Moment

The rth moment of a set {x1, x2,…, xn} about the point a is the following sum:

$\displaystyle {\frac{{1}}{{n}}}$$\displaystyle \sum_{{i=1}}^{{n}}$$\displaystyle \left(\vphantom{ x_{i}-a}\right.$xi - a$\displaystyle \left.\vphantom{ x_{i}-a}\right)^{{r}}_{}$

Thus, the mean is also known as the first moment about zero. The second moment about zero is the quantity μ2 + σ2, where μ is the mean and σ2 is the variance of the data. The rth moment about the mean is the sum

$\displaystyle {\frac{{1}}{{n}}}$$\displaystyle \sum_{{i=1}}^{{n}}$$\displaystyle \left(\vphantom{ x_{i}-\frac{1}{n}\sum_{j=1}^{n}x_{j}}\right.$xi - $\displaystyle {\frac{{1}}{{n}}}$$\displaystyle \sum_{{j=1}}^{{n}}$xj$\displaystyle \left.\vphantom{ x_{i}-\frac{1}{n}\sum_{j=1}^{n}x_{j}}\right)^{{r}}_{}$



\begin{example}
The $3$rd and $4$th moments of the set $\left\{
2,4,6,8,10,12,...
...}%
\vspace{6pt} &=&\frac{3776}{3}\approx 1258.7
\end{eqnarray*}
\end{example}

$\blacktriangleright$ Statistics + Moment

$\left[\vphantom{
\begin{array}{r}
8.5 \\
-5.5 \\
-3.7 \\
3.5
\end{array}
}\right.$$\begin{array}{r}
8.5 \\
-5.5 \\
-3.7 \\
3.5
\end{array}$$\left.\vphantom{
\begin{array}{r}
8.5 \\
-5.5 \\
-3.7 \\
3.5
\end{array}
}\right]$,
First moment about 0 :  .7
Second moment about 0 :  32.11

$\left(\vphantom{ .123,.703,.445,.284}\right.$.123,.703,.445,.284$\left.\vphantom{ .123,.703,.445,.284}\right)$,
First moment about 0 :  .389
Second moment about 0 :  .197

$\left(\vphantom{ .123,.703,.445,.284}\right.$.123,.703,.445,.284$\left.\vphantom{ .123,.703,.445,.284}\right)$,
First moment about the mean :   0
Second moment about the mean : 4.59×10-2