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Arsenic in Toenails 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, \(, 9\) 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: \(\begin{array}{llllll}0.8 & 1.9 & 2.7 & 3.4 & 3.9 & 7.1\end{array}\) \(\begin{array}{ll}11.9 & 26.0\end{array}\) (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. 2.62 Fiber in the Diet The number of grams of fiber eaten in one day for a sample of ten people are \(\begin{array}{ll}10 & 11\end{array}\) \(11 \quad 14\) \(\begin{array}{llllll}15 & 17 & 21 & 24 & 28 & 115\end{array}\) (a) Find the mean and the median for these data. (b) The value of 115 appears to be an obvious outlier. Compute the mean and the median for the nine numbers with the outlier excluded. (c) Comment on the effect of the outlier on the mean and on the median.

Step by step solution

01

Arsenic levels analysis

First, sort the arsenic levels in increasing order: 0.8, 1.9, 2.7, 3.4, 3.9, 7.1, 11.9, 26.0.

02

Calculate Arsenic levels mean

To compute the mean (average), add up all the numbers and divide by the total count. The mean arsenic level is: \(\frac{0.8 + 1.9 + 2.7 + 3.4 + 3.9 + 7.1 + 11.9 + 26.0}{8} = 7.34 \mathrm{mg/kg}\) .

03

Calculate Arsenic levels median

To find the median (middle value), pick the number in the middle of the sorted list. If there are an even number of observations, it is the average of the two middle values. The median arsenic level is: \(\frac{3.9 + 7.1}{2} = 5.5 \mathrm{mg/kg}\) .

04

Fiber consumption analysis

First, sort the fiber consumption levels in increasing order: 10, 11, 11, 14, 15, 17, 21, 24, 28, 115.

05

Calculate Fiber consumption mean

The mean fiber consumption is: \(\frac{10 + 11 + 11 + 14 + 15 + 17 + 21 + 24 + 28 + 115}{10} = 26.6 \) grams.

06

Calculate Fiber consumption median

The median fiber consumption is: \(\frac{15 + 17}{2} = 16 \) grams.

07

Excluding the outlier

Remove the value 115 which appears to be an outlier in the fiber consumption data, so the list becomes: 10, 11, 11, 14, 15, 17, 21, 24, 28.

08

Calculate mean and median without the outlier

The mean fiber consumption without the outlier is: \(\frac{10 + 11 + 11 + 14 + 15 + 17 + 21 + 24 + 28}{9} = 16.9 \) grams. The median fiber consumption without the outlier is: 15 grams.

09

Effect of the outlier

The outlier greatly increased the mean. The median, on the other hand, was not affected by the outlier, as it is the middle value and is therefore less sensitive to extreme values.

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Key Concepts

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

Mean vs Median

Understanding the difference between mean and median is crucial in statistical data analysis. The mean, often referred to as the average, is calculated by summing all the numbers in a dataset and then dividing by the total number of values. In contrast, the median is the middle value when the numbers are arranged in order, or the average of the two middle values if there's an even number of observations.

This distinction is significant when analyzing data because the mean is sensitive to extreme values, or outliers, while the median is more robust in the face of these outliers. As we saw in the exercises provided, the arsenic levels had a mean of 7.34 mg/kg and a median of 5.5 mg/kg, while the fiber intake data had a mean substantially skewed by an outlier (115 grams) that did not affect the median as much.

For this reason, the median can often provide a better sense of the 'typical' value in a dataset, particularly when dealing with skewed distributions or when outliers are present. The choice between mean and median depends on the nature of the data and the specific analysis objectives.

Data Outliers

Outliers are data points that differ significantly from other observations. They can arise due to variability in the measurement or may indicate experimental errors. In the fiber intake example, the value of 115 grams is an obvious outlier. When dealing with such data, it is imperative to consider the impact of outliers on the analysis.

Outliers can skew the results of statistical analysis, particularly the mean. They can mislead conclusions about the dataset as a whole if not appropriately managed. In our exercise, excluding the outlier in the fiber data changed the mean from 26.6 grams to 16.9 grams, showing a significant shift. However, the median remained unchanged at 15 grams, demonstrating its resistance to the influence of outliers.

It's essential to investigate outliers to decide whether they are random occurrences or if they signify a pattern that could be important to the analysis.

Arsenic Toxicity Measurement

Arsenic is a toxic element that can be found in various environmental sources, including drinking water, food, and soil. Measuring the levels of arsenic toxicity is critical for assessing exposure and potential health risks. In the given exercise, arsenic levels were measured using toenail clippings, which can reflect long-term exposure due to arsenic's accumulation in keratin-rich tissues.

Determining arsenic concentration in biological samples such as toenails can provide valuable information on the extent of exposure. Statistically analyzing this data can help identify potential health risks and the necessity for public health interventions. For instance, when the mean arsenic levels are high, it may indicate widespread contamination that requires immediate attention.

Dietary Fiber Intake Analysis

Dietary fiber intake is a key indicator of diet quality and has numerous health benefits. Accurate analysis of fiber intake is essential for nutritional epidemiology studies, dietary recommendations, and for assessing an individual's dietary health. The problem we've reviewed shows the significance of identifying and handling outliers when analyzing such data.

High fiber intake can be associated with lower risks of heart disease, diabetes, and certain types of cancer. When assessing the fiber intake data of a population or study group, both the mean and median provide insightful information. The fiber data from our exercise suggests that, without the outlier, the average intake is closer to the median, possibly offering a more accurate representation of the 'typical' diet of the studied population.

Analysis of dietary fiber intake, considering the proper statistical measures, enables researchers and health professionals to more effectively evaluate nutritional status and develop dietary interventions.