Question

Pulse Rates A group of 50 biomedical students recorded their pulse rates by counting the number of beats for 30 seconds and multiplying by \(2 .\) \(\begin{array}{llllllllll}80 & 70 & 88 & 70 & 84 & 66 & 84 & 82 & 66 & 42 \\\ 52 & 72 & 90 & 70 & 96 & 84 & 96 & 86 & 62 & 78 \\ 60 & 82 & 88 & 54 & 66 & 66 & 80 & 88 & 56 & 104 \\ 84 & 84 & 60 & 84 & 88 & 58 & 72 & 84 & 68 & 74 \\\ 84 & 72 & 62 & 90 & 72 & 84 & 72 & 110 & 100 & 5888\end{array}\) a. Why are all of the measurements even numbers? b. Draw a stem and leaf plot to describe the data, splitting each stem into two lines. c. Construct a relative frequency histogram for the data. d. Write a short paragraph describing the distribution of the student pulse rates.

Short answer

Answer: All the pulse rate measurements are even numbers because they are obtained by counting the number of beats for 30 seconds and then multiplying by 2. Since the number of beats in 30 seconds will always be a whole number, and we are doubling that number, the result will always be an even number. The distribution of the student pulse rates is slightly skewed to the right, with the majority of the data points lying between 60 and 99 bpm. The range of the data excluding the outlier (5888) is from 42 to 110 bpm. The distribution shows that the pulse rates for the biomedical students tend to cluster around 60 to 99 bpm, with a few students having pulse rates lower or higher than this range.

Step-by-step solution

a. Explanation for Even Numbers

The pulse rates are measured by counting the number of beats for 30 seconds and then multiplying by 2. Since the number of beats in 30 seconds will always be a whole number, and we are doubling that number, the result will always be an even number.

b. Stem and Leaf Plot

To create a stem and leaf plot with each stem split into two lines, follow these steps: 1. Sort the data in ascending order. 2. Divide the data into stems based on the first digits and leaves based on the second digits. 3. Create two lines for each stem: the first line will contain leaves for the first half of the digits, and the second line will contain leaves for the second half of the digits. Sorted data: 42, 52, 54, 56, 58, 60, 60, 62, 62, 66, 66, 66, 68, 70, 70, 70, 72, 72, 72, 72, 74, 78, 80, 80, 82, 82, 84, 84, 84, 84, 84, 84, 84, 86, 88, 88, 88, 88, 90, 90, 96, 96, 100, 104, 110, 5888 Stem and leaf plot: 4 | 2 5 | 2 4 6 8 6 | 0 0 2 2 6 6 6 8 7 | 0 0 0 2 2 2 2 4 8 8 | 0 0 2 2 4 4 4 4 4 4 4 6 9 | 05 6 6 10 | 048 58 | 88

c. Relative Frequency Histogram

To create a relative frequency histogram, follow these steps: 1. Organize the data into intervals (bins). 2. Count the number of data points in each bin. 3. Calculate the relative frequency of each bin by dividing the count by the total number of data points. 4. Draw a histogram with the bins on the x-axis and the relative frequencies on the y-axis. Here, we choose the bins as follows: 40-59, 60-79, 80-99, 100-119, and 5880-5899 (to account for the outlier), and calculate the relative frequencies: 40-59: 5/50 = 0.1 60-79: 20/50 = 0.4 80-99: 22/50 = 0.44 100-119: 2/50 = 0.04 5880-5899: 1/50 = 0.02 Plot the histogram with the above values on the y-axis and the bin ranges on the x-axis.

d. Description of the Distribution

The distribution of the student pulse rates is slightly skewed to the right, with the majority of the data points lying between 60 and 99 bpm. The range of the data excluding the outlier (5888) is from 42 to 110 bpm. The outlier, 5888 bpm, is an error and should be investigated further or removed from the analysis, as it is significantly higher than the other measurements and not representative of a typical pulse rate. Overall, the distribution shows that the pulse rates for the biomedical students tend to cluster around 60 to 99 bpm, with a few students having pulse rates lower or higher than this range.

Key concepts

Stem and Leaf Plot

A stem and leaf plot is a convenient way to visualize data and see its shape and distribution. It's like a simpler form of a histogram that retains the actual numerical values of the data.
To create this plot, we split numbers into a "stem" (all but the last digit) and a "leaf" (the final digit). This method helps to see patterns, including gaps, clusters, and outliers directly.

In the pulse rate exercise, the plot was made by first arranging the data into ascending order. The stems represented the tens digit of the pulse rates, and the leaves represented the units digit. The trick here was to use two lines per stem to better view the data spread. For instance, pulse rates from 80 to 89 had a stem of 8; all rates in this range were represented by adding their leaves on either the first or second line, helping to show whether numbers were closer to the lower or higher end of that decade.

This setup can easily show how most pulse rates cluster. The exercise revealed a concentration around 60 to 89 beats per minute (bpm), with very few numbers outside this span, except for an outlier, 5888 bpm, which was an error. This plotting method makes it simple to identify values frequently occurring in a dataset.

Relative Frequency Histogram

A relative frequency histogram is a graphical representation that shows how often each data point occurs relative to the total number of points. It's useful for understanding the distribution's shape and comparing datasets of different sizes.
To create one, we first choose data ranges or "bins". Each bin groups a range of data points. We then count how many data points fall into each bin and divide by the total to get the relative frequency.

For the pulse rates, the data was divided into bins such as 40-59, 60-79, etc., each interval representing different frequency ranges. The relative frequency for each bin was calculated by dividing the count of data points in that bin by the total number of students (50, excluding the outlier). This gave values like 0.44 for the 80-99 bpm range, meaning 44% of students fall into this category.

These relative frequencies were plotted on the vertical axis, with the bins on the horizontal. This histogram showed the central peak around 60 to 99 bpm, indicating where most pulse rates lay. It highlighted the concentration of pulse rates in the biomedical students, using a visual and comprehensible format.

Pulse Rates Distribution

Pulse rate distribution provides insights into the spread and central tendency of data. Analyzing distribution enables us to understand common pulse rates and identify any anomalies.
In our analysis of biomedical students, we primarily dealt with numbers ranging from 42 to 110 bpm. Understanding why the distribution is skewed or peaked is essential for identifying trends.

The distribution was slightly skewed to the right, meaning more students had pulse rates on the lower end of the spectrum. Most students had pulse rates between 60 and 99 bpm, showing where the natural center of this dataset lies.
It's crucial to address outliers like the 5888 bpm reading, which could significantly skew results and misrepresent actual trends. Such an outlier is generally considered an error and must be reviewed to avoid bias. Ignoring or adjusting this value in calculations helps reveal a more accurate depiction of student health.

Overall, understanding distribution is key in data analysis. It allows us to describe the dataset comprehensively and spot areas needing further investigation or data cleaning.