Question

Make a stem-and-leaf plot for the data. Use the result to list the data in increasing order. $$ \begin{aligned} &\text 48,10,48,25,40,42,44,23\\\ &\begin{array}{l} 21,13,50,17,18,19,21,57, \\ 35,33,25,50,13,12,46 \end{array} \end{aligned} $$

Short answer

The data in increasing order, using the stem-and-leaf plot, is: 10, 12, 13, 13, 17, 18, 19, 21, 21, 23, 25, 25, 33, 35, 40, 42, 44, 46, 48, 48, 50, 50, 57.

Step-by-step solution

Arrange the Data

First, the data is arranged in increasing order: 10, 12, 13, 13, 17, 18, 19, 21, 21, 23, 25, 25, 33, 35, 40, 42, 44, 46, 48, 48, 50, 50, 57. This is a necessary step before creating a stem-and-leaf plot.

Identify the Stems and Leaves

The stem-and-leaf plot splits each data point into a 'stem' and a 'leaf'. In this case, the tens digit of each data point will form the 'stem', and the ones digit will form the 'leaf'. For example, in the data point '10', '1' is the stem and '0' is the leaf.

Create the Stem-and-Leaf Plot

Now create the stem-and-leaf plot, listing the stems in increasing order (from the smallest value to the greatest), and listing the leaves for each stem in increasing order. Here is the resulting plot: \[\begin{align*}1 & | 0, 2, 3, 3, 7, 8, 9 \2 & | 1, 1, 3, 5, 5 \3 & | 3, 5 \4 & | 0, 2, 4, 6, 8, 8 \5 & | 0, 0, 7\end{align*}\]

Key concepts

Data Arrangement

Understanding how to effectively organize data is crucial for analysis and interpretation. Data arrangement involves sorting and structuring data in an order that makes it more comprehensible. In the context of a stem-and-leaf plot, this means ordering the data points from least to greatest before creating the plot.

This preliminary step is essential because it simplifies the process of identifying stems and leaves. The stems typically represent the leading digits, and the leaves represent the trailing digits. For instance, in the number 42, '4' would be the stem, and '2' the leaf. By arranging the data, we ensure that we can efficiently assign each data point to its proper position within the plot.

A well-arranged data set not only helps create an accurate stem-and-leaf plot but also allows for easier identification of patterns, trends, and outliers within the data. This organizational process translates complex datasets into a format that is manageable and ready for further statistical analysis.

Data Visualization

Data visualization is the graphical representation of information. It uses visual elements like charts, graphs, and maps to present data in a way that is accessible, understandable, and actionable. A stem-and-leaf plot is one such tool that offers a quick way to visualize data in a semi-tabular form.

In the exercise provided, a stem-and-leaf plot allows students to see the distribution of data at a glance. Each 'leaf' clusters around the 'stem', offering a visual breakdown of how many data points fall into various ranges. This representation helps in easily spotting clusters of data and gaps within the dataset. For example, the plot might illustrate a concentration of data points in the lower ranges and sparser data in higher ranges, indicating a specific trend.

Visualizing data this way simplifies complex information and aids in making informed decisions based on the dataset. It bridges the gap between raw data and the ability to interpret it meaningfully.

Statistical Plots

Statistical plots are essential tools in data analysis that provide a visual representation of the distribution, trends, and patterns in data. They range from simple scatter plots and line graphs to more complex boxplots and stem-and-leaf plots. The key to using statistical plots effectively is choosing the right type for the data you are analyzing.

The stem-and-leaf plot is a statistical tool that displays data in a form similar to a histogram, while retaining the original data points. It is particularly useful for small to medium-sized data sets where a quick visual of the data distribution is needed. Unlike bar graphs or histograms which group data into ranges, stem-and-leaf plots provide a more detailed display showing each individual data point.

This type of plot is beneficial for identifying the shape of a distribution, central tendency, and the spread of the data. It can be helpful in the classroom to reinforce the understanding of numerical data and to introduce the concept of statistical distributions in a straightforward manner.