Sample median

Statistical measure of central tendency

The sample median is a measure of central tendency that represents the middle value of a data set when it is ordered in ascending or descending order. It is a robust statistic that is less affected by outliers and skewed data compared to the sample mean. The sample median is widely used in various fields, including medicine, economics, and social sciences, to summarize data and make inferences about the population.

Definition

The sample median is defined as follows:

• For a data set with an odd number of observations, the sample median is the middle value when the data is arranged in order.
• For a data set with an even number of observations, the sample median is the average of the two middle values.

Mathematically, if a sample consists of \( n \) observations \( x_1, x_2, \ldots, x_n \) sorted in non-decreasing order, the sample median \( \tilde{x} \) is given by:

\[ \tilde{x} = \begin{cases}

 x_{(\frac{n+1}{2})}, & \text{if } n \text{ is odd} \\
 \frac{x_{(\frac{n}{2})} + x_{(\frac{n}{2} + 1)}}{2}, & \text{if } n \text{ is even}

\end{cases} \]

Properties

• Robustness: The sample median is robust to outliers, meaning that extreme values have little effect on its value.
• Non-parametric: The sample median does not assume any specific distribution for the data, making it a non-parametric statistic.
• Breakdown point: The sample median has a high breakdown point of 50%, meaning that up to 50% of the data can be contaminated without affecting the median.

Applications

In medical research, the sample median is often used to describe the central tendency of biological measurements, such as blood pressure or cholesterol levels, where data may be skewed or contain outliers. It is also used in clinical trials to compare the effectiveness of treatments.

Calculation Example

Consider a sample data set: \( \{3, 1, 4, 1, 5, 9, 2, 6, 5, 3\} \).

1. Sort the data: \( \{1, 1, 2, 3, 3, 4, 5, 5, 6, 9\} \) 2. Since there are 10 observations (an even number), the sample median is the average of the 5th and 6th values:

  \[
  \tilde{x} = \frac{3 + 4}{2} = 3.5
  \]

Comparison with Other Measures

The sample median is often compared with the sample mean and sample mode. While the mean is sensitive to extreme values, the median provides a better measure of central tendency for skewed distributions. The mode, on the other hand, represents the most frequently occurring value in the data set.

Also see

• Sample mean
• Sample mode
• Robust statistics
• Central tendency
• Quantile