Question

A researcher applied the carcinogenic (cancercausing) compound benzo(a)pyrene to the skin of five mice, and measured the concentration in the liver tissue after 48 hours. The results (nmol/gm) were as follows: \(^{21}\) $$\begin{array}{lllll}6.3 & 5.9 & 7.0 & 6.9 & 5.9\end{array}$$ Determine the mean and the median.

Short answer

The mean concentration is 6.4 nmol/gm, and the median concentration is 6.3 nmol/gm.

Step-by-step solution

List the Data

Begin by listing the given concentrations of benzo(a)pyrene in the liver tissue of the mice, measured in nmol/g: 6.3, 5.9, 7.0, 6.9, 5.9.

Calculate the Mean

To calculate the mean (average) concentration, sum all the concentrations and divide by the number of data points. Mean = (6.3 + 5.9 + 7.0 + 6.9 + 5.9) / 5.

Organize the Data in Ascending Order

For calculating the median, the data must be ordered from smallest to largest. Here's the ordered set: 5.9, 5.9, 6.3, 6.9, 7.0.

Calculate the Median

The median is the middle value in the ordered set. Since there is an odd number of data points, the median is the third number: 6.3.

Key concepts

Mean Calculation

The mean, often referred to as the average, is a fundamental concept in statistics that provides a simple summary of a set of numerical data. To calculate it, you sum up all the values in a dataset and then divide that total by the number of values.

In the context of our exercise, we have five measurements of benzo(a)pyrene concentration in the liver tissue of mice: 6.3, 5.9, 7.0, 6.9, and 5.9 nmol/g. By adding these together, we get a total of 32.0 nmol/g. Dividing this sum by the number of measurements, which is 5, we get the mean concentration of 6.4 nmol/g.

The mean is especially useful for understanding the central tendency of data, but it's sensitive to outliers, which are atypical values that differ significantly from other observations in the dataset. Despite its susceptibility to extreme values, the mean is a reliable indicator of the middle of a dataset when the values are relatively evenly distributed.

Median Calculation

The median, another measure of central tendency, is the value at the center of an ordered dataset. Unlike the mean, it is not influenced by outliers, which makes it a more robust representation of a dataset's central point in certain cases. To compute the median, we first need to arrange the data in ascending or descending order.

If there is an odd number of observations, the median is the number that sits exactly in the middle. If instead there is an even number, the median is obtained by averaging the two central numbers. For our dataset, when we arrange the values (5.9, 5.9, 6.3, 6.9, 7.0) in order, we can see that the median is 6.3, as it is the third value in the list. This median value is particularly insightful when dealing with skewed data or when you want to find a typical value that isn't affected by outliers.

Data Analysis

Data analysis is an overarching term for the various methods and techniques used to inspect, clean, and model data with the aim of discovering useful information, drawing conclusions, and supporting decision making. Descriptive statistics, like the mean and median we've just computed, are vital tools in data analysis.

In analyzing our dataset, we started by summarizing the measurements of a carcinogenic compound in the liver tissues of mice. Through descriptive statistics, we are able to convert raw data into meaningful information—such as identifying the typical concentration level (using the mean) and the central tendency that is unaffected by extreme values (using the median). It's the initial stage of data analysis that can guide further inquiry and helps us understand what questions to ask next. Whether in biostatistics or other fields, data analysis lays the foundation for informed and evidence-based conclusions.

Biostatistics

Biostatistics is the application of statistics to a wide range of topics in biology, and it's particularly crucial in the fields of medicine, public health, and ecology. It includes the design of biological experiments and the collection, summarization, analysis, interpretation, and presentation of data related to biological phenomena.

In our exercise, we used biostatistical methods to determine the central tendency of concentrations of a chemical in animal tissue. Such analyses are essential in biostatistics as they can inform further research and policy decisions, like evaluating the safety and efficacy of new drugs or assessing the impact of environmental toxins on health. In our dataset, knowing the mean and median concentrations can help researchers understand distribution patterns and potential risks associated with the compound under study.