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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

In a USA TODAY survey, registered dietitians with the American Dietetic Association were asked, "What is the major reason people want to lose weight?" The responses were classified as Health (H), Cosmetic (C), and Other (O). Suppose a random sample of 20 dietitians is taken and these dietitians are asked the same question. Their responses are as follows. $$ \begin{array}{llllllllll} \mathrm{H} & \mathrm{H} & \mathrm{C} & \mathrm{H} & \mathrm{O} & \mathrm{C} & \mathrm{C} & \mathrm{H} & \mathrm{C} & \mathrm{O} \\ \mathrm{O} & \mathrm{H} & \mathrm{C} & \mathrm{H} & \mathrm{H} & \mathrm{C} & \mathrm{H} & \mathrm{H} & \mathrm{O} & \mathrm{H} \end{array} $$ a. Prepare a frequency distribution table. b. Compute the relative frequencies and percentages for all categories. c. What percentage of these dietitians gave Health as the major reason for people to lose weight? d. Draw a pie chart for the percentage distribution.

The following data give the political party of each of the first 30 U.S. presidents. In the data, D stands for Democrat, DR for Democratic Republican, F for Federalist, \(\mathrm{R}\) for Republican, and \(\mathrm{W}\) for Whig. $$ \begin{array}{lllllllll} \text { F } & \text { F } & \text { DR } & \text { DR } & \text { DR } & \text { DR } & \text { D } & \text { D } & \text { W } & \text { W } \\ \text { D } & \text { W } & \text { W } & \text { D } & \text { D } & \text { R } & \text { D } & \text { R } & \text { R } & \text { R } \\ \text { R } & \text { D } & \text { R } & \text { D } & \text { R } & \text { R } & \text { R } & \text { D } & \text { R } & \text { R } \end{array} $$ a. Prepare a frequency distribution table for these data. b. Calculate the relative frequency and percentage distributions. c. Draw a bar graph for the relative frequency distribution and a pie chart for the percentage distribution. d. What percentage of these presidents were Whigs?

Statisticians often need to know the shape of a population to make inferences. Suppose that you are asked to specify the shape of the population of weights of all college students. a. Sketch a graph of what you think the weights of all college students would look like. b. The following data give the weights (in pounds) of a random sample of 44 college students (F and M indicate female and male, respectively). $$ \begin{array}{llllllll} 123 \mathrm{~F} & 195 \mathrm{M} & 138 \mathrm{M} & 115 \mathrm{~F} & 179 \mathrm{M} & 119 \mathrm{~F} & 148 \mathrm{~F} & 147 \mathrm{~F} \\ 180 \mathrm{M} & 146 \mathrm{~F} & 179 \mathrm{M} & 189 \mathrm{M} & 175 \mathrm{M} & 108 \mathrm{~F} & 193 \mathrm{M} & 114 \mathrm{~F} \\ 179 \mathrm{M} & 147 \mathrm{M} & 108 \mathrm{~F} & 128 \mathrm{~F} & 164 \mathrm{~F} & 174 \mathrm{M} & 128 \mathrm{~F} & 159 \mathrm{M} \\ 193 \mathrm{M} & 204 \mathrm{M} & 125 \mathrm{~F} & 133 \mathrm{~F} & 115 \mathrm{~F} & 168 \mathrm{M} & 123 \mathrm{~F} & 183 \mathrm{M} \\ 116 \mathrm{~F} & 182 \mathrm{M} & 174 \mathrm{M} & 102 \mathrm{~F} & 123 \mathrm{~F} & 99 \mathrm{~F} & 161 \mathrm{M} & 162 \mathrm{M} \\ 155 \mathrm{~F} & 202 \mathrm{M} & 110 \mathrm{~F} & 132 \mathrm{M} & & & & \end{array} $$ i. Construct a stem-and-leaf display for these data. ii. Can you explain why these data appear the way they do? c. Now sketch a new graph of what you think the weights of all college students look like. Is this similar to your sketch in part a?

The following table gives the frequency distribution for the numbers of parking tickets received on the campus of a university during the past week for 200 students. $$ \begin{array}{cc} \hline \text { Number of Tickets } & \text { Number of Students } \\ \hline 0 & 59 \\ 1 & 44 \\ 2 & 37 \\ 3 & 32 \\ 4 & 28 \\ \hline \end{array} $$ Draw two bar graphs for these data, the first without truncating the frequency axis and the second by truncating the frequency axis. In the second case, mark the frequencies on the vertical axis starting with 25 . Briefly comment on the two bar graphs.

The following data give the number of turnovers (fumbles and interceptions) by a college football team for each game in the past two seasons. $$ \begin{array}{llllllllllll} 3 & 2 & 1 & 4 & 0 & 2 & 2 & 1 & 0 & 3 & 2 & 3 \\ 0 & 2 & 3 & 1 & 4 & 1 & 3 & 2 & 4 & 0 & 1 & 2 \end{array} $$ a. Prepare a frequency distribution table for these data using single-valued classes. b. Calculate the relative frequencies and percentages for all classes. c. In how many games did the team commit two or more turnovers? d. Draw a bar graph for the frequency distribution of part a.
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