Question

A discrete variable can take on only the values \(0,1,\) or \(2 .\) Use the set of 20 measurements on this variable to answer the questions. $$ \begin{array}{lllll} 1 & 2 & 1 & 0 & 2 \\ 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 1 & 0 \\ 0 & 1 & 2 & 1 & 1 \end{array} $$ How could you define the stem and leaf for this data set? Draw the stem and leaf plot.

Short answer

Answer: 10

Step-by-step solution

Identify the stem and leaf

Since the data set consists of discrete values, we can use these values as the leaves. In this case, our leaves will be 0, 1, and 2. The stem will just be a single value which represents the category of the data set.

Organize the data

Arrange the data set in ascending order: $$0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2$$

Create the stem and leaf plot

We will now create the stem and leaf plot showing the frequency of occurrence of each value in the data set: Stem | Leaf -----|----- 0 | 00000 1 | 1111111111 2 | 22222 The stem and leaf plot can be interpreted as follows: - There are five 0's, ten 1's, and five 2's in the data set.

Key concepts

Discrete Variable

In the world of statistics, a discrete variable is one that can take on a limited number of distinct values within a given range. These values are countable, much like the number of students in a class or the number of blueberries in a muffin. Discrete variables stand in contrast to continuous variables, which can take on an infinite number of values within a range, like the temperature on a summer day or the height of students.

For example, in the exercise, the discrete variable could only assume the values 0, 1, or 2. These are specific, isolated points on the number line, with no possibility of having an in-between value, like 1.5 or 2.1. The essence of a discrete variable is its distinct and separate nature, making it easy to count and tabulate within datasets, just as students could count the number of apples in a basket.

Frequency Distribution

The term frequency distribution is a fundamental concept in statistics, referring to an organized tabulation of the number of occurrences of each unique value of a variable. It's similar to creating a list of how many students got each possible grade on a test. A frequency distribution can take the form of a table, a histogram, or in this case, a stem and leaf plot.

In our exercise, the frequency distribution is evident in the stem and leaf plot. The plot displays how often the discrete variables 0, 1, and 2 appear in the dataset in a simple and clear manner. By looking at the plot, one can tell at a glance that the value 1 occurred ten times, which is the highest frequency, followed by the values 0 and 2, each occurring five times. This visualization provides a quick way to understand the distribution of values in the set and is particularly useful for datasets with a small range of discrete values.

Data Visualization

The practice of data visualization is akin to painting a picture with numbers. It involves representing data graphically to enhance comprehension and reveal insights that might not be evident from raw data alone. Data visualization helps in simplifying complex concepts and allows for a more intuitive understanding of the information being presented.

Stem and leaf plots are a type of data visualization particularly well-suited for small datasets with discrete variables. In our exercise, the stem and leaf plot visualizes the frequency distribution of the dataset, clearly depicting the number of times each value occurs. It's a straightforward and efficient method to convey statistical information, making it possible to see patterns and trends at a glance. For learners, such visual tools are invaluable as they transform abstract numbers into concrete visuals, facilitating better understanding and comparison.

Improving Interpretation

For even clearer interpretation, the stem and leaf plot could be supplemented with additional annotations, such as indicating the mode, or most frequent value, and providing a key or legend if the format of the plot features any non-standard elements.