Question

Construct a relative frequency histogram for these 20 measurements on a discrete variable that can take only the values \(0, I,\) and 2. Then answer the questions. $$ \begin{array}{lllll} 1 & 2 & 1 & 0 & 2 \\ 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 1 & 0 \\ 0 & 1 & 2 & 1 & 1 \end{array} $$ Describe the shape of the distribution. Do you see any outliers?

Short answer

Based on the given data, create a frequency distribution table, calculate the relative frequencies, construct a relative frequency histogram, analyze its shape, and determine if there are any outliers.

Step-by-step solution

Create a frequency distribution table

To create a frequency distribution table, count the occurrences of each unique value (0, 1, and 2) in the given data. $$ \begin{array}{c|c} Values & Frequency\\\hline 0 & 6 \\ 1 & 8 \\ 2 & 6 \\ \end{array} $$

Calculate Relative Frequencies

Divide the frequency of each value by the total number of values (20) to get the relative frequency of each value. $$ \begin{array}{c|c|c} Values & Frequency & Relative \ Frequency\\\hline 0 & 6 & \frac{6}{20}=0.3 \\ 1 & 8 & \frac{8}{20}=0.4 \\ 2 & 6 & \frac{6}{20}=0.3 \\ \end{array} $$

Construct a Relative Frequency Histogram

Draw a histogram with the x-axis representing the values (0, 1, and 2) and the y-axis representing the relative frequencies (0.3, 0.4, and 0.3). Plot bars for each value, and make sure the bars have the same width.

Analyze the Shape of the Distribution

By examining the histogram, we can see that the distribution is fairly symmetric, with values 0 and 2 having equal relative frequencies (0.3) and value 1 having a slightly higher relative frequency (0.4).

Identify Outliers

Since the distribution is symmetric, with no significant gaps between the bars, and all values occur within the range of the discrete variable (0, 1, and 2), we can conclude that there are no outliers in this dataset.

Key concepts

Frequency Distribution

When working with data, understanding its organization is essential. Frequency distribution is a method used to represent data in an organized format. It involves listing the possible values of a variable along with the frequency, or number of times each value occurs. In the given exercise, we constructed a frequency distribution table for the discrete variable with values 0, 1, and 2.

Here's how you create a frequency distribution table:

• Identify the unique values in the dataset. In this case, the values are 0, 1, and 2.
• Count how often each value appears. For instance, the value 1 appears 8 times in the dataset.
• Display these counts in a table format, aligning each value with its frequency.

This table helps visualize the concentration of data points, offering a glance at which values are more common in the dataset. It's the building block for analyzing data distributions further.

Discrete Variable

In statistics, variables are categorized into different types, such as discrete and continuous. A discrete variable is characterized by a finite number of distinct values. Unlike continuous variables, which can take any value within a range, discrete variables jump from one value to another.

In our exercise, the discrete variable is limited to three distinct values: 0, 1, and 2. This means the variable can only take these specific values and not any in-between values such as 1.5. Discrete variables are often used in situations where measurements are counted rather than measured.

Some key points to remember about discrete variables include:

• They often represent counts or items, like the number of students in a class.
• Each value is separate and distinct.
• They can be easily displayed using methods like bar charts or histograms.

Understanding the nature of discrete variables is crucial in choosing the right methods to analyze and visualize data effectively.

Histogram Analysis

A histogram is a graphical representation that organizes a group of data points into user-specified ranges. It's similar to a bar chart but suitable for displaying the frequency distribution of a dataset. In particular, with a relative frequency histogram, like in our exercise, each bar represents the relative frequency of a data point rather than just the count.

To construct a relative frequency histogram:

• The x-axis represents the discrete variable values, which are 0, 1, and 2 in this case.
• The y-axis represents the relative frequencies, calculated by dividing the frequency of each value by the total number of observations.
• Bars of equal width are drawn, each representing a different value of the discrete variable, and their heights correspond to their relative frequencies.

Analyzing the histogram involves interpreting the shape of the distribution. In our example, the distribution is symmetric, with similar frequencies for values 0 and 2, and a slightly higher frequency for value 1. This central tendency suggests a balanced dataset. Since no significant gaps exist between the bars, the histogram indicates no outliers are present.