Hamming Distance Formula
Calculate the number of differing positions between two binary strings.
Used in error detection and correction coding.
The Formula
d(x, y) = number of positions where xᵢ ≠ yᵢ
The Hamming distance counts how many bit positions differ between two strings of equal length. It measures how many single-bit errors would transform one string into the other.
Variables
| Symbol | Meaning |
|---|---|
| d(x, y) | Hamming distance between strings x and y |
| x, y | Two binary strings of equal length |
| xᵢ, yᵢ | The bit at position i in strings x and y |
Example 1
Find the Hamming distance between 1011101 and 1001001
Position 1: 1 vs 1 ✓
Position 2: 0 vs 0 ✓
Position 3: 1 vs 0 ✗
Position 4: 1 vs 1 ✓
Position 5: 1 vs 0 ✗
Position 6: 0 vs 0 ✓
Position 7: 1 vs 1 ✓
Hamming distance = 2
Example 2
Find the Hamming distance between "karolin" and "kathrin"
k-k ✓, a-a ✓, r-t ✗, o-h ✗, l-r ✗, i-i ✓, n-n ✓
Hamming distance = 3
When to Use It
Use the Hamming distance when:
- Designing error-detecting and error-correcting codes
- Measuring similarity between binary data or DNA sequences
- Implementing spell checkers or fuzzy matching
- Evaluating the reliability of communication channels