Question

From the StudentSurvey dataset, select any categorical variable and select any quantitative variable. Use technology to create side-by-side boxplots to examine the relationship between the variables. State which two variables you are using and describe what you see in the boxplots. In addition, use technology to compute comparative summary statistics and compare means and standard deviations for the different groups.

Short answer

Gender ('Male', 'Female') and 'GPA' were selected as categorical and quantitative variables, respectively. In the boxplots, suppose the median GPA for males is higher than that for females leading to inference of males having better average GPA. Analysis of computed summary statistics further confirms this, if mean GPA of males exceeds that of females and standard deviation of scores is lower for males, implying better and consistent performance.

Step-by-step solution

Select Variables

Choose two variables from the StudentSurvey dataset, specifically one categorical and one quantitative variable. For this example, let's select 'Gender' as the categorical variable and 'GPA' as the quantitative variable.

Create Side-by-Side Boxplots

Use software tools like R, Python, or Excel to create side-by-side boxplots for the chosen variables. The categorical variable (Gender) should be represented on the x-axis and the quantitative variable (GPA) on the y-axis. The boxplots will visually show the distribution and dispersion of data for each category (Male, Female).

Analyze the relationship

Examine the boxplots to analyze the relationship between Gender and GPA. Look for features such as the median (line in the box), interquartile range (size of the box), and outliers (dots outside the whiskers). For instance, if the median GPA for males is higher than that for females, it can be inferred that males have better average grades.

Compute Comparative Summary Statistics

With the help of statistical software, compute the mean and standard deviation of GPA for each gender category. These values will allow further analysis and comparison of GPA performance.

Compare Means and Standard Deviations

Compare the computed means and standard deviations for males and females. If males have a higher mean GPA and a lower standard deviation than females, it can be assumed that males generally perform better and with less variability in their performance.

Key concepts

Categorical Variable

A categorical variable is a type of data that can be grouped into different categories or groups that are qualitative in nature. Unlike numerical or quantitative data, categorical data represent characteristics such as a person’s gender, marital status, hometown, or the brand of a product. These variables are essential in research and statistics because they allow us to classify subjects into distinct groups to see how they compare across different levels.

For instance, in the exercise from the StudentSurvey dataset, 'Gender' is the chosen categorical variable. This means that we can categorize survey participants by their gender—often male, female, or other—and it doesn’t make sense to apply mathematical operations to these categories. Gender is nominal, with no inherent order. By using categorical variables in side-by-side boxplots, we can visually compare different groups and analyze variations in a related quantitative variable.

Quantitative Variable

In contrast to categorical variables, quantitative variables are numerical and can be measured on a numeric scale. They are data points that can represent amounts or quantities and allow us to perform mathematical computations. Quantitative variables can be either continuous or discrete. Continuous variables represent measurements and can take any value within a range, such as height, weight, or time. Discrete variables, on the other hand, represent counts and can only take specific values, such as the number of siblings a person has.

The 'GPA' from our StudentSurvey dataset is an example of a continuous quantitative variable because Grade Point Averages can theoretically take on any value between the minimum and maximum possible scores. Analyzing quantitative variables, such as GPA, alongside categorical variables, like Gender, helps reveal trends and patterns, and boxplots are an ideal way to visually display this information.

Summary Statistics

Summary statistics provide a way to summarize and describe the main features of a dataset using a few numbers. These statistics include measures of central tendency, like the mean (average) and median (middle value), and measures of dispersion, like the range (difference between the highest and lowest values), interquartile range (middle 50% of values), variance, and standard deviation (how spread out the values are).

When creating side-by-side boxplots, summary statistics such as the median are visually depicted by the line in the center of the box, while the interquartile range is represented by the box itself. These statistics are crucial for interpreting the data because they help us to quickly understand the distribution, central tendency, and variability of a quantitative variable across different categories of a categorical variable.

Compare Means and Standard Deviations

Comparing means and standard deviations is essential in statistical analysis when we want to understand differences between groups. The mean provides a measure of central tendency, offering a snapshot of the 'typical' value in a dataset, while the standard deviation reveals the variability or spread of the data around this mean.

In our exercise, comparing the means and standard deviations of GPAs between different genders shows us not only which group has the higher average GPA but also which group displays more consistency in their academic performance. A higher standard deviation in one group could indicate a wider range of abilities or a greater diversity in the performance of its members. This comparison can lead to deeper insights, especially when visual tools like side-by-side boxplots are used, because these plots illustrate such statistics in a clear and interpretable way.