91Ó°ÊÓ

Refer to the data set shown in Table \(15-12 .\) The table shows the scores on a Chem 103 test consisting of 10 questions worth 10 points each. $$ \begin{array}{c|c|c|c} \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \text { Score } & \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \text { Score } \\ \hline 1362 & 50 & 4315 & 70 \\ \hline 1486 & 70 & 4719 & 70 \\ \hline 1721 & 80 & 4951 & 60 \\ \hline 1932 & 60 & 5321 & 60 \\ \hline 2489 & 70 & 5872 & 100 \\ \hline 2766 & 10 & 6433 & 50 \\ \hline 2877 & 80 & 6921 & 50 \\ \hline 2964 & 60 & 8317 & 70 \\ \hline 3217 & 70 & 8854 & 100 \\ \hline 3588 & 80 & 8964 & 80 \\ \hline 3780 & 80 & 9158 & 60 \\ \hline 3921 & 60 & 9347 & 60 \\ \hline 4107 & 40 & & \end{array} $$ Suppose that the grading scale for the test is \(\mathrm{A}: 80-100 ; \mathrm{B}:\) \(70-79 ; \mathrm{C}: 60-69 ; \mathrm{D}: 50-59 ;\) and \(\mathrm{F}: 0-49\) (a) Make a frequency table for the distribution of the test grades. (b) Draw a relative frequency bar graph for the test grades.

Step by step solution

01

Construct Frequency Table

Firstly, sort the scores into grade bins A, B, C, D, F. Count the number of scores for each grade bin and fill the frequency table. For example, if there are four scores within the range 80-100, the frequency for grade A is 4.

02

Calculate Total Number of Scores

Add up all the frequencies. This is the total number of scores, which is also the number of students.

03

Construct Relative Frequency Table

Calculate the relative frequency for each grade by dividing the frequency of each grade by the total number of scores. For example, the relative frequency for grade A is the frequency of grade A divided by the total number of scores.

04

Draw Bar Graph

The x-axis represents the grades A, B, C, D, F, and the y-axis represents the relative frequencies. For each grade, draw a bar from the x-axis to the height corresponding to its relative frequency. Make sure that the bars do not touch each other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Table

A frequency table is a simple yet powerful tool used in statistics to organize data values into classes and record their corresponding frequencies. In the context of the Chem 103 test scores, constructing a frequency table helps to categorize the scores based on the defined grading scale: A (80-100), B (70-79), C (60-69), D (50-59), and F (0-49).
To create this table, you start by tallying the number of scores that fall within each grade range, known as bins. For example, if there are four student scores that are between 80 and 100, the frequency for the grade "A" would be 4.

Frequency tables are fundamental in summarizing large amounts of data in an easily interpretable form. They lay the groundwork for further statistical analysis by providing clear insight into how the data values are distributed across the defined categories.

Relative Frequency

The concept of relative frequency takes the raw data from a frequency table a step further by allowing you to express each frequency as a proportion of the total number of observations. This step is crucial when you want to understand the distribution of data in a more relatable way.

• To find the relative frequency, divide the frequency of each category by the total number of scores.
• For instance, if the frequency of grade "A" is 4 and there are 20 students in total, the relative frequency of grade "A" is calculated as \( \frac{4}{20} = 0.2 \).
• Relative frequencies are often expressed as percentages by multiplying the decimal by 100.

This approach provides a clearer perspective of how significant each grade category is relative to the whole data set, helping you evaluate patterns and trends effectively.

Bar Graph

A bar graph is a visual representation of data where rectangular bars of lengths proportional to the values they represent are used. They are incredibly useful when you want to compare different categories at a glance.
In the context of the Chem 103 test scores, a relative frequency bar graph is used. Each bar corresponds to a grade, with the bar height indicating the relative frequency of that grade.

• The x-axis of the graph would display the grades: A, B, C, D, F.
• The y-axis represents the relative frequencies of each grade.
• Bars should be evenly spaced and distinct, so they do not touch each other.

This graphical representation makes it easy to see which grades were most and least common among students, providing a visual snapshot of data distribution.

Grading Scales

Grading scales are sets of standards used to assess student performance by dividing scores into various categories. In this exercise, the grading scale is explicitly defined:
- A: 80-100
- B: 70-79
- C: 60-69
- D: 50-59
- F: 0-49

By categorizing student scores against these ranges, educators can easily convey students' performances in a clear, standardized manner. Each grade represents a specific range of scores, providing a comprehensive interpretation beyond raw test scores.
Understanding grading scales allows students to know how their scores relate to the overall performance expectations and can help guide their future studies by explicitly showing areas that require improvement or have been mastered. Such scales are key in ensuring fairness and consistency in evaluating student performance across educational settings.