8.9.3: Formula Review
8.3 Mean, Median and Mode
Suppose we have a set of data with \(n\) values, ordered from smallest to largest. If \(n\) is odd, then the median is the data value at position \(\frac{n+1}{2}\). If \(n\) is even, then we find the values at positions \(\frac{n}{2}\) and \(\frac{n}{2}+1\). If those values are named \(a\) and \(b\), then the median is defined to be \(\frac{a+b}{2}\).
8.4 Range and Standard Deviation
\[s=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}} \nonumber \]
Here, \(s\) is the standard deviation, \(x\) represents each data value, \(\bar{x}\) is the mean of the data values, \(n\) is the number of data values, and the capital sigma \((\Sigma)\) indicates that we take a sum.
8.6 The Normal Distribution
If \(x\) is a member of a normally distributed dataset with mean \(\mu\) and standard deviation \(\sigma\), then the standardized score for \(x\) is
\[z=\frac{x-\mu}{\sigma} \nonumber \]
If you know a \(z\)-score but not the original data value \(x\), you can find it by solving the previous equation for \(x\) :
\[x=\mu+z \times \sigma \nonumber \]
8.8 Scatter Plots, Correlation, and Regression Lines
If a line has slope \(m\) and passes through a point \(\left(x_0, y_0\right)\), then the point-slope form of the equation of the line is:
\[y=m\left(x-x_0\right)+y_0 \nonumber \]
Suppose \(x\) and \(y\) are explanatory and response datasets that have a linear relationship. If their means are \(\bar{x}\) and \(\bar{y}\) respectively, their standard deviations are \(s_x\) and \(s_y\) respectively, and their correlation coefficient is \(r\), then the equation of the regression line is:
\[y=r\left(\frac{s_y}{s_x}\right)(x-\bar{x})+\bar{y} \nonumber \]