The median is a robust measure of central tendency that represents the middle value in an ordered
dataset. Unlike the mean, which uses all data values, the median depends only on the position of
values in the ordered list. This makes it highly resistant to extreme values (outliers) and provides
a better representation of the "typical" value when data contains anomalies or skewed distributions.
Median Calculation:
Odd number of data points (n odd): Median = value at position
$\frac{n+1}{2}$
Even number of data points (n even): Median = average of values at
positions $\frac{n}{2}$ and $\frac{n}{2} + 1$
Formula: $Median = \begin{cases} x_{\frac{n+1}{2}} & \text{if } n \text{ is odd}
\\ \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2} & \text{if } n \text{ is even}
\end{cases}$
Where:
$n$ = number of data points
$x_k$ = k-th value in ordered data
Interactive Median Calculator
Data Visualization
6.5
Median
41
Median position
6
Data Points
Worked Example 1: Odd Number of Values
Find the median of: 12, 7, 15, 9, 21
Solution:
Step 1: Sort the data: 7, 9, 12, 15, 21
Step 2: n = 5 (odd), so median position = (5+1)/2 = 3rd value
Step 3: Median = 12
Worked Example 2: Even Number of Values
Find the median of: 8, 12, 15, 7, 10, 22
Solution:
Step 1: Sort the data: 7, 8, 10, 12, 15, 22
Step 2: n = 6 (even), so median positions = 3rd and 4th values
Step 3: Median = (10 + 12) / 2 = 11
Worked Example 3: Median with Outliers
Compare mean and median for: 2, 4, 6, 8, 100
Solution:
Sorted: 2, 4, 6, 8, 100
Median = 6 (middle value)
Mean = (2+4+6+8+100)/5 = 120/5 = 24
The outlier (100) greatly affects the mean but not the median.
Properties of the Median
Robustness: Unlike the mean, the median is not affected by extreme values or
outliers.
Position-based: The median depends only on the relative ordering of data, not
on the actual values.
Break-even Point: In an ordered dataset, 50% of values are below the median and
50% are above it.
Geometric Interpretation: The median minimizes the sum of absolute deviations
from any point.
Median vs Mean: When to Use Each
Use Median When:
Data contains outliers or extreme values
Distribution is skewed
Data is ordinal (rank-based)
Robustness is important
Income, property values, test scores with anomalies
Use Mean When:
All data points are equally important
Mathematical calculations are needed
Data is normally distributed
Precision is more important than robustness
Physical measurements, balanced data
Practice Problems
Problem 1: Find the median of: 15, 8, 12, 6, 19, 23, 10
Solution:
Sorted: 6, 8, 10, 12, 15, 19, 23
n = 7 (odd), median position = (7+1)/2 = 4th value
Median = 12
Problem 2: Find the median of: 4, 7, 2, 9, 5, 8
Solution:
Sorted: 2, 4, 5, 7, 8, 9
n = 6 (even), median positions = 3rd and 4th values
Median = (5 + 7) / 2 = 6
Problem 3: The median of 9 numbers is 25. What can you say about the numbers?
Solution:
For 9 numbers (odd), median is the 5th value when sorted
At least 5 numbers ≤ 25, and at least 5 numbers ≥ 25
At least 4 numbers ≤ 25, and at least 4 numbers ≥ 25
Problem 4: Find the median of: 1, 3, 3, 3, 5, 7, 9
Solution:
Sorted: 1, 3, 3, 3, 5, 7, 9
n = 7 (odd), median position = 4th value
Median = 3
Problem 5: Compare the mean and median of: 1, 2, 3, 4, 100
Solution:
Sorted: 1, 2, 3, 4, 100
Median = 3
Mean = (1+2+3+4+100)/5 = 110/5 = 22
The outlier (100) increases the mean but doesn't affect the median.
Key Takeaways
Definition: The median is the middle value in an ordered dataset - 50% of
values are below it, 50% are above it.
Calculation: For odd n: median = value at position (n+1)/2; For even n: median
= average of values at positions n/2 and n/2+1.
Robustness: Unlike the mean, the median is resistant to extreme values and
outliers.
Position-based: The median depends only on the ordering of data, not on the
magnitude of values.
Applications: Preferred for skewed distributions, ordinal data, and when
outliers are present (income, house prices, test scores).
Comparison with Mean: Mean uses all data points and is affected by outliers;
median is robust but ignores the magnitude of non-central values.
Break-even Property: The median minimizes the sum of absolute deviations from
any point in the dataset.