Moments

October 4, 2020

The mean is an another name for the average. You can think of the mean as a typical representative value of all the observations in a set. The standard deviation is a measure of the “spread” of the data. For example, if in a set of observations, the data was not concentrated around the mean, then that dataset would have a high standard deviation. If all the data, was concentrated around some single value, then that dataset would have a low standard deviation. The standard deviation squared is the variance. Another name for these measurements are moments.

The following are formal methods of computing the sample mean and sample variance, respectively

$$\begin{gathered} \overline{x}=\frac{x_1+x_2+\cdot \cdot \cdot +x_n}{n} \\ {s}^2={\displaystyle \frac{1}{n-1}\sum _{i=0}^n(x_i-\overline{x}{)}^2} \end{gathered}$$

It should be noted that these calculutions are for a sample. If the entire population is used to compute these values then these equations would look like this:

$$\begin{gathered} \mu =\frac{x_1+x_2+\cdot \cdot \cdot +x_N}{N} \\ {\sigma }^2={\displaystyle \frac{1}{N}\sum _{i=0}^N(x_i-\mu {)}^2} \end{gathered}$$

Note that the greek letters have replaced the symbols for sample mean and sample variance; the population size, $N$ has replaced the sample size, $n$ to symbolize we are talking about population parameters. We do not subtract unity because we are no longer estimating.

Use the R summary widget to compute the first moment for sample datasets. A standard deviation widget is used to compute the second moment.