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In article you’ll about Measures of Distance in Data Mining, **Euclidean Distance**, **Manhattan Distance** and **Jaccard Index** and more.

**Clustering** consists of grouping certain objects that are similar to each other, it can be used to decide if two items are similar or dissimilar in their properties.

In a Data Mining sense, the similarity measure is a distance with dimensions describing object features. That means if the distance among two data points is **small** then there is a **high** degree of similarity among the objects and vice versa. The similarity is **subjective** and depends heavily on the context and application. For example, similarity among vegetables can be determined from their taste, size, colour etc.

## Measures of Distance in Data Mining

Most clustering approaches use distance measures to assess the similarities or differences between a pair of objects, the most popular distance measures used are

### 1. Euclidean Distance

Euclidean distance is considered the traditional metric for problems with geometry. It can be simply explained as the **ordinary distance** between two points. It is one of the most used algorithms in the cluster analysis. One of the algorithms that use this formula would be **K-mean**. Mathematically it computes the **root of squared differences** between the coordinates between two objects.

### 2. Manhattan Distance

This determines the absolute difference among the pair of the coordinates.

Suppose we have two points P and Q to determine the distance between these points we simply have to calculate the perpendicular distance of the points from X-Axis and Y-Axis.

In a plane with P at coordinate (x1, y1) and Q at (x2, y2).

Manhattan distance between P and Q = |x1 – x2| + |y1 – y2|

Here the total distance of the **Red** line gives the Manhattan distance between both the points.

### 3. Jaccard Index

The Jaccard distance measures the similarity of the two data set items as the **intersection** of those items divided by the **union** of the data items.

**4. **Minkowski distance

It is the **generalized** form of the Euclidean and Manhattan Distance Measure. In an** N-dimensional space**, a point is represented as,

(x1, x2, ..., xN)

Consider two points P1 and P2:

P1:(X1, X2, ..., XN)P2:(Y1, Y2, ..., YN)

Then, the Minkowski distance between P1 and P2 is given as:

- When
**p = 2**, Minkowski distance is same as the**Euclidean**distance. - When
**p = 1**, Minkowski distance is same as the**Manhattan**distance.

**5. **Cosine Index

Cosine distance measure for clustering determines the **cosine** of the angle between two vectors given by the following formula.

Here (**theta**) gives the angle between two vectors and A, B are n-dimensional vectors.