notes/education/statistics/Correlation and Regression.md

(Chapter 8, STAT 1040)

Correlation

Scatter Diagrams

A scatter diagram or scatter plot shows the relationship between two variables. One variable is on the X axis, the other on the Y axis.

If a scatter diagram is football shaped, it can be summarized using the 5-number summary:

Variable	Description
\bar{x}	The average of the set graphed along the X axis
\sigma_x	The standard deviation of set graphed along the X axis
\bar{y}	The average of the set graphed along the Y axis
\sigma_y	The standard deviation of the set graphed along the Y axis
r	The correlation coefficient, or how closely clustered the datapoints are in a line

The intersection of the averages of x and y will be the center of an oval shaped scatter diagram. Draw lines 2\sigma (will contain ~95% of all data) from the center along each axis to generalize the shape of a scatter plot.

You can approximate the mean by trying to find the upper bound and the lower bound of 2\sigma deviation to either side of the mean, then finding the middle of those two points to find \bar{x}. You can divide the range between the two points by 4 to find \sigma.

Association

• Positive association is demonstrated when the dots are trend upward as x increases (r is positive).
• Negative association is demonstrated when the the dots trend downward as x increases (r is negative).
• Strong association is demonstrated when dots are clustered tightly together along a line (|r| is closer to 1).
• Weak association is demonstrated when dots are not clustered tightly. (|r| is closer to 0)

Correlation

Correlation is between -1 and 1. Correlation near 1 means tight clustering, and correlation near 0 means loose clustering. r is -1 if the points are on a line with negative slope, r is positive 1 if the points are on a line with a positive slope. As |r| gets closer to 1, the line points cluster more tightly around a line.

If x is above average, we expect the y to be above average if there's a strong correlation coefficient

Calculating r by hand

Put the x values into L1, put the y values into L2.

1. Convert the x each x value in the list to standard units(z). Convert each y value to standard units.

 z = \frac{x-\bar{x}}{\sigma_x} 

2. Multiply the standard units for each (x, y) pair in the sets, giving you a third list, named p in this example.

 x * y = p

3. Find the average of the values from step 3, this is r.

 \bar{x}(p) 

https://www.thoughtco.com/how-to-calculate-the-correlation-coefficient-3126228

Terminology

Term	Definition
r	Correlation Coefficient
Linear Correlation	Measures the strength of a line

(Chapter 9, STAT 1040)

Correlation contd.

Notes

• r is a pure number, and does not have units
• r does not change if you:
  • switch the x and y values
  • add the same number to every x or y value
  • multiply each x or y value by a positive number
• The correlation coefficient can be misleading in the presence of outliers or nonlinear association.
• r only shows the strength of a linear relationship.
• High r only indicates high correlation, not causation.
• r is not a linear, perfect scale. An r of 0.8 does not mean twice as much linearity as an r of 0.4

Ecological Correlations

• Sometimes each point on the plot represents the average or rate for a whole group of individuals.
• Ecological correlations are artificially strong
• The size of points may indicate the size of the datapoints