What Are the Main Properties of Median in Statistics?

The median of a statistical data set is the value that divides the ordered data into two equal parts, with half the data below and half above. It serves as a central tendency measure, particularly useful for unevenly distributed data.

Mathematical Definition and Computation of the Median

Definition: In an ordered data set of $n$ values, the median is the $\left(\dfrac{n+1}{2}\right)^{\text{th}}$ value if $n$ is odd, and the mean of the $\left(\dfrac{n}{2}\right)^{\text{th}}$ and $\left(\dfrac{n}{2}+1\right)^{\text{th}}$ values if $n$ is even.

For a discrete data set $x_1, x_2, \ldots, x_n$ arranged in ascending order, compute the median by identifying the positional value as described above. For continuous grouped data, the median is obtained using interval methods.

Principal Properties Characterizing the Median

The following are well established properties that define the behavior and reliability of the median across data sets of varying composition and distribution:

• Resistant to outliers and extreme values
• Divides ordered data set into two equal halves
• May or may not be an actual data point
• Exact value unaffected by magnitude of extreme observations
• Well defined for both discrete and continuous variables
• Suitable for ordinal, interval, and ratio scales
• Uniqueness in data sets (single value or mean of two middle values)
• Sum of absolute deviations from median is minimum

Algebraic Statement: Median Minimizes Absolute Deviations

Let $c$ be any real number. The sum $S(c) = \sum_{i=1}^n |x_i - c|$ is minimized when $c$ equals the median of the data. The median thus possesses an optimal property of minimum absolute deviation, distinguishing it from other central tendencies.

Median in Symmetric and Skewed Distributions

For perfectly symmetric distributions, the mean and median coincide. In positively skewed (right-tailed) data, the mean exceeds the median; for negatively skewed (left-tailed) data, the median exceeds the mean. For further analysis, see Difference Between Mean, Median And Mode.

Exam Illustration: Direct Computation of Median for Discrete Data

Example: Determine the median for $10, 11, 12, 8, 14, 9, 6$.

Arrange in ascending order: $6, 8, 9, 10, 11, 12, 14$.

There are $n=7$ data points (odd). Median is the $4^{\text{th}}$ value: $10$.

Stepwise Calculation: Median for Grouped Frequency Data

For a frequency distribution divided into $k$ intervals, the median is located using:

Median $= l + \left(\dfrac{\dfrac{n}{2} - F}{f}\right) \times h$

where $l$ is the lower class boundary of the median class, $F$ is the cumulative frequency before the median class, $f$ is the frequency of the median class, $h$ is the class width, and $n$ is the total frequency.

This formula ensures that the median divides the area of the histogram into two equal halves. For comprehensive calculation procedures, reference Understanding Median In Statistics.

Median versus Mean: Comparative Robustness

The median is robust against outliers, reflecting only the middle position, while the mean is influenced by every data value. This distinction leads to the characteristic of stability in the median, making it a "good average" for skewed distributions. For comparative details, consult Statistics And Probability.

Common Exam Patterns Involving Median

Standard examination queries on the median include direct computation for ordered data, calculation from grouped frequencies, and deduction of skewness from mean-median relations. Practice sets are provided in Important Questions On Statistics.

Further Study: Median in Probability Distributions

In probability distribution contexts, the median is that value $m$ for which $P(X \leq m) \geq \dfrac{1}{2}$ and $P(X \geq m) \geq \dfrac{1}{2}$. This extends the median's applicability beyond simple data arrangements to theoretical statistics.

Statistics And Probability Revision Notes provides summary resources for the topic, facilitating final revision and quick access to formulae and worked examples.