61 lines
3.4 KiB
Markdown
61 lines
3.4 KiB
Markdown
(Chapter 8, STAT 1040)
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# Correlation
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## Scatter Diagrams
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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.
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If a scatter diagram is football shaped, it can be summarized using the 5-number summary:
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| Variable | Description |
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| -- | -- |
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| $\bar{x}$ | The average of the set graphed along the X axis |
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| $\sigma_x$| The standard deviation of set graphed along the X axis |
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| $\bar{y}$ | The average of the set graphed along the Y axis |
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| $\sigma_y$ | The standard deviation of the set graphed along the Y axis |
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| $r$ | The correlation coefficient, or how closely clustered the datapoints are in a line |
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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.
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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 $\sigma$.
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### Association
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- Positive association is demonstrated when the dots are trend upward as $x$ increases ($r$ is positive).
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- Negative association is demonstrated when the the dots trend downward as $x$ increases ($r$ is negative).
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- Strong association is demonstrated when dots are clustered tightly together along a line ($|r|$ is closer to 1).
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- Weak association is demonstrated when dots are not clustered tightly. ($|r|$ is closer to 0)
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## Correlation
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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.
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If $x$ is above average, we expect the $y$ to be above average if there's a strong correlation coefficient
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## Calculating $r$ by hand
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Put the $x$ values into $L1$, put the $y$ values into $L2$.
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1. Convert the $x$ each x value in the list to standard units($z$). Convert each $y$ value to standard units.
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$$ z = \frac{x-\bar{x}}{\sigma_x} $$
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2. Multiply the standard units for each ($x$, $y$) pair in the sets, giving you a third list, named $p$ in this example.
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$$ x * y = p$$
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3. Find the average of the values from step 3, this is $r$.
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$$ \bar{x}(p) $$
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https://www.thoughtco.com/how-to-calculate-the-correlation-coefficient-3126228
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# Terminology
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| Term | Definition |
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| -- | -- |
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| $r$ | Correlation Coefficient |
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| Linear Correlation | Measures the strength of a line |
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(Chapter 9, STAT 1040)
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# Correlation contd.
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## Notes
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- $r$ is a pure number, and does not have units
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- $r$ does not change if you:
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- switch the $x$ and $y$ values
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- add the same number to every x or y value
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- multiply each $x$ or $y$ value by a positive number
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- The correlation coefficient can be misleading in the presence of outliers or nonlinear association.
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- $r$ only shows the strength of a linear relationship.
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- High $r$ only indicates high correlation, not causation.
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- $r$ is not a linear, perfect scale. An $r$ of 0.8 does not mean twice as much linearity as an $r$ of 0.4
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### Ecological Correlations
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- Sometimes each point on the plot represents the average or rate for a whole group of individuals.
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- Ecological correlations are artificially strong |