Median

Median: Definition, Calculation, and When to Use It

What is the median?

The median is the middle value in a list of numbers sorted from smallest to largest (or largest to smallest). It marks the point where 50% of observations lie below and 50% lie above. Because it depends only on the order of values, the median is less sensitive to outliers than the mean (average).

Why the median matters

• It provides a robust measure of central tendency when data include extreme values or are skewed.
• Commonly used in economics (e.g., median household income) and other fields where a typical value is needed without distortion from extremes.
• Serves as the center line for quartiles and many summary statistics.

How to calculate the median

1. Sort the data in ascending (or descending) order.
2. If the number of observations (n) is odd: median = the value at position (n + 1) / 2.
3. If n is even: median = average of the two middle values at positions n/2 and (n/2) + 1.

Examples:
– Odd count: For 3, 13, 2, 34, 11, 26, 47 → sorted = 2, 3, 11, 13, 26, 34, 47. n = 7, median position = (7 + 1)/2 = 4 → median = 13.
– Even count: For 3, 13, 2, 34, 11, 17, 27, 47 → sorted = 2, 3, 11, 13, 17, 27, 34, 47. n = 8 → middle values = 13 and 17 → median = (13 + 17) / 2 = 15.

Quick step-by-step:
– Sort values.
– Find middle position(s).
– Take the middle value (odd n) or average the two middle values (even n).

Median vs. mean

• Mean (arithmetic average): sum of values divided by number of values.
• Median: middle value in ordered list.

Example showing difference:
Dataset: 3, 5, 7, 19
– Mean = (3 + 5 + 7 + 19) / 4 = 34 / 4 = 8.5
– Median = average of middle two values (5 and 7) = (5 + 7) / 2 = 6

When data are skewed or include outliers, the mean can be pulled toward the extremes while the median remains near the center of the bulk of observations.

Example with skew/outliers:
0, 0, 0, 1, 1, 2, 10, 10
– Mean = 24 / 8 = 3
– Median = average of 4th and 5th values = (1 + 1) / 2 = 1
Here the median better reflects the typical value.

Median in distributions

• In a perfectly symmetric (normal) distribution, mean = median = mode and they coincide at the center of the bell curve.
• In skewed distributions, the median and mean differ; the median indicates the central position of the population while the mean indicates the arithmetic center.

Relation to quartiles and other partitions

• The median is the second quartile (Q2), dividing data into lower and upper halves.
• Other partitions include quartiles (4 parts), quintiles (5 parts), and deciles (10 parts).

Key takeaways

• The median is the middle value of an ordered dataset and is robust to outliers.
• Use the median when you need a typical value unaffected by extreme observations.
• For odd n, pick the center value; for even n, average the two center values.
• In symmetric distributions mean = median; in skewed distributions they differ.

References

• Purdue Online Writing Lab — Descriptive Statistics
• University of Illinois — Quartiles and Box Plots
• National Cancer Institute — Normal Distribution