Overview

Mean and median are two foundational statistics used to describe the center of a numerical dataset. While both aim to capture a “typical” value, they behave differently depending on the distribution of the data.

Core Idea

The mean averages all values and is sensitive to extreme observations, whereas the median identifies the middle value and is robust to outliers. Choosing between them depends on the data’s distribution and the analytical goal.

Formal Definition (if applicable)

For a dataset $x_1, x_2, \dots, x_n$:

  • Mean (arithmetic mean):
    The sum of all values divided by the number of values:
    $$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i $$

  • Median:
    The middle value after sorting the data; if $n$ is even, it is the average of the two central values.

Intuition

The mean asks, “What value would make the total balance out evenly?”
The median asks, “What value splits the data into two equal halves?”
As a result, the mean is pulled toward extremes, while the median resists them.

Examples

  • Dataset: ${2, 3, 3, 4, 100}$
    Mean = 22.4, Median = 3.
    The large outlier inflates the mean but leaves the median largely unchanged.
  • In a symmetric distribution without outliers, mean and median are equal or very close.

Common Misconceptions

  • “The mean always represents the typical value.”
    This fails in skewed distributions.
  • “Median ignores most of the data.”
    While it depends on order rather than magnitude, it still reflects the dataset’s central position.
  • Mode
  • Skewness
  • Outliers
  • Trimmed mean
  • Robust statistics

Applications

  • Income and wealth: Median income better reflects typical earners in skewed distributions.
  • Quality control: Mean is useful when deviations in either direction matter equally.
  • Exploratory data analysis: Comparing mean and median helps diagnose skewness.

Criticism / Limitations

Neither statistic captures variability or distribution shape. Relying solely on mean or median can obscure multimodality or dispersion, requiring complementary measures such as variance or quantiles.

Further Reading

  • Introductory statistics textbooks on descriptive statistics
  • Discussions of robust measures of central tendency in statistical methodology