- Green Line: Manhattan Distance
- Red Line: Euclidean Distance

Visual Representation of the Difference Between Manhattan and Euclidean Distances

The Manhattan distance is found by using the absolute value of the sum of the difference between two points. Its formula is given by:

		$\sum_{i = 1}^k {|x_i-y_i|}$ 

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Often times, statistical methods rely on measuring the distance between quantitative data points. However, there are various ways to measure this distance. These include:

1. Euclidean distance 
2. Manhattan distance
3. Hamming Distance

Measuring Distance Between Quantitative Data Points

The reference below has a good reference to KNN, as well as different ways in which distance is measured.

Brownlee, J. (2019, August 12). K-Nearest Neighbors for Machine Learning. Retrieved March 23, 2020, from [machinelearningmastery.com/k-nearest-neighbors-for-machine-learning/](https://machinelearningmastery.com/k-nearest-neighbors-for-machine-learning/) 

K-Nearest Neighbors for Machine Learning (Reference)

In order to find the Euclidean distance, one has to take the square root of the sum of the squared difference between two points. The formula for calculating the Euclidean distance is given by:

	 $\sqrt{ \sum_{i=1}^k {(x_i - y_i)}^2 }$


Euclidean Distance

Manhattan Distance

If you are using categorical variables, then using the Hamming distance will be helpful. If two points are the same, then the difference between them is 0. If they are not the same, the difference between them is 1. Its formula is given by:

		$D_H =  \sum_{i=1}^k {|x_i - y_i|}$
		$if x = y \Rightarrow 0$
		$if x \neq y \Rightarrow 1$

Hamming Distance

We need to specify a distance metric in our future space, so we know how to properly select the nearby neighbors. In the fruit classification example, we use the simple straight line, or Euclidean distance to measure the distance between points.

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