Problem 17 - Calculus for Biology and Medicine

Question

The following table contains the scores of 25 students on a certain exam: \(\begin{array}{rrrrr}7 & 8 & 8 & 3 & 2 \\ 5 & 6 & 9 & 10 & 6 \\ 8 & 8 & 7 & 6 & 9 \\ 10 & 4 & 4 & 8 & 6 \\ 9 & 10 & 5 & 5 & 8\end{array}\) (a) Find the relative frequency distribution. (b) Compute the average value by (i) averaging the values in the table directly and (ii) using the relative frequency distribution obtained in (a).

Step-by-Step Solution

Answer

(a) Relative frequencies: 2: 0.04, 3: 0.04, 4: 0.08, 5: 0.12, 6: 0.16, 7: 0.08, 8: 0.24, 9: 0.12, 10: 0.12. (b) Average: 7.04 from both methods.

Step 1: Organize Data for Frequency Distribution

First, list all unique scores from the table in ascending order: 2, 3, 4, 5, 6, 7, 8, 9, 10. Count how many times each score appears to create a frequency distribution.

Step 2: Calculate Frequency of Each Score

The frequency of each score is: - 2 occurs 1 time - 3 occurs 1 time - 4 occurs 2 times - 5 occurs 3 times - 6 occurs 4 times - 7 occurs 2 times - 8 occurs 6 times - 9 occurs 3 times - 10 occurs 3 times

Step 3: Determine Relative Frequency

Relative frequency is calculated by dividing the frequency of each score by the total number of scores, 25. - 2: \( \frac{1}{25} = 0.04 \)- 3: \( \frac{1}{25} = 0.04 \)- 4: \( \frac{2}{25} = 0.08 \)- 5: \( \frac{3}{25} = 0.12 \)- 6: \( \frac{4}{25} = 0.16 \)- 7: \( \frac{2}{25} = 0.08 \)- 8: \( \frac{6}{25} = 0.24 \)- 9: \( \frac{3}{25} = 0.12 \)- 10: \( \frac{3}{25} = 0.12 \)

Step 4: Average Directly from Data

Add up the scores directly from the table and divide by the number of scores (25). \(\text{Average} = \frac{7 + 8 + 8 + 3 + 2 + 5 + 6 + 9 + 10 + 6 + 8 + 8 + 7 + 6 + 9 + 10 + 4 + 4 + 8 + 6 + 9 + 10 + 5 + 5 + 8}{25} = \frac{176}{25} = 7.04\)

Step 5: Average Using Relative Frequency

Multiply each score by its relative frequency and sum the results for the average. \(\text{Average} = (2 \times 0.04) + (3 \times 0.04) + (4 \times 0.08) + (5 \times 0.12) + (6 \times 0.16) + \)\((7 \times 0.08) + (8 \times 0.24) + (9 \times 0.12) + (10 \times 0.12) = 7.04\)

Key Concepts

Understanding Average Value Calculation

Calculating an average is like finding the center point of a data set. It gives you a single value to represent a group of numbers. There are a couple of ways to calculate the average, also known as the mean, especially when dealing with data. Let's explore two methods: direct calculation from the raw data and using relative frequency distribution.

• **Direct Method**: Add up all the values and divide by the total number of values. In the problem, the scores add up to 176. Since there are 25 scores, divide 176 by 25 which results in an average of 7.04.
• **Relative Frequency Method**: Multiply each score by its relative frequency (which is a fraction showing how often each score occurred compared to the total) and then sum these products. This still gives us the same result as directly averaging the scores.

Both methods should, and do, yield the same average if done correctly. This is useful because it means we can use whichever method is more convenient for our situation.

Exploring Frequency Distribution

Frequency distribution is a tool that helps visualize data to see how often each value in a dataset occurs. It's like grouping similar items together to understand the overall picture quickly.

• First, identify all unique numbers in your data set.
• Count how many times each number appears. This count is known as the frequency.

In the problem, the frequency of scores ranged from 2 to 10. For example, the score of 8 occurred six times. This shows us through frequency distribution how the different scores are spread out within a group of students. Seeing which scores appear most often can tell us a lot about the dataset, like the most common outcome or which scores are rare.

The beauty of a frequency distribution is that it simplifies potentially complex data into something understandable. It's a way of organizing data which can later allow us to perform further analysis like calculating relative frequencies.

The Role of Data Analysis

Data analysis is the process of evaluating data to extract insightful conclusions. The objective is to bring clarity into understanding the data, help in identifying trends, and solve questions

Data analysis begins with organizing the data, as we did in our frequency distribution, to see the bigger picture easily. After determining the frequency and relative frequencies, as we did earlier, deeper insights can be revealed:

• **Highlight Trends or Patterns**: Data analysis might show trends, like whether scores are bunched around a particular value or if certain scores are missing.
• **Decision Making**: Knowing that the average score is 7.04 helps educators understand student performance levels.

In our problem's context, we carried out basic descriptive data analysis steps. Calculating averages and frequency is just the beginning. A complete analysis might look into why certain scores are more frequent and others less so, tapping into the potential for further inquiry or enhancing overall understanding, such as potential areas for teaching focus.