Question

Construct a stem and leaf plot for these 50 measurements and answer the questions. $$ \begin{array}{llllllllll} 3.1 & 4.9 & 2.8 & 3.6 & 2.5 & 4.5 & 3.5 & 3.7 & 4.1 & 4.9 \\ 2.9 & 2.1 & 3.5 & 4.0 & 3.7 & 2.7 & 4.0 & 4.4 & 3.7 & 4.2 \\ 3.8 & 6.2 & 2.5 & 2.9 & 2.8 & 5.1 & 1.8 & 5.6 & 2.2 & 3.4 \\ 2.5 & 3.6 & 5.1 & 4.8 & 1.6 & 3.6 & 6.1 & 4.7 & 3.9 & 3.9 \\ 4.3 & 5.7 & 3.7 & 4.6 & 4.0 & 5.6 & 4.9 & 4.2 & 3.1 & 3.9 \end{array} $$ Find the eighth and ninth largest observations.

Short answer

Answer: The eighth largest observation is 4.9, and the ninth largest observation is 4.9.

Step-by-step solution

Understand Stem and Leaf Plot Components

A stem and leaf plot is a data representation method that separates numbers into stems (tens place) and leaves (units place). It is a simple and convenient way to sort and visualize the distribution of data points. In this dataset, our measurements are given in tenths, so our stems will represent whole numbers (1, 2, etc.), and our leaves will represent tenths (0.1, 0.2, etc.).

Find Range of Stems

Look at the data to determine the lowest and highest values, which will form the range for our stems. In this case, the lowest value is 1.6, and the highest is 6.2. So, our stems will range from 1 to 6.

Separate Leaves from Stems

For each measurement, separate the stem from the leaf. We will keep the leaf value as the tenths after the decimal point and drop the decimals. For example, for the measurement 3.1, the stem is 3, and the leaf is 1.

Create the Stem and Leaf Plot

Arrange the stems vertically from smallest to largest and list the leaves associated with each stem in ascending order: ``` 1 | 6 8 2 | 1 2 5 5 5 7 8 8 9 9 3 | 1 1 4 5 5 6 6 7 7 7 8 9 9 9 9 4 | 0 0 1 2 2 3 4 5 6 7 8 9 9 9 5 | 1 1 6 6 7 6 | 1 2 ```

Find the Eighth and Ninth Largest Observations

To find the eighth and ninth largest observations, we will work from the right-hand side of the plot (largest numbers) and count our way to the eighth and ninth spots. If we count from the right, we encounter the following observations in order: 6.2, 6.1, 5.7, 5.6, 5.6, 5.1, 5.1, 4.9, and 4.9. Therefore, the eighth largest observation is 4.9, and the ninth largest observation is 4.9.

Key concepts

Data Representation

A stem and leaf plot is a fantastic way to represent data. It gives you a quick visual of numbers without losing the original raw data. This type of plot is especially useful when you want to see all data points and compare them easily.
Each number in your dataset is divided into a 'stem' and a 'leaf'. The stem is typically the main part of the number, such as the tens digit, while the leaf is the rest, like the units or tenths.

• For example, in the number 3.7, '3' is the stem (the whole number), and '.7' is the leaf (the decimal part).
• The stem and leaf plot keeps each original number intact, which truly allows for data retention while organizing it.

Constructing this plot helps to visualize how numbers fall into different categories, such as different scores or measurements.

Distribution Visualization

A stem and leaf plot also excels at showing the distribution of your dataset. By stacking numbers vertically, it's easy to see where most data points lie.
For instance, by examining the distribution of numbers between our stems from 1 to 6, you can quickly determine which numbers are more common in the dataset.

• With stems listed, the leaves give you a clear picture of how numbers are spread across the different ranges.
• The longer the list of leaves under a particular stem, the more frequently numbers in that range appear.

This visualization helps to identify trends, such as peaks and valleys within the data, making it a useful tool for understanding distribution at a glance.

Ordering and Ranking Data

Ordering and ranking data becomes more intuitive with a stem and leaf plot. Once numbers are neatly categorized, you can easily rank your data to find specific values.
For example, finding the eighth and ninth largest observations is straightforward once you arrange leaves in ascending order. You observe the stem and leaf plot from the largest categories to the smallest.

• This plot style makes it easy to identify the biggest and smallest numbers and everything in between.
• By counting the leaves from either the largest to smallest or vice versa, you can quickly find rank positions of interest, such as the eighth or ninth largest number.

Understanding how to order and rank using a stem and leaf plot simplifies the process of analyzing data sets, making it both straightforward and efficient.