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Chapter 11: Problem 6

The following are the grades of an IB course with 40 students (two sections) on a 100 -point test. Use the graphical methods you have learned so far to describe the grades. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline 61 & 62 & 93 & 94 & 91 & 92 & 86 & 87 & 55 & 56 \\ \hline 63 & 64 & 86 & 87 & 82 & 83 & 76 & 77 & 57 & 58 \\\ \hline 94 & 95 & 89 & 90 & 67 & 68 & 62 & 63 & 72 & 73 \\ \hline 87 & 88 & 68 & 69 & 65 & 66 & 75 & 76 & 84 & 85 \\ \hline \end{array}$$

Short Answer

Expert verified
Use a histogram to display frequency of grades and optionally a box plot for data spread.

Step by step solution

01

Organize Data into Classes

Divide the data range into intervals or "classes". Since the scores range from 55 to 95, we can use intervals of 5 points: 55-59, 60-64, 65-69, 70-74, 75-79, 80-84, 85-89, 90-94, and 95-99.
02

Count Frequencies

Count how many scores fall into each class interval. For example, the interval 55-59 has scores such as 55, 56, 57, and 58, so it has a frequency of 4.
03

Create a Frequency Table

Create a table with two columns; one for the class intervals and one for the frequency of scores within each interval. Populate the table using the data counts from Step 2.
04

Draw a Histogram

Use the frequency table to draw a histogram. On the horizontal axis, display the class intervals and on the vertical axis, display the frequencies. Each bar represents the number of scores within that interval.
05

Create a Box Plot (Optional)

For a different graphical representation, you can also generate a box plot. Find the minimum, first quartile (Q1), median, third quartile (Q3), and maximum to construct the plot. This will show data spread and potential outliers.

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Key Concepts

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

Histogram
A histogram is a powerful graphical tool used to display the distribution of numerical data. It resembles a bar chart, with adjacent bars. Each bar represents the frequency of data within certain intervals, known as "bins". To create a histogram, first, determine the range of your data. In this example, the scores range from 55 to 95.
  • Divide the data range into equal-sized intervals. For instance, you can use intervals of 5 points: 55-59, 60-64, and so on.
  • Count how many data points fall into each interval. These counts are the frequencies, which are plotted on the vertical axis.
  • The horizontal axis shows the class intervals.
Each bar height reflects the number of observations within the interval it covers. Thus, a histogram shows the shape and spread of the data distribution. It helps identify the mode (most common interval) and any patterns like skewness or uniformity in the data.
Frequency Table
A frequency table is an organized tabular representation of data. It helps you summarize large datasets by showing the frequency, or count, of each outcome in a dataset. Here's how you can create it:
  • First, identify the class intervals corresponding to your data range. For our example, intervals like 55-59, 60-64, etc., fit well.
  • Next, count how many data points fall into each interval, like 4 scores in the interval 55-59.
  • Finally, write these counts in a table with two columns: one for intervals and one for the frequencies.
This table provides a clear visual of the data's frequency distribution, making it easy to analyze and interpret. Frequency tables are foundational for creating histograms or other graphical presentations, offering a simple yet effective way to understand the underlying data trends.
Box Plot
A box plot, or a whisker plot, visually displays the central tendency, spread, and any potential outliers in a dataset. It's particularly useful for comparing distributions between several groups or datasets. Here's how to construct one:
  • Calculate the five-number summary of your data: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
  • Draw a "box" from Q1 to Q3 with a line at the median. This box captures the interquartile range (middle 50% of your data).
  • Extend "whiskers" from the box to the minimum and maximum data points. This shows the range of your data.
Outliers, which are data points significantly higher or lower than the rest, can also be shown as individual points. Box plots provide a comprehensive snapshot of a dataset's variability and highlight any deviations from "normal" ranges, offering insights that might be less apparent in histograms or frequency tables.

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Most popular questions from this chapter

For a large class of 60 students, 12 points are added to each grade to boost the student's score on a relatively difficult test. a) Knowing that \(\sum(x+12)=4404,\) find the mean score (without the 12 points) of this group of 60 students. b) Another section of the class has 40 students and their average score is 67.4 Find the average of the whole group of 100 students.

At 5: 30 p.m. during the holiday season, a toy shop counted the number of items sold and the revenue collected for that day. The result was \(n=90\) toys with a total revenue of \(\sum x=鈧� 4460\) a) Find the average amount spent on each toy that day. Shortly before the shop closed at 6 p.m. two new purchases of \(鈧� 74\) and \(鈧� 60\) were made. b) Calculate the new mean of the sales per toy that day.

The following are the grades earned by 25 students on a 50 -mark test in statistics. 26,27,36,38,23,26,20,35,19,24,25,27,34,27,26,42,46,18,22,23,24,42,46 33,40 a) Calculate the mean of the grades. b) Draw a stem plot of the grades. Use the plot to estimate where the median is. c) Draw a histogram of the grades. d) Develop a cumulative frequency graph of the grades. Use your graph to estimate the median.

Aptitude tests sometimes use jigsaw puzzles to test the ability of new applicants to perform precision assembly work in electronic instruments. One such company that produces the computerized parts of video and CD players gave the following results: $$\begin{array}{|c|c|} \hline \text { Time to finish the puzzle (nearest second) } & \text { Number of employees } \\ \hline 30-35 & 16 \\ \hline 35-40 & 24 \\ \hline 40-45 & 22 \\ \hline 45-50 & 26 \\ \hline 50-55 & 38 \\ \hline 55-60 & 36 \\ \hline 60-65 & 32 \\ \hline 65-70 & 18 \\ \hline \end{array}$$a) Draw a histogram of the data. b) Draw a cumulative frequency curve and estimate the median and IQR. c) Calculate the estimates of the mean and standard deviation of all such participants.

The following table gives the number of DVD players owned by a sample of 50 typical families in a large city in Germany. $$\begin{array}{|l|c|c|c|c|}\hline \text { Number of DVD players } & 0 & 1 & 2 & 3 \\\\\hline \text { Number of households } & 12 & 24 & 8 & 6 \\\\\hline\end{array}$$ Find the average and the median number of DVD players. Which measure is more appropriate here? Explain.
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