Paired data shows up in statistics when we are observing a link between two varaibles. One of the most basic relationships that could exist between these variables is a linear one, or straight line. To determine which line is the best one through the points, we construct what is known as the least squares regression line. This least squares line has connections to other statistical calculations. One example of this is that the slope of the least squares line is related to the correlation coefficient.

Most practical applications of statistics require some kind of aid for calculations. There are a number of statistical software packages available for use. One of these is Microsoft Excel.

Some of the more basic calculations in statistics are that of the mean and standard deviation. See how to use Excel to calculate the mean of a data set with the AVERAGE function and standard deviation with the STDEV.S function.

One area of statistics, called mathematical statistics, examines the theoretical underpinnings of statistics. This requires careful, logical proof and results from other areas of mathematical, such as calculus. One operation in calculus is called the derivative. There are a number of formulas for derivatives including:

• The chain rule
• The power rule
• The product rule
• The quotient rule

These derivative rules can be applied to questions in statistics, such as determining the inflection points of the normal distribution, or to perform calculations with the gamma function.

One area of statistics involves the use of pure mathematics to prove theorems involving statistics. For example, we can use results from calculus such as the power rule and product rule to prove the location of the inflection points of the probability density function of the normal distribution. Inflection points are the points where the curvature of the graph changes.