91Ó°ÊÓ

Consider a sample \(x_{1}, \ldots, x_{n}\) with \(n\) even. Let \(\bar{x}_{L}\) and \(\bar{x}_{U}\) denote the average of the smallest \(n / 2\) and the largest \(n / 2\) observations, respectively. Show that the mean absolute deviation from the median for this sample satisfies $$ \Sigma\left|x_{i}-\tilde{x}\right| / n=\left(\bar{x}_{U}-\bar{x}_{L}\right) / 2 $$ Then show that if \(n\) is odd and the two averages are calculated after excluding the median from each half, replacing \(n\) on the left with \(n-1\) gives the correct result. [Hint: Break the sum into two parts, the first involving observations less than or equal to the median and the second involving observations greater than or equal to the median.]

Step by step solution

01

Understanding the Median for Even n

For an even number of observations, the median \( \tilde{x} \) is typically the average of the middle two observations. When sorted, the smallest \( n/2 \) observations form the lower half, and the largest \( n/2 \) observations form the upper half.

02

Calculating Mean Absolute Deviation for Even n

The mean absolute deviation from the median \( \tilde{x} \) is given by \( \Sigma|x_i - \tilde{x}| / n \). This sum is split into two parts: one for the observations from the lower half (each deviating by \( \tilde{x} - x_i \)) and the other for the upper half (each deviating by \( x_i - \tilde{x} \)). Since \( x_i \leq \tilde{x} \) in the lower half and \( x_i \geq \tilde{x} \) in the upper half, \( |x_i - \tilde{x}| \) simplifies to appropriate expression depending on the section.

03

Breaking Down Sum for Even n

For the lower half, the total deviation is \( /2 (\tilde{x} - \bar{x}_L) \) and for the upper half, it's \( n/2 (\bar{x}_U - \tilde{x}) \). The combined mean absolute deviation is \( (n/2 (\tilde{x} - \bar{x}_L) + n/2 (\bar{x}_U - \tilde{x}))/n = (\bar{x}_U - \bar{x}_L)/2 \).

04

Understanding the Median for Odd n

For an odd number of observations, the median is a single value, and the lower half now includes one less observation than when \( n \) was even, excluding the median.

05

Calculating Mean Absolute Deviation for Odd n

For \( n \) odd, the mean absolute deviation with \( n-1 \) replaces \( n \) in the formula, as each half has one less observation. Again, break the sum into deviations below and above the median.

06

Calculating New Averages for Odd n

Calculate \( \bar{x}_L \) and \( \bar{x}_U \) excluding the median, so they are smaller by one observation in their calculations.

07

Final Result for Odd n

With the adjusted halves and calculated averages, the equation \( \Sigma|x_i - \tilde{x}| / (n-1) = (\bar{x}_U - \bar{x}_L) / 2 \) holds true by setting it just as with even \( n \) but adjusted for one less element in each half.

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Key Concepts

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

Median

The median is a crucial concept in statistics, serving as a measure of central tendency. It separates a dataset into two equal halves. For a given sample, arranging the data in order from smallest to largest helps find the middle point. If the sample size, represented by \( n \), is even, the median \( \tilde{x} \) is calculated by averaging the two central numbers: \( \tilde{x} = \frac{x_{(n/2)} + x_{((n/2) + 1)}}{2} \). The median provides a more robust central point than the mean, especially in datasets with outliers or skewed distributions. Its resilience against extreme values makes it incredibly useful in statistical analysis.

Sample Observations

Sample observations refer to the individual data points in a statistical sample. Understanding each observation's role aids in comprehending the overall data structure and summarizing the dataset effectively. The individual observations in a dataset are denoted as \( x_1, x_2, \ldots, x_n \). These observations form the basis from which we calculate the median, mean, and other statistical metrics. They are integral in calculations like the mean absolute deviation, helping to measure how much each data point deviates from the median.

Even and Odd n

The distinction between even and odd \( n \) is essential in calculating statistical measures, particularly the median and mean absolute deviation.

• For even \( n \), the dataset divides neatly into two equal halves, requiring the median to be the average of the two middle values.
• For odd \( n \), the median is simply the middle value itself, as the dataset inherently has a single, central number.

This difference affects how we handle the deviation calculations, influencing whether we include or exclude the median in subsequent calculations like finding the mean absolute deviation.

Deviation Calculation

Deviation calculation explores how far each observation in a dataset deviates from a central measure, such as the median. The mean absolute deviation (MAD) focuses on average deviation magnitude from a specified center, disregarding the direction of deviation.The formula for MAD in the context of the median is: \[\text{MAD} = \frac{\Sigma |x_i - \tilde{x}|}{n}\]Where \( n \) might adjust to \( n-1 \) if excluding the median. This deviation reflects variability or spread in the dataset. Understanding these deviations aids in making data-driven decisions, providing insights into consistency and dispersion within a dataset.