Question

The red blood cell count of a healthy person was measured on each of 15 days. The number recorded is measured in millions of cells per microliter ( \(\mu \mathrm{L}\) ). \(\begin{array}{lllll}5.4 & 5.2 & 5.0 & 5.2 & 5.5\end{array}\) \(\begin{array}{lllll}5.3 & 5.4 & 5.2 & 5.1 & 5.3\end{array}\) \(\begin{array}{lllll}5.3 & 4.9 & 5.4 & 5.2 & 5.2\end{array}\) a. Use a stem and leaf plot to describe the data. b. Describe the shape and location of the red blood cell counts. c. If the person's red blood cell count is measured today as 5.7 million cells per microliter, would this be unusual? What conclusions might you draw?

Short answer

Answer: The shape of the red blood cell counts is slightly left-skewed with a minimum of 4.9 million cells per microliter, a maximum of 5.5 million cells per microliter, and a median of 5.2 million cells per microliter. A red blood cell count of 5.7 million cells per microliter would be considered unusual since it is significantly higher than the maximum value in the dataset.

Step-by-step solution

Organize the data

First, let's organize the data in ascending order: 4.9, 5.0, 5.1, 5.2, 5.2, 5.2, 5.2, 5.2, 5.3, 5.3, 5.3, 5.4, 5.4, 5.4, 5.5

Create the stem and leaf plot

Now, we'll create the stem and leaf plot where stems represent the digits before the decimal point and the leaves represent the digits after the decimal point. Stem | Leaves ------------ 4 | 9 5 | 0 1 2 2 2 2 2 3 3 3 4 4 4 5

Describe the shape and location

By examining the stem and leaf plot, we can describe the shape and location of the red blood cell counts: - The distribution is slightly left-skewed, as most values are concentrated at the higher end. - The data have a Minimum value of 4.9 million cells per microliter and a Maximum value of 5.5 million cells per microliter. - The median can be found as there are 15 data points, with the middle value 5.2 (5th, 6th, and 7th values).

Analyze the case of 5.7 million cells per microliter

Given the person's red blood cell count at 5.7 million cells per microliter today, it seems to be an unusual observation since it's significantly higher than the Maximum value in our dataset (5.5 million cells per microliter). Possible conclusions: 1. The measurement error occurred, and the count is not accurate. 2. The person's health changed, resulting in a higher red blood cell count. 3. This higher count might be an outlier, and measuring the red blood cell count a few more times might give us a clearer understanding of whether this count is typical or not.

Key concepts

Stem and Leaf Plot

A stem and leaf plot is a simple yet powerful tool that can help us visualize numerical data in a structured way. Each number is divided into two parts:

• Stem - The part before the decimal point.
• Leaf - The part after the decimal point.

In the exercise data, numbers are listed in the form millions of cells per microliter. By creating a stem and leaf plot, we can easily see the distribution of data, and it is particularly useful for detecting clusters, gaps, or outliers.

Let's take the red blood cell count example. Here, the 'stem' represents the whole number, while the 'leaf' displays the decimal part. For instance, a cell count of 5.4 is represented with the stem "5" and the leaf "4".

• The stem "4" has a leaf of "9", indicating one data point, 4.9.
• The stem "5" has multiple leaves, "0, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5", showcasing variability and clusters around certain values.

This plot helps to clearly see which figures occur most frequently and outlines a visual representation of the data.

Data Distribution

Data distribution is about how the data values are spread or dispersed across the range. The exercise dataset, showing the red blood cell counts, suggests certain patterns in the data. When analyzing the stem and leaf plot, we notice a slightly left-skewed distribution:

• Most data points lie towards the higher end of our range.
• The minimum and maximum values are clear, with the lowest being 4.9 and the highest being 5.5.

As the values range from 4.9 to 5.5, understanding the distribution can help in making sense of data trends and deviations. A central tendency, like the median, offers insights into where most of our data points lie. In this case, the median is 5.2 as indicated by the middle values in the dataset. Knowing the spread aids in interpreting how the values compare and contrast with each other.

Outliers

Outliers are data points that differ significantly from other observations in the dataset. They can provide critical insights or indicate potential anomalies. In analyzing the given exercise, the typical range of red blood cell counts is from 4.9 to 5.5.

An observed value of 5.7 million cells per microliter stands out as an outlier, not fitting within the range of our 15-day observations. Several factors might be at play:

• An outlier might signal error in data collection or reporting.
• A natural variation due to unusual conditions on that measurement day.
• An indicator of a biological change or condition affecting the person’s health.

Determining whether 5.7 is an outlier requires further data collection and analysis. This would clarify if the count is repeated across multiple days or if it's simply a random occurrence.

Statistical Analysis

Statistical analysis is a method to examine and interpret data to identify patterns, trends, and insights. With the current dataset of red blood cell counts, statistical analysis aids in identifying the normal range and spotting deviations or anomalies such as outliers.

When performing statistical analysis, several steps contribute to a comprehensive understanding:

• Organizing data systematically for clarity.
• Identifying median, mode, and mean to understand central tendencies.
• Analyzing distribution patterns to detect skewness or symmetry.
• Identifying outliers and assessing their potential impact.

In this exercise, the process led us to identify 5.7 as an unusual number. Based on statistical insights, it was analyzed as potentially significant, suggesting either measurement error, a unique health condition, or simply an outlier. These steps showcase how statistical analysis not only explains data behaviors but also aids in informed decision-making based on quantitative evidence.