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. $$ \begin{array}{lllll} 1 & 2 & 1 & 0 & 2 \\ 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 1 & 0 \\ 0 & 1 & 2 & 1 & 1 \end{array} $$ What proportion of the measurements are less than \(2 ?\)

Short answer

Answer: The proportion of measurements less than 2 is 0.75 or 75%.

Step-by-step solution

Count the occurrences of each value

Count the number of times that each value (0, 1, and 2) appears in the dataset: $$ \begin{array}{lllll} 1 & 2 & 1 & 0 & 2 \\ 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 1 & 0 \\ 0 & 1 & 2 & 1 & 1 \end{array} $$ Occurrences: - Value \(0\) appears \(5\) times - Value \(1\) appears \(10\) times - Value \(2\) appears \(5\) times

Calculate the relative frequencies

Since there are \(20\) measurements in total, we can calculate the relative frequency of each value by dividing the number of occurrences of each value by the total number of measurements: Relative frequencies: - Value \(0\): \(\frac{5}{20}=\frac{1}{4}=0.25\) - Value \(1\): \(\frac{10}{20}=\frac{1}{2}=0.5\) - Value \(2\): \(\frac{5}{20}=\frac{1}{4}=0.25\)

Create the relative frequency histogram

We'll create a histogram with three bars representing values \(0\), \(1\), and \(2\). The height of each bar corresponds to the relative frequency of each value: | Value | Relative Frequency | |-------|--------------------| | 0 | 0.25 | | 1 | 0.5 | | 2 | 0.25 |

Calculate the proportion of measurements less than \(2\)

To find the proportion of measurements that are less than \(2\), we'll add the relative frequencies of values \(0\) and \(1\): Proportion less than \(2\) = \(0.25 + 0.5 = 0.75\) Thus, \(75\%\) of the measurements are less than \(2\).

Key concepts

Descriptive Statistics

Descriptive statistics involves summarizing and organizing data so that it can be easily understood. This field uses statistics and graphic representations to describe the features of a data set, including measures of central tendency such as mean, median, and mode, and measures of variability like standard deviation and variance.

For example, the problem provided allows us to apply descriptive statistics to understand a dataset of 20 measurements that take on values of 0, 1, or 2. By counting the occurrence of each value and calculating the relative frequencies, we can summarize the data effectively. The histogram is then used to visualize these frequencies in a way that instantly communicates the proportion of each value in the dataset.

When interpreting the calculated relative frequencies (0.25 for value 0, 0.5 for value 1, and 0.25 for value 2), it becomes clear that the value '1' appears most frequently, making it the mode of the dataset. Descriptive statistics provide a foundation for all statistical reasoning and the basis for both data visualization and understanding probability distributions.

Data Visualization

Data visualization is the graphical representation of information and data. By using visual elements like charts, graphs, and maps, data visualization provides an accessible way to see and understand trends, outliers, and patterns in data. In our exercise, a relative frequency histogram serves as the data visualization tool.

Creating a relative frequency histogram, as in this exercise, involves displaying the relative frequencies of different values in a dataset. Each bar's height reflects the proportion of the data that corresponds to the value on the x-axis. With a glance, you can see that the relative frequencies for values 0 and 2 are the same, and value 1 is twice as frequent. This graphic representation makes the analysis much more intuitive than comparing raw numbers.

A well-constructed histogram can not only convey the distribution of the data but also highlight the central tendencies and the amount of variability in the data. In educational settings, reinforcing the connection between numerical data and visual representation deepens understanding and aids in memory retention of statistical concepts.

Probability Distribution

Probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. It is fundamental to the field of probability and statistics as it provides a mathematical description of the real world in terms of the outcomes of a random process.

In the context of the relative frequency histogram from our exercise, we can see that the values follow a uniform probability distribution - this is because values 0 and 2 have the same frequency and value 1 is exactly double that frequency, with no other values present. If we were to generalize this to a probability distribution, we would say that there is a 25% chance of drawing a 0, 50% chance of drawing a 1, and 25% chance of drawing a 2 from the data set.

Understanding probability distributions is essential for making predictions based on data. It helps in determining the likelihood of different outcomes, which is critical in many fields like finance, insurance, and areas of engineering. In educational materials, conveying the real-world application of probability distributions can make the concept more engaging and relevant to students.