91Ó°ÊÓ

O The risk of developing iron deficiency is especially high during pregnancy. Detecting such a deficiency is complicated by the fact that some methods for determining iron status can be affected by the state of pregnancy itself. Consider the following data on transferrin receptor concentration for a sample of women with laboratory evidence of overt iron-deficiency anemia ("Serum Transferrin Receptor for the Detection of Iron Deficiency in Pregnancy," American Journal of Clinical Nutrition [1991]: \(1077-1081):\) \(\begin{array}{lllllllll}15.2 & 9.3 & 7.6 & 11.9 & 10.4 & 9.7 & 20.4 & 9.4 & 11.5\end{array}\) \(\begin{array}{lll}16.2 & 9.4 & 8.3\end{array}\) Compute the values of the sample mean and median. Why are these values different here? Which one do you regard as more representative of the sample, and why?

Step by step solution

01

Compute Mean

First, add together each number in the data set, giving a total. Take this total and divide by the number of items in the data set. In this instance, the sum of the numbers is \(15.2+9.3+7.6+11.9+10.4+9.7+20.4+9.4+11.5+16.2+9.4+8.3 = 139.3\). There are 12 numbers, so the mean is \(139.3/12 = 11.61\).

02

Compute Median

To find the median, list the numbers in numerical order. If there is an odd number of values, the median is the middle value. If there is an even number of numbers, the median is the mean of the two middle numbers. In this instance, when organized from lowest to highest, the values are \[7.6, 8.3, 9.3, 9.4, 9.4, 9.7, 10.4, 11.5, 11.9, 15.2, 16.2, 20.4\]. Since there is an even number of values, take the average of the two middle ones: \((9.7+10.4)/2 = 10.05\)

03

Difference and Representation

The difference between the mean and median exists because data can be skewed by extreme values, in this case, 20.4 is notably higher than the other values. The median is less influenced by outliers and might provide a more accurate idea of the typical value in this case, especially since the data isn't normally distributed.

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Key Concepts

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

Mean

The mean, often referred to as the average, is a fundamental concept in statistics. It provides a single value representing the center of a dataset by taking into account all values.To calculate the mean:

• Add up all the values in the dataset.
• Divide the sum by the number of values.

In the given exercise, the transferrin receptor values sum up to 139.3, and as there are 12 data points, the mean is calculated as \( \frac{139.3}{12} = 11.61 \).The mean can be very informative; however, it can also be sensitive to outliers—extreme values that do not represent the general trend of the dataset. This can sometimes make the mean not entirely reflective of the typical data point when the distribution is skewed.

Median

The median is another measure of central tendency. It is the value that sits in the middle of a dataset when arranged in order.To find the median:

• Arrange the data points in ascending order.
• If the dataset has an odd number of values, the median is the middle one.
• If there's an even number of values, the median is the average of the two central numbers.

For the dataset in question, the ordered values are \[7.6, 8.3, 9.3, 9.4, 9.4, 9.7, 10.4, 11.5, 11.9, 15.2, 16.2, 20.4\]. Since there are 12 values, the median is calculated as \( \frac{9.7 + 10.4}{2} = 10.05 \).The median gives a more representative measure of the center, especially when the dataset includes outliers, as it isn't skewed by extremely high or low values.

Outliers

Outliers are values in a dataset that deviate significantly from the other observations. They can affect statistical calculations and interpretations because they can pull the mean in one direction or another. In this particular dataset, the value 20.4 is significantly higher than the other numbers and acts as an outlier. This is why the mean (11.61) is higher than the median (10.05) since the mean considers this large value. Outliers can occur due to:

• Variability in measurements.
• Experimental errors.
• Novelty in the context being measured.

Recognizing outliers is crucial when analyzing data as it helps in choosing the most appropriate measure of central tendency to represent the dataset.

Data Distribution

Data distribution refers to how the values in a dataset are spread out or clustered together. A dataset may be symmetrical, having a bell curve, or it can be skewed, where one tail is longer than the other. In this exercise, the data is not symmetrically distributed due to the presence of standout high values (outliers), which causes skewness. Understanding data distribution helps in:

• Choosing the right statistical tools to analyze data.
• Deciding which measure of central tendency (mean or median) is more representative of the data.
• Recognizing patterns, trends, and outliers.

In cases where the data distribution is skewed, like in the provided dataset, the median is often a better measure of the central tendency since it is not affected by extreme values, unlike the mean.