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To understand better the effects of exercise and aging on various circulatory functions, the article "Cardiac Output in Male Middle-Aged Runners" (Journal of Sports Medicine [1982]: 17-22) presented data from a study of 21 middle- aged male runners. The following data set gives values of oxygen capacity values (in milliliters per kilo- gram per minute) while the participants pedaled at a specified rate on a bicycle ergometer: \(\begin{array}{lllllll}12.81 & 14.95 & 15.83 & 15.97 & 17.90 & 18.27 & 18.34 \\\ 19.82 & 19.94 & 20.62 & 20.88 & 20.93 & 20.98 & 20.99 \\ 21.15 & 22.16 & 22.24 & 23.16 & 23.56 & 35.78 & 36.73\end{array}\) a. Compute the median and the quartiles for this data set. b. What is the value of the interquartile range? Are there outliers in this data set? c. Draw a modified boxplot, and comment on the interesting features of the plot.

Step by step solution

01

Calculate the Median and Quartiles

First, sort the values from lowest to highest. To find the median value (Q2), which splits the data into two halves, find the middle value. If there is an even number of observations, take the mean of the two middle numbers. For the first quartile (Q1), find the middle value between the smallest number and the median. For the third quartile (Q3), find the median between the largest number and the median.

02

Calculation of Interquartile Range and Identification of Outliers

The interquartile range (IQR) can be found by subtracting Q1 from Q3. Outliers can be identified by using the formula \(Q1 - 1.5*IQR\) and \(Q3 + 1.5*IQR\). Any value outside of these values could be considered as an outlier.

03

Create a Modified Boxplot and Discuss its Features

To create a modified boxplot, first draw a rectangular box from Q1 to Q3 and draw a line at the median. Then, draw lines (called whiskers) from the box to the maximum and minimum values that are not outliers. Outliers should be denoted separately, typically with a cross or a different symbol. In terms of interesting features, it should be noted where most of the data lies (i.e., what portion of the box the median line is closest to). Additionally, note the existence and location of any outliers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Median

The median is a crucial statistic in understanding data, as it represents the middle value in a sorted data set.
This can give you a clearer picture of the data distribution, especially when compared to the mean, which can be skewed by outliers.
To find the median, arrange the data in ascending order and look for the center value.

• If your data set has an odd number of observations, the median is the middle number.
• If your data set has an even number of observations, the median is the average of the two middle numbers.

In the case of the oxygen capacity values provided, the median helps determine the central tendency without being affected by extreme values like 35.78 or 36.73.

Quartiles

Quartiles split your data into four equal parts, helping to understand your data at different percentile points.
The most important quartiles are the first quartile (Q1) and the third quartile (Q3).

• Q1 (First Quartile): This is the median of the first half of the data. It represents the 25th percentile, meaning 25% of the data points fall below or at this value.
• Q3 (Third Quartile): This is the median of the second half of the data and denotes the 75th percentile, indicating that 75% of the data is below or at this value.

Identifying quartiles is essential for calculating the interquartile range and understanding the spread of data.

Interquartile Range

The interquartile range (IQR) measures the range within which the central half of your data lies.
It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1):\[ \text{IQR} = Q3 - Q1 \]This value helps to understand the spread and variability of the middle 50% of your data.
A smaller IQR indicates that the data points are closer together, while a larger IQR signifies more spread out data points.
When considering data like oxygen capacity values, the IQR gives insights into how consistent or variable the runners' oxygen capacities are under specific conditions.

Boxplot

A boxplot is a graphical representation of data that displays the quartiles and median, allowing for a quick visual assessment of distribution. To create a boxplot:

• Draw a box from Q1 to Q3.
• Draw a line across the box at the median value (Q2).
• Add lines, or whiskers, from the box to the smallest and largest observations that are not outliers.
• Identify outliers with individual markers such as dots or stars.

Boxplots are particularly useful for spotting anomalies and getting an intuitive sense of the data's dispersion and symmetry.
For example, in analyzing the oxygen capacity data, a boxplot can instantly reveal skewness towards the higher or lower values, and highlight any anomalies.

Outliers

Outliers are data points that differ significantly from other observations in the data.
They can distort statistical analyses, making it important to identify and understand them. Outliers can be identified using the IQR:

• Calculate the boundaries for outliers:
  
  Any value outside of \[ Q1 - 1.5 \times \text{IQR} \text{ \ and \ } Q3 + 1.5 \times \text{IQR} \] could be considered an outlier.

In the context of the oxygen capacity study, certain values like 35.78 and 36.73 would flag as outliers, as they significantly deviate from the rest.
Understanding these can help identify unusual conditions or errors in data collection or just genuine variability in a biological parameter.