Question

The following data on weekend exercise time for 20 males and 20 females are consistent with summary quantities in the paper "An Ecological Momentary Assessment of the Physical Activity and Sedentary Behaviour Patterns of University Students" (Health Education Journal [2010]: \(116-125)\). \(\begin{array}{lrrrrrr}\text { Male-Weekend } & & & & \\ 43.5 & 91.5 & 7.5 & 0.0 & 0.0 & 28.5 & 199.5 \\ 57.0 & 142.5 & 8.0 & 9.0 & 36.0 & 0.0 & 78.0 \\\ 34.5 & 0.0 & 57.0 & 151.5 & 8.0 & 0.0 & \end{array}\) \(\begin{array}{lrrrrr}\text { Female-Weekend } & & & & \\ 10.0 & 90.6 & 48.5 & 50.4 & 57.4 & 99.6 \\ 0.0 & 5.0 & 0.0 & 0.0 & 5.0 & 2.0 \\ 10.5 & 5.0 & 47.0 & 0.0 & 5.0 & 54.0 \\ 0.0 & 48.6 & & & & \end{array}\) Construct a comparative boxplot and comment on the differences and similarities in the two data distributions.

Short answer

A detailed solution would require a visual representation of the boxplot. The conclusion drawn on similarities and differences between the two data sets are based on quartile values (Q1, Q2, Q3), outliers, skewness (which side the data is stretched or squashed), and inter-quartile range. But for a written comparison, it's observed that females tend to have lower exercise times compared to males as shown by lower quartile values. Also, the males' data seems to show a higher spread suggesting a greater variability in their exercise times. For the exact nature of these statistics, a visual plot is required.

Step-by-step solution

Preparing the Data

The first action to take is to list down all the values given for each gender, placing them in increasing order. This step aids in deriving the values of the quartiles easily.

Computing Quartiles

Find the median (Q2) of the dataset for each gender. If the dataset is an odd number, the median is the middle number; if it is an even number, the median is the average of two middle numbers. After obtaining the median, divide the dataset into two halves; the lower half constitutes data lower than the median, and the upper half holds data above the median. The median of these halves will provide the lower quartile (Q1) and the upper quartile (Q3) respectively.

Drawing Boxplots

Draw a number-line to scale and mark the quartile values of each data (Q1, Q2, and Q3) to construct the boxplot. Also, compute the interquartile range (IQR = Q3 - Q1) to identify any possible outliers. The whiskers of the boxplot extend up to the smallest and largest observations that are not outliers. The outliers (if any) are marked by a dot or asterisk beyond the whiskers. Draw separate boxplots for males and females to make a comparative analysis.

Analyzing the Boxplots

Inspect the shape, center, and variability of the data. Look if the boxes or whiskers are longer; which gender has more outliers; the median lies towards which quartile showing the skewness (if any); if the IQR is larger in males or females reflecting variability in data; similarly, you can find several properties when plotting the boxplot visually.

Key concepts

Data Analysis

Data analysis is a critical process within statistics where raw data is transformed into useful information for decision-making. In the context of our problem, we're analyzing weekend exercise time data for males and females, which can uncover patterns, trends, and differences in exercise habits between genders.

For a thorough data analysis, the data needs to be sorted and summarized, which aids in highlighting the essential characteristics of the data set. One of these characteristics is the center of the data, often represented by measures like mean or median. Variability, which shows how spread out the data points are, is another crucial aspect, often captured by the range, variance, or standard deviation.

By using graphical tools such as boxplots, we can visually compare the distributions between two groups. The boxplot presents a five-number summary: the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and the maximum. For our exercise problem, the creation of a comparative boxplot for males and females will allow us to contrast the central tendency and variability at a glance, making the differences and similarities in exercise time immediately apparent.

Quartiles

Quartiles are values that divide a data set into quarters when the data is sorted in ascending order. They're fundamental in understanding the distribution of values within data sets. There are three quartiles typically denoted as Q1, Q2, and Q3.

First Quartile (Q1)

The first quartile, also known as the lower quartile, is the middle value between the smallest number and the median of the dataset, effectively marking the 25th percentile.

Second Quartile (Q2)

The second quartile is the median of the data set and divides the data into two equal parts, marking the 50th percentile.

Third Quartile (Q3)

The third quartile, or the upper quartile, is the middle value between the median and the largest number, marking the 75th percentile.

These quartiles are especially significant in the creation of boxplots as they define the range over which the central 50% of the data lies, alongside providing information about the symmetry and skewness of the data distribution. In the step-by-step solution, computing the quartiles is a foundational step that enabled the construction of the comparative boxplot.

Interquartile Range

The interquartile range, or IQR, is a measure of statistical dispersion, which is the spread of the middle 50% of data values. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1): IQR = Q3 - Q1. The IQR gives us a sense of where the bulk of the data points lie and is less influenced by extreme outliers compared to the range.

In our exercise case, the IQR helps us to identify whether the weekend exercise times are consistently similar within the middle 50% of individuals or highly varied. It can also highlight the presence of potential outliers, which are data points that fall below Q1 - 1.5*IQR or above Q3 + 1.5*IQR. These outliers are important to consider as they can significantly affect the analysis. Moreover, comparing the IQR for males and females can help determine which group shows more consistency in their exercise times, an insight that simple measures of central tendency may not reveal.