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Chapter 2: Problem 51

Arsenic is toxic to humans, and people can be exposed to it through contaminated drinking water, food, dust, and soil. Scientists have devised an interesting new way to measure a person's level of arsenic poisoning: by examining toenail clippings. In a recent study, \({ }^{29}\) scientists measured the level of arsenic (in \(\mathrm{mg} / \mathrm{kg}\) ) in toenail clippings of eight people who lived near a former arsenic mine in Great Britain. The following levels were recorded: \(\begin{array}{ll}0.8 & 1.9\end{array}\) \(\begin{array}{llll}3.9 & 7.1 & 11.9 & 26.0\end{array}\) \(\begin{array}{ll}2.7 & 3.4\end{array}\) (a) Do you expect the mean or the median of these toenail arsenic levels to be larger? Why? (b) Calculate the mean and the median.

Short Answer

Expert verified
The mean arsenic level is 7.34 mg/kg and the median arsenic level is 3.65 mg/kg. You would expect the mean to be larger because it is more sensitive to outliers or extremely high values in the data.

Step by step solution

01

Understand what mean and median are

The mean is the sum of the sample divided by the number of elements in the sample. The median is the middle value of a list when it is ordered in either increasing or decreasing order. If there's an even number of observations, the median will be the average of the two middle numbers.
02

Order the data and calculate the median

First, arrange the data in ascending order: 0.8, 1.9, 2.7, 3.4, 3.9, 7.1, 11.9, 26.0. Since we have 8 observations, the median is the average of the fourth and fifth observations. Therefore, the median is (3.4+3.9)/2 = 3.65 mg/kg.
03

Calculate the mean

Calculate the mean by summing all the data points and dividing by the number of data points: (0.8 + 1.9 + 2.7 + 3.4 + 3.9 + 7.1 + 11.9 + 26.0) / 8 = 7.34 mg/kg.
04

Compare the mean and the median

Comparing these results, it is clear that the mean (7.34 mg/kg) is larger than the median (3.65 mg/kg). A one-off high value of 26.0 mg/kg drives the mean up. However, because the median only considers the middle data points in an ordered list, it is not so affected by potentially skewed data points, making it a more balanced representation of the given data in this case.

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Key Concepts

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

Mean vs Median
When we begin analyzing data, it's crucial to understand two key measures of central tendency: the mean and the median. The mean, often referred to as the average, is computed by summing up all the values in a dataset and then dividing that sum by the number of data points. It represents the central point of a data set.

The median, on the other hand, is the middle value when a data set is ordered from smallest to largest (or vice versa). If there's an even number of observations, the median is the midway point between the two central numbers, found by computing their average.

In the context of the arsenic levels in toenail clippings, we would expect the mean to be higher if there are outliers or extremely high values, as these would skew the mean upwards. The median, unaffected by such anomalies, would likely provide a more representative value of the central tendency in this scenario.
Data Analysis
Data analysis encompasses a variety of techniques to inspect, clean, transform, and model data with the goal of discovering useful information, informing conclusions, and supporting decision-making. One of the primary steps in data analysis is understanding the distribution of data points.

For the arsenic exposure study, by calculating both the mean and median, scientists can gain insights about the data distribution. If the mean is significantly higher than the median, as is the case with the arsenic levels, it suggests that the data is right-skewed and that there are a few very high values affecting the mean.
Measures of Central Tendency
Measures of central tendency are statistical tools used to summarize a set of data by identifying the center point of its distribution. The three main measures are the mean, median, and mode. The mean provides a mathematical average, the median presents the middle value, and the mode is the most frequently occurring value in a data set.

The choice between these measures depends on the nature of the data and the specific information sought. For data with outliers or a skewed distribution, the median can often be a more reliable measure than the mean, as it doesn't get distorted by extreme values.
Outliers in Data
Outliers are data points that differ significantly from other observations. They can occur due to variability in the measurement or possibly due to experimental error. Outliers can have a pronounced effect on the mean, pulling it toward their value and potentially resulting in a misleading interpretation of the data.

In our exercise, the high arsenic level of 26.0 mg/kg acts as an outlier, causing the mean to misrepresent the typical arsenic exposure. Identifying outliers is an essential part of data analysis, as it can influence which measure of central tendency (mean, median, or mode) best represents the typical data point within the dataset.

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Most popular questions from this chapter

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Scientists are working to train dogs to smell cancer, including early stage cancer that might not be detected with other means. In previous studies, dogs have been able to distinguish the smell of bladder cancer, lung cancer, and breast cancer. Now, it appears that a dog in Japan has been trained to smell bowel cancer. \({ }^{12}\) Researchers collected breath and stool samples from patients with bowel cancer as well as from healthy people. The dog was given five samples in each test, one from a patient with cancer and four from healthy volunteers. The dog correctly selected the cancer sample in 33 out of 36 breath tests and in 37 out of 38 stool tests. (a) The cases in this study are the individual tests. What are the variables? (b) Make a two-way table displaying the results of the study. Include the totals. (c) What proportion of the breath samples did the dog get correct? What proportion of the stool samples did the dog get correct? (d) Of all the tests the \(\operatorname{dog}\) got correct, what proportion were stool tests?

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