Question

On January \(14,2012,\) Andrew Brown of Great Britain set the world record time ( 40 days) for rowing solo across the northern Atlantic Ocean. On March 14 , 2010, Katie Spotz of the United States became the youngest person to ever row solo across the Atlantic when she completed it in 70 days at the age of 22 years old. Table 1.3 shows times for males and females who rowed solo across the Atlantic Ocean in the last few years. \(^{10}\) (a) How many cases are there in this dataset? How many variables are there and what are they? Is each categorical or quantitative? (b) Display the information in Table 1.3 as a dataset with cases as rows and variables as columns. $$ \begin{array}{ll} \hline \text { Male times: } & 40,87,78,106,67 \\ \text { Female times: } & 70,153,81 \\ \hline \end{array} $$

Short answer

There are 8 cases within this dataset. There are two variables, 'sex' which is a categorical variable and 'time' which is a quantitative variable. The dataset structured with cases as rows and variables as columns is completed as per part (b) of the exercise.

Step-by-step solution

Identify the Cases and Variables

Inspect the exercise. Recall that cases are the individuals about whom we have data. In this exercise, the cases are the rowers who have rowed solo across the Atlantic Ocean, represented by the males and females in the data set. The variables are the two types of information collected about these rowers: their gender (sex), which is categorical data, and their time for crossing the Atlantic Ocean, which is quantitative data.

Count the Cases and Variables

Count the cases present in the data. From the data set, there are 5 male rower times and 3 female rower times, so there are 8 (=5+3) cases in total. As we've determined above, there are 2 variables (sex and time).

Display the Information as a Dataset

In accordance with part (b) of the exercise, we'll display the information with the cases represented as rows and the variables as columns. | Sex | Time (days) ||------- |-------------|| Male | 40 || Male | 87 || Male | 78 || Male | 106 || Male | 67 || Female | 70 || Female | 153 || Female | 81 | This has the cases (the rowers) represented as rows and the variables (sex and time) represented as columns.

Key concepts

Cases and Variables in Data

Understanding the concepts of cases and variables is crucial when dealing with data sets in statistics. Cases, sometimes referred to as units or subjects, are the entities about which data is collected. In our exercise, the cases are the individual rowers who have completed the solo Atlantic crossing.

Variables, on the other hand, provide information about each case. There are two main types of variables: categorical and quantitative. In the rowing data set, we have one of each type — 'sex' is a categorical variable indicating the gender of the rower, and 'time' is a quantitative variable indicating the number of days it took to complete the crossing.

It's essential to clearly identify cases and variables as they lay the foundation for any analysis. The number of cases and variables can determine the type of statistical methods that will be used for analyzing the data. In the context of our exercise, with the data neatly broken into cases and variables, we can proceed to analyze patterns, such as whether females take longer on average than males to row across the Atlantic and other interesting questions that the data might hold.

Categorical vs Quantitative Data

Distinguishing between categorical and quantitative data is key to understanding how to handle and interpret different types of information. Categorical data represent characteristics and can be divided into groups or categories. For instance, in our Atlantic crossing data set, 'sex' with categories 'Male' and 'Female' is a categorical variable.

Quantitative data, however, are numeric and can be measured or counted. They often answer questions like 'how much?' or 'how many?'. The 'time' taken to cross the Atlantic, measured in days, is a quantitative variable since it can be numerically analyzed to find, say, averages or ranges.

Why does the distinction matter?

It informs the statistical techniques used for analysis. For categorical data, you might use modes or chi-squared tests, while for quantitative data, you could look at mean, median, standard deviation, or employ regression analysis. Making the right distinction impacts the insights you can obtain from the data.

Data Representation

Presenting data in a clear and organized manner allows for better understanding and analysis. Data representation is particularly important because it directly affects how the subsequent data analysis will be performed and interpreted. The step-by-step solution showed how information from Table 1.3 was transformed into a dataset format with rows representing cases and columns representing variables.

Creating a structured data set as demonstrated can help in visualizing and performing descriptive statistics more effectively. We often use tables, graphs, and charts to represent data visually. For example, to represent our rowing data, we could use a bar graph to compare the average times of males and females, or a scatterplot to display each individual's time.

Practical Tips for Data Representation:

• Ensure accuracy: Mistakes in representation can lead to incorrect conclusions.
• Choose the right form: Different types of data and analysis may require different forms of representation.
• Clarity is key: The goal of data representation is to make the information understandable at a glance.

Effective data representation is an art that, when done correctly, can tell a story as insightful and complex as the data itself.