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

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

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

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

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)

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.

Specifying a distance metric for k-Nearest Neighbors algorithm

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

University of Michigan - Ann Arbor

Quantitative data is referred to as Continuous or Discrete data because it comprised of numerical data. This data is commonly stored in data types of Integer, Float, and Double. When using Quantitative data for Statistical learning it is often used in regression problems to predict a numerical output such as percent increase or decrease. This data is a an objective measurement.

Quantitative Data 

Measuring Distance Between Quantitative Data Points

Here is a video on the differences between discrete and continuous data.

[MooMoo Math and Science]. (2017, March 24). Continuous vs Discrete Data [Video File]. Retrieved from [www.youtube.com/watch?v=g30cI4Kbhb4](https://www.youtube.com/watch?v=g30cI4Kbhb4) 

Continuous vs Discrete Data (Reference)

The case of quantitative analysis of quantitative data applies when we use numeric data to analyze the data statistically and mathematically.  

Quantitative Analysis of Quantitative Data

The case of quantitative analysis of qualitative data applies when we use qualitative data to analyze the data statistically and mathematically.  


Quantitative Analysis of Qualitative Data

The case of qualitative analysis of quantitative data applies when we use quantitative data to interpret (qualitatively analyze) the data.  It can involve searching for patterns in the data to understand the "meaning."  

Qualitative Analysis of Quantitative Data

If substance A leads to biological reaction B, then in general (but not 100%), a larger dose of A leads to the stronger reaction B. 


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