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: $$ \begin{equation} R^2 = \frac{1}{n^2} \left [ \sum_{i=1}^n \frac{ ( x_i - \bar{x} ) ( y_i - \bar{y} ) }{ \sigma_x \sigma_y } \right ] \end{equation} $$

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