Question

Use the stem-and-leaf plot below to order the data set. $$\begin{array}{l|lllllll}1 & 6 & 3 & 8 & 1 & 7 & 5 & 3 \\\2 & 8 & 2 & 5 & 0 & 7 & 7 & 0 \\\3 & 8 & 3 & 1 & 4 & 7 & &\end{array}$$ Key: \(3 | 8=38\)

Short answer

The ordered data set is: 11, 13, 13, 15, 16, 17, 18, 20, 20, 22, 25, 27, 27, 28, 31, 33, 34, 37, 38.

Step-by-step solution

Understand the stem-and-leaf plot

The key \(3 | 8=38\) tells us how to interpret the plot. The stem, the number on the left of the pipeline, represents the tens place of the numbers. The numbers on the right side of the pipeline, the leaves, represent the ones place. Therefore, each leaf joined with its corresponding stem gives a number in the data set.

Extract the numbers from the plot

Now, extract individual numbers from the plot by joining the stems with each of their corresponding leaves. The numbers are: 16, 13, 18, 11, 17, 15, 13, 28, 22, 25, 20, 27, 27, 20, 38, 33, 31, 34, 37.

Order the extracted numbers

Once the numbers are extracted, they can be put in ascending order. The ordered data set is: 11, 13, 13, 15, 16, 17, 18, 20, 20, 22, 25, 27, 27, 28, 31, 33, 34, 37, 38.

Key concepts

Data Visualization

When it comes to understanding numerical data, a visual representation is often more digestible than rows of numbers. Data visualization plays a critical role in exhibiting data in a more approachable and comprehensible manner. A common method of data visualization for small datasets is the stem-and-leaf plot, which exhibits information in a semi-graphical format. This type of visual helps to maintain the original data while also showing the distribution of the numbers.

For example, the stem-and-leaf plot provided in the exercise is a form of data visualization that simplifies the observation of a set of numerical data. It is simple yet informative, providing insights into not only individual data values but also the shape and spread of the data set as a whole. This visualization method makes it easier for students to quickly interpret the dataset without losing sight of the individual data points.

Ordering Data Sets

The act of ordering data sets is a fundamental practice in statistics, allowing for a better understanding of the data by sorting values in a specific sequence, typically ascending or descending. With ordered data sets, it becomes clearer to identify patterns, make comparisons, and even spot outliers or unusual values.

In the context of the stem-and-leaf plot, ordering involves arranging the 'leaves' (the digits on the right side of the bar) alongside their corresponding 'stems' (the digits on the left side of the bar) to form a sequence of numbers. By carefully analyzing the plot and combining stems with leaves, one can order the given data set systematically, making it more interpretable and useful for further statistical analysis.

Interpreting Graphs

The skill of interpreting graphs is crucial for students as it forms the bridge between raw data and real-world implications. Being able to interpret graphs means that students can extract meaningful information and trends from a visual representation of data, such as a stem-and-leaf plot. It involves understanding what each part of the graph signifies and being able to discern the story the data tells.

The stem-and-leaf plot in the given exercise, when interpreted correctly, allows students to rapidly ascertain the range of the data, the frequency of certain values, and the distribution across the number line. For example, when interpreting this plot, one can see that the data is concentrated in the 'teens and twenties, with some data points in the thirties. Regular practice in interpreting such graphs is key in developing strong analytical skills, which have broad applications across various fields.