## Mode

The mode is the value that occurs most frequently in a sample. eg, the mode of ( 1, 7, 7, 1, 3, 7, 3 ) is 7.

## Median

In a (sorted) sample, the median is the value that has an equal number of samples both less than and greater than it. eg, the median of ( 5, 1, 3, 2, 4 ) is 3.

## (Arithmetic) Mean

The mean ($\bar{x}$) is the arithmetic average.

## Variance

The variance ($\sigma^2$) of an entire population $N$ is $$\sigma^2 = \frac{1}{N} \sum_{i=1}^N ( x_i - \bar{x} )^2$$.

The variance of a sample of a population $n$ is $$s^2 = \frac{1}{n - 1} \sum_{i=1}^n ( x_i - \bar{x} )^2$$.

## Standard Deviation

The standard deviation is the square root of the variance.

## Linear Least Squares Regression

"http://stattrek.com/AP-Statistics-1/Regression.aspx?Tutorial=Stat"

### Coefficient Of Determination (Correlation Coefficient)

The coefficient of determination ($R^2$) is the proportion of the variance in $y$ that is predictable from $x$, ranges from 0 to 1, and is calculated by: $$$$R^2 = \frac{1}{n^2} \left [ \sum_{i=1}^n \frac{ ( x_i - \bar{x} ) ( y_i - \bar{y} ) }{ \sigma_x \sigma_y } \right ]$$$$

$R^2 = 1$ means that $y$ can be predicted perfectly from $x$. The residual is dmjp

### Linear Least Squares Regression Example

Sample$x_i$$y_i$$x_i - \bar{x}$$y_i - \bar{y}$$(x_i - \bar{x})^2$$(y_i - \bar{y})^2$$(x_i - \bar{x})(y_i - \bar{y})$
1958517828964136
2859571849324126
380702-7449-14
47065-8-126414496
56070-18-732449126
Sum390385730630470
Mean7877

the slope of the best fit line is $$$$R^2 = \frac{1}{n^2} \left [ \sum_{i=1}^n \frac{ ( x_i - \bar{x} ) ( y_i - \bar{y} ) }{ \sigma_x \sigma_y } \right ]$$$$ the intercept is $$$$R^2 = \frac{1}{n^2} \left [ \sum_{i=1}^n \frac{ ( x_i - \bar{x} ) ( y_i - \bar{y} ) }{ \sigma_x \sigma_y } \right ]$$$$ so $$\begin{eqnarray} y & = & 26.768 + 0.644x \\ \sigma_x & = & \sqrt{\frac{730}{5}} \\ & = & 12.083 \\ \sigma_y & = & \sqrt{\frac{630}{5}} \\ & = & 11.225 \\ R & = & \frac{1}{5} \frac{470}{12.083 \times 11.225 } \\ & = & \frac{94}{135.632} \\ & = & 0.693 \\ R^2 & = & 0.48 \\ \end{eqnarray}$$