Question

Make a stem-and-leaf plot for the data. Use the result to list the data in increasing order. $$\begin{aligned}&4,31,22,37,39,24,2,28\\\&1,26,28,30,28,3,20,20,5\end{aligned}$$

Short answer

The stem-and-leaf plot would appear as follows: \[ \begin{array}{r|l} 0 & 1,2,4,5 \ 1 & \ 2 & 0,0,2,4,6,8,8,8 \ 3 & 0,1,7,9 \ 4 & \end{array} \]. And the list in increasing order is: 1,2,4,5,20,20,22,24,26,28,28,28,30,31,37,39.

Step-by-step solution

Identify the Stems and Leaves

For this set of data, the 'stem' will be the tens digit of the number and the 'leaf' will be the unit digit. For single digit numbers, the 'stem' will be 0 and the 'leaf' will be the actual number itself.

List of Stems

Create a list of 'stems' in increasing order. For the given data, the stems include 0, 1, 2, 3, and 4.

Attach Leaves to corresponding Stems

Write each 'leaf' next to its corresponding 'stem'. Arrange the leaves in increasing order for each stem.

Creating the stem-and-leaf plot

Compile the stems and attached leaves to create a stem-and-leaf plot. This will appear as a table, with the stem on the left side and the leaves on the right.

Key concepts

Data Representation

Data representation is crucial in statistics as it makes complex data more understandable and interpretable. A common challenge faced by students and researchers alike is how to visually convey numerical data in a clear and cohesive manner. This is where statistical tools like stem-and-leaf plots come into play.

A stem-and-leaf plot is particularly effective in representing smaller sets of data. In our exercise, the data set is not too large, making it a perfect candidate for this type of plot. The plot consists of two parts: the 'stem', which represents the leading digit or digits of each number, and the 'leaf', which represents the last digit. To construct the plot, every number is split into a stem and a leaf. Such a visual breakdown enables a quick analysis of the data, helping students identify patterns, outliers, and frequencies within the data with ease.

Statistical Plots

Statistical plots provide a visual representation of data distributions, which can aid in understanding the underlying structure of the data and in making inferences. There are various types of statistical plots, such as histograms, scatter plots, box-and-whisker plots, and stem-and-leaf plots.

The stem-and-leaf plot is particularly unique as it retains the actual data values, unlike other plots that provide a summary or an abstraction of the data. In the step-by-step solution, compiling the stem-and-leaf plot organizes the data in an easily readable format, while still preserving the individual data points. This level of detail can be helpful when comparing individual elements or when the exact values are significant for further analysis.

Organizing Data

Organizing data is a foundational step in statistical analysis and it can greatly influence the result and interpretation of the study. Proper organization helps us to see patterns, compare different data sets, and make informed conclusions. When we look at the stem-and-leaf plot from our exercise, we see that organizing the data into stems and leaves provides us with a clear, concise view of the dataset.

Organizing data is not just about cleanliness; it’s about setting the stage for accurate analysis and decision-making. With the ordered stems and sorted leaves, such as those constructed in Step 2 and Step 3 of the exercise, we can easily spot trends and outliers in the dataset. This pre-analysis step is often what makes or breaks quality conclusions in the field of statistics.