Problem 5 - Algebra 2 and Trigonometry

Question

Organize the data in a stem-and-leaf diagram. The heights, in centimeters, of 25 ten-year-old children: \(\begin{array}{llllllllllll}{137} & {134} & {130} & {144} & {131} & {141} & {136} & {140} & {137} & {129} & {139} & {137} & {144} \\ {127} & {147} & {143} & {132} & {132} & {142} & {142} & {131} & {129} & {138} & {151} & {137}\end{array}\)

Step-by-Step Solution

Answer

Organize the data as stems 12, 13, 14, 15 with leaves for each height. Ensure duplicates are shown as repeated leaves.

Step 1: Understand the Stem-and-Leaf Plot

A stem-and-leaf plot is a way to organize data visually. The "stem" represents the leading digit or digits of each number, while the "leaf" represents the last digit.

Step 2: Identify Stems

Look at the provided data and identify the range of values. The heights range from 127 cm to 151 cm, so the stems will be the tens digits: 12, 13, 14, and 15.

Step 3: Organize Data by Stems

List the numbers under their corresponding stems. For example, the number 137 has a stem of 13 and a leaf of 7.

Step 4: Create the Stem-and-Leaf Plot

Write each stem in a vertical column, and next to each stem, list the leaves in ascending order that correspond to that stem. If a number appears more than once, its leaf will also appear multiple times.

Step 5: Verify and Finalize

Ensure all numbers from the data set are included and correctly placed in the plot. Double-check that the leaves are sorted within each stem.

Key Concepts

Data Visualization

Data visualization is an essential tool in statistics for representing information clearly and effectively. It allows us to see patterns, trends, and outliers that might not be immediately apparent from raw data alone.

The stem-and-leaf plot is a simple form of data visualization that provides an immediate visual impression of the distribution of data. This type of plot is particularly useful because it retains the original data points while showing how they are distributed. By splitting each number into a stem (representing the highest place value) and a leaf (representing the lowest place value), you can see how values cluster and spread across a dataset.

This visualization method helps identify the frequency of data points. For instance, a glance at a completed stem-and-leaf plot reveals which heights are most common among children in the exercise data. This form of visualization is especially useful when dealing with compact datasets like small sample sizes, where insights into the data's shape can still be gleaned without needing more sophisticated statistical tools.

Organizing Data

Organizing data is a fundamental step in statistical analysis. It simplifies complex data, making it easier to interpret and understand. By arranging data into manageable formats, such as a stem-and-leaf plot, we can glean meaningful insights quickly.

In the context of a stem-and-leaf plot, organizing data involves splitting each data point into a stem and a leaf. Consider a dataset with values like those in the exercise: several heights ranging from 127 cm to 151 cm. Each height can be divided into a 'stem' representing the hundreds and tens digit (for example, '13' for 137 cm) and a 'leaf' representing the unit digit (like '7' in 137).

By placing these in a structured format, we can transform a list of seemingly haphazard numbers into an organized visualization. This not only makes the data more digestible but also supports further statistical analysis by highlighting data distribution and frequency.

Statistical Representation

Statistical representation is about using various methods and tools to present data for analysis and interpretation. Each method has different strengths depending on the kind of data and what needs to be understood from it.

In a stem-and-leaf plot, statistical representation is achieved by displaying the data in a way that maintains individual data points while also revealing the overall shape and distribution of the dataset. This dual representation allows students to see individual variations and collective trends simultaneously.

For the given exercise, the stem-and-leaf plot maintains each child’s height in a concise format conducive to analysis. It clarifies how many children fall within each height range and identifies the most and least common heights among them. By providing a clear and organized view of the data, this method facilitates a deeper understanding of the underlying statistics, aiding in drawing conclusions and making informed decisions based on the data.