Median: Definition, Calculation, and Examples

Learn about the median, an important statistical measure. Understand its definition, calculation methods, and practical examples.

The median is a crucial measure of central tendency in statistics, representing the middle value in a sorted list. When data is ordered in ascending or descending sequence, the median offers a central value that can provide a more accurate reflection of the data set compared to the mean, especially in the presence of outliers.

Definition

The median is the middle number when a set of values is arranged in order. If the data set has an odd number of observations, the median is the central value. For an even number of observations, the median is the average of the two central numbers.

$$\text{Median} = \begin{cases} \text{Middle value}, & \text{if the number of observations is odd} \\ \frac{value_{n/2} + value_{(n/2) + 1}}{2}, & \text{if the number of observations is even} \end{cases}$$

Types of Data Sets

• Odd Number of Observations:
  • Example: For the data set {3, 5, 9}, the median is 5.
• Even Number of Observations:
  • Example: For the data set {10, 20, 30, 40}, the median is (20 + 30) / 2 = 25.

Calculation Steps

1. Odd Number of Observations

• Arrange the data in either ascending or descending order.
• Identify the middle value.
• Example: {8, 1, 3} → {1, 3, 8}, Median = 3.

2. Even Number of Observations

• Arrange the data in order.
• Calculate the average of the two middle values.
• Example: {15, 47, 22, 4} → {4, 15, 22, 47}, Median = (15 + 22) / 2 = 18.5

Special Considerations

• Outliers: The median is resistant to extreme values. Unlike the mean, it won’t be overly influenced by very large or very small numbers, making it a better measure in skewed distributions.
• Applications: The median is often used in areas such as economics (e.g., median household income), and real estate (e.g., median home prices).

Practical Examples

• Income Distribution:

  • Data: {25,000, 30,000, 45,000, 50,000, 200,000}
  • Ordered: {25,000, 30,000, 45,000, 50,000, 200,000}
  • Median: 45,000

• Class Scores:

  • Data: {55, 80, 90, 95}
  • Ordered: {55, 80, 90, 95}
  • Median: (80 + 90) / 2 = 85

Historical Context

The concept of the median was introduced by Francis Galton, an English polymath, in the late 19th century. It has since become a fundamental aspect of statistical analysis.

Related Terms

• Mean: The arithmetic average of a set of values.
• Mode: The most frequently occurring value in a data set.
• Quartiles: Values that divide a data set into four equal parts.

FAQs

Q1. When is it better to use the median instead of the mean?

• Whenever the data set contains outliers or is skewed, as the median provides a more accurate measure of central tendency.

Q2. Can the median be used for categorical data?

• No, the median is only applicable to ordinal, interval, and ratio data, but not nominal data.

References

• Galton, F. (1879). “The Geometrical Mean, in Vital and Social Statistics.” Proceedings of the Royal Society of London.
• Lane, D. M. (2003). “Measures of Central Tendency.” HyperStat Online Textbook.

Summary

In statistics, the median is a valuable measure of central tendency that offers advantages over the mean in skewed distributions or data with outliers. Its simplicity in calculation and robustness to extreme values make it a vital tool in various fields such as economics, finance, and social sciences. By understanding its definition and calculation, and through practical examples, one can effectively use the median to interpret data sets.