yours is throwing darts. If they're a good player, most of their darts will cluster tightly around the bullseye; this tight cluster has a low standard deviation. However, if their darts are scattered all around the dartboard, they would have a high standard deviation. In the context of exam scores, a lower standard deviation would mean that most students scored similarly, while a higher standard deviation would indicate varied performance.

To calculate standard deviation, we start by finding the mean, then for each score, we calculate the difference between that score and the mean and square the result (to eliminate negative values). Multiplying these squared differences by the number of students who scored each grade, we get a total that we then divide by the number of students to find the average squared difference. The square root of this average gives us the standard deviation, representing how much the students' scores deviate, on average, from the mean score.

Probability and Statistics

Probability and statistics are branches of mathematics that deal with data collection, analysis, interpretation, and presentation. Probability is about measuring the likelihood of various outcomes, while statistics involves summarizing and making sense of collected data.

In the realm of education, especially in courses like the IB economics exam, understanding these concepts is crucial for interpreting test results. For instance, with the provided exam scores, we can calculate the probability of a student achieving a certain grade based on past performances. If 4 out of 17 students scored a 6, we can say the probability of a student scoring a 6 is roughly 23.5%.

Moreover, statistics such as the mean and standard deviation give us a quantitative insight into the performance variability among students. This can be particularly helpful for educators looking to improve their teaching strategies or identify topics where students may need additional support. Always remember that the goal is not just to compute these values but to understand what they say about the student's learning and the effectiveness of instruction.