Question

Use the table that lists the weights (in ounces) and operating times (in minutes) for 10 cellular telephones. $$\begin{array}{|c|c|}\hline \text { Wompor } & \text { Caresesum } \\\\\hline 8.8 & 140 \\\\\hline 7.7 & 80 \\\\\hline 4.8 & 90 \\\\\hline 7.0 & 90 \\\\\hline 7.1 & 90 \\\\\hline 8.4 & 70 \\\\\hline 3.8 & 60 \\\\\hline 3.6 & 260 \\\\\hline 6.7 & 75 \\\\\hline 6.2 & 135 \\\\\hline\end{array}$$ Make a stem-and-leaf plot to order the operating times of the phones.

Short answer

The Stem-and-Leaf plot for the operating times is: \\ \[ \begin{array}{rl} 6: & 0 \\ 7: & 0, 5 \\ 8: & 0, 0, 0 \\ 9: & 0, 0 \\ 13: & 5 \\ 14: & 0 \\ 26: & 0 \end{array} \]

Step-by-step solution

Identify the stem and leaf values

In a stem-and-leaf plot, the data is divided into 'stem' and 'leaf' values. Here, our data are the operating times of the phones. We will take the first digit(s) as the stem and the last digit as the leaf. For instance, for the operating time 140, the stem is 14 and the leaf is 0.

Arrange the stems vertically

Starting from the stem with the smallest value, write down all stems in a column. Each unique stem should only appear once, and the sequence should be in ascending order. The stems here are: 6, 7, 8, 9, 13, 14, and 26.

Assign leaf values to the respective stems

Attach each leaf to its corresponding stem. If a stem has multiple leaves, write them in ascending order. For example, the stem 9 corresponds to the leaves 0, 0, 0 which are the last digits of 90, 90, 90 respectively.

Construct the stem-and-leaf plot

Draw a vertical line to separate the stem and leaf values. The finished plot looks like this: \\ \[ \begin{array}{rl} 6: & 0 \\ 7: & 0, 5 \\ 8: & 0, 0, 0 \\ 9: & 0, 0 \\ 13: & 5 \\ 14: & 0 \\ 26: & 0 \end{array} \] Each row now represents an operating time: the value is created by combining the stem and each of the leaf digits visually.

Key concepts

Data Representation

When we talk about data representation, we are referring to the various ways in which numerical or categorical information can be organized and displayed for analysis. A critical part of data analysis is ensuring that data is represented in a manner that is easy to interpret and understand. One such method of representation is the stem-and-leaf plot.

This plot is a useful tool, especially when dealing with small to moderate sets of data. It's similar to a histogram, as it provides a visual insight into the distribution of data. However, the advantage of the stem-and-leaf plot over other methods, such as bar graphs or line charts, is that it includes the actual data values, not just an aggregate summary, preserving the original data while providing a graphic representation.

For learners to make the most out of their stem-and-leaf plots, it’s important to ensure that the separation between stems and leaves is clear and that each data point is accurately represented. This precision helps in identifying patterns, such as clusters or gaps within the data, and also aids in comparing different data sets effectively.

Statistical Graphs

Statistical graphs are visuals that statistically showcase data. There are many types of statistical graphs, each designed for specific types of data and analyses. Common examples include bar charts, line graphs, pie charts, and, as highlighted in our exercise, stem-and-leaf plots.

Stem-and-leaf plots are particularly beneficial when you want to explore and present the data's shape, central value, and spread. They allow a quick assessment of data distribution while keeping the individual data values accessible. This plot type is unparalleled in its ability to maintain the granular level detail of quantitative information, which something like a line or bar graph might overlook.

To improve upon the basic understanding of statistical graphs, students should practice reading and interpreting different graph types. They should compare the pros and cons of each graph type in representing different kinds of data, thus developing a deeper insight into the right circumstances and data sets that warrant the use of a stem-and-leaf plot.

Organizing Data

Organizing data efficiently is a cornerstone of good data analysis. It enhances comprehension and enables the analyst to identify trends, make comparisons, and draw meaningful interpretations. Initially, one might organize data into tables, but to extract more depth, graphical tools such as stem-and-leaf plots are used.

Organizing data into a stem-and-leaf plot requires a methodical approach. The first step often involves sorting the data into ascending order, which then makes the delineation of stems and leaves straightforward. The plot also helps in spotting outliers and understanding the distribution – whether the data is skewed or symmetric, for instance.

For students to become proficient at organizing data, they should practice sorting data sets and creating stem-and-leaf plots from scratch, as in our exercise. They should learn to define criteria for the 'stems' and 'leaves' based on the data set's range and immediately notice how the layout provides instant insights into the data's attributes.