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Chapter 2: Problem 45

What advantage does preparing a stem-and-leaf display have over grouping a data set using a frequency distribution? Give one example.

Short Answer

Expert verified
The main advantage of preparing a stem-and-leaf display over the grouping of a data set using a frequency distribution is that a stem-and-leaf display retains the original data points while providing a representation of their frequency. An example demonstrating this advantage can be the case of student test scores, where a stem-and-leaf display can provide the exact scores along with their occurrences.

Step by step solution

01

Understanding Stem-and-Leaf Display

A stem-and-leaf display is a statistical technique to present a set of data. Each number in the data set is separated into a stem (first digit or digits) and a leaf (last digit). This method allows not only viewing the frequency of data but also the actual data points. The data in a stem-and-leaf plot is organized, which can save space while keeping the original data points.
02

Understanding Frequency Distributions

A frequency distribution is a summary of a set of data that displays the number of times each value or range of values occurs. It is typically represented in the form of a table or graph. While this method provides a clear picture of the frequency of data points, it does not represent the actual data points. The original data points are lost in this method.
03

Identifying the Advantage

The major advantage of a stem-and-leaf display over a frequency distribution is that it retains the original data points while also indicating the frequency of data points. This is particularly advantageous when the actual data points are important and we do not want to lose them within ranges or groups as in a frequency distribution.
04

Giving an Example

For example, consider a small dataset of student test scores out of 10: [10, 9, 9, 8, 9, 7, 7, 6, 6, 6]. In a frequency distribution, this might be grouped and displayed as 6-7: 4 students, 8-9: 4 students, 10: 2 students. In this case, you do not have the exact test scores for each student. In a stem-and-leaf display, however, each test score would be shown. For instance, stem 0: 6 6 6 7 7; stem 1: 0 8 9 9 9. Here, exact scores of each test can be seen along with their frequency.

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Most popular questions from this chapter

How are the relative frequencies and percentages of classes obtained from the frequencies of classes? Illustrate with the help of an example.

The following data give the amounts spent on video rentals (in dollars) during 2009 by 30 households randomly selected from those who rented videos in 2009. $$ \begin{array}{rrrrrrrrr} 595 & 24 & 6 & 100 & 100 & 40 & 622 & 405 & 90 \\ 55 & 155 & 760 & 405 & 90 & 205 & 70 & 180 & 88 \\ 808 & 100 & 240 & 127 & 83 & 310 & 350 & 160 & 22 \\ 111 & 70 & 15 & & & & & & \end{array} $$ a. Construct a frequency distribution table. Take \(\$ 1\) as the lower limit of the first class and \(\$ 200\) as the width of each class. b. Calculate the relative frequencies and percentages for all classes. c. What percentage of the households in this sample spent more than \(\$ 400\) on video rentals in \(2009 ?\)

Briefly explain the three decisions that have to be made to group a data set in the form of a frequency distribution table.

Eighty adults were asked to watch a 30 -minute infomercial until the presentation ended or until boredom became intolerable. The following table lists the frequency distribution of the times that these adults were able to watch the infomercial. $$ \begin{array}{lc} \hline \begin{array}{c} \text { Time } \\ \text { (minutes) } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Adults } \end{array} \\ \hline 0 \text { to less than } 6 & 16 \\ 6 \text { to less than } 12 & 21 \\ 12 \text { to less than } 18 & 18 \\ 18 \text { to less than } 24 & 11 \\ 24 \text { to less than } 30 & 14 \\ \hline \end{array} $$ Draw two histograms for these data, the first without truncating the frequency axis. In the second case, mark the frequencies on the vertical axis starting with 10 . Briefly comment on the two histograms.

Thirty adults were asked which of the following conveniences they would find most difficult to do without: television (T), refrigerator (R), air conditioning (A), public transportation (P), or microwave (M). Their responses are listed below. $$ \begin{array}{cccccccccc} \mathrm{R} & \mathrm{A} & \mathrm{R} & \mathrm{P} & \mathrm{P} & \mathrm{T} & \mathrm{R} & \mathrm{M} & \mathrm{P} & \mathrm{A} \\ \mathrm{A} & \mathrm{R} & \mathrm{R} & \mathrm{T} & \mathrm{P} & \mathrm{P} & \mathrm{T} & \mathrm{R} & \mathrm{A} & \mathrm{A} \\ \mathrm{R} & \mathrm{P} & \mathrm{A} & \mathrm{T} & \mathrm{R} & \mathrm{P} & \mathrm{R} & \mathrm{A} & \mathrm{P} & \mathrm{R} \end{array} $$ a. Prepare a frequency distribution table. b. Calculate the relative frequencies and percentages for all categories. c. What percentage of these adults named refrigerator or air conditioning as the convenience that they would find most difficult to do without? d. Draw a bar graph for the relative frequency distribution.
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