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education/statistics/Correlation and Regression.md
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@ -8,15 +8,15 @@ If a scatter diagram is football shaped, it can be summarized using the 5-number
| Variable | Description | | Variable | Description |
| -- | -- | | -- | -- |
| $\mu_x$ | The average of the set graphed along the X axis | | $\bar{x}$ | The average of the set graphed along the X axis |
| $\sigma_x$| The standard deviation of set graphed along the X axis | | $\sigma_x$| The standard deviation of set graphed along the X axis |
| $\mu_y$ | The average of the set graphed along the Y 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 | | $\sigma_y$ | 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 | | $r$ | 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. 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 $\mu$. You can divide the range between the two points by 4 to find $\sigma$. 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 ### Association
- Positive association is demonstrated when the dots are trend upward as $x$ increases ($r$ is positive). - 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). - Negative association is demonstrated when the the dots trend downward as $x$ increases ($r$ is negative).
@ -28,12 +28,12 @@ Correlation is between `-1` and `1`. Correlation near 1 means tight clustering,
## Calculating $r$ by hand ## Calculating $r$ by hand
Put the $x$ values into $L1$, put the $y$ values into $L2$. Put the $x$ values into $L1$, put the $y$ values into $L2$.
1. Convert the $x$ values to standard units ($z$). Convert the $y$ values to standard units. 1. Convert the $x$ each x value in the list to standard units($z$). Convert each $y$ value to standard units.
$$ z = \frac{x-\bar{x}}{\sigma_x} $$ $$ z = \frac{x-\bar{x}}{\sigma_x} $$
2. Multiply the standard units for each ($x$, $y$) pair in the sets 2. Multiply the standard units for each ($x$, $y$) pair in the sets, giving you a third list, named $p$ in this example.
$$ x * y = $$ $$ x * y = p$$
3. Find the average of the values from step 3, this is $r$. 3. Find the average of the values from step 3, this is $r$.
$$ z_x = \frac{L_1-\bar{x}}{\sigma_x} $$ $$ \bar{x}(p) $$
https://www.thoughtco.com/how-to-calculate-the-correlation-coefficient-3126228 https://www.thoughtco.com/how-to-calculate-the-correlation-coefficient-3126228
# Terminology # Terminology