Variance and Standard Deviation

The sample variance for x1, x2,…, xn is the sum of the squares of differences with the mean, divided by n - 1.

$\displaystyle {\frac{{\dsum\limits_{i=1}^{n}\left( x_{i}-\frac{\sum_{j=1}^{n}x_{j}}{%
\stackrel{\vspace{2pt}}{n}}\right) ^{2}}}{{n-1}}}$

$\blacktriangleright$ To compute variance

1.
Place the insertion point in a list of data, in a vector, or in a matrix.

2.
From the Statistics submenu, choose Variance.

$\blacktriangleright$ Statistics + Variance

5, 1, 89, 4, 29, 47, 18, Variance(s): 1002.6

$\left[\vphantom{
\begin{array}{r}
18.1 \\
5.3 \\
7.6
\end{array}
}\right.$$\begin{array}{r}
18.1 \\
5.3 \\
7.6
\end{array}$$\left.\vphantom{
\begin{array}{r}
18.1 \\
5.3 \\
7.6
\end{array}
}\right]$, Variance(s): 46.563

$\left[\vphantom{
\begin{array}{rrr}
23 & 5 & -6 \\
18 & 23 & -22 \\
5 & 0 & 0
\end{array}
}\right.$$\begin{array}{rrr}
23 & 5 & -6 \\
18 & 23 & -22 \\
5 & 0 & 0
\end{array}$$\left.\vphantom{
\begin{array}{rrr}
23 & 5 & -6 \\
18 & 23 & -22 \\
5 & 0 & 0
\end{array}
}\right]$, Variance(s): $\left[\vphantom{ 86.333,146.33,129.33}\right.$86.333, 146.33, 129.33$\left.\vphantom{ 86.333,146.33,129.33}\right]$

$\left(\vphantom{
\begin{array}{cc}
x & y \\
a & b \\
c & d
\end{array}
}\right.$$\begin{array}{cc}
x & y \\
a & b \\
c & d
\end{array}$$\left.\vphantom{
\begin{array}{cc}
x & y \\
a & b \\
c & d
\end{array}
}\right)$, Variance(s): $\left[\vphantom{ \left( .5a-.5c\right) ^{2}+\left(
.5c-.5a\right) ^{2},\left( .5b-.5d\right) ^{2}+\left( .5d-.5b\right)
^{2}}\right.$$\left(\vphantom{ .5a-.5c}\right.$.5a - .5c$\left.\vphantom{ .5a-.5c}\right)^{{2}}_{}$ + $\left(\vphantom{
.5c-.5a}\right.$.5c - .5a$\left.\vphantom{
.5c-.5a}\right)^{{2}}_{}$,$\left(\vphantom{ .5b-.5d}\right.$.5b - .5d$\left.\vphantom{ .5b-.5d}\right)^{{2}}_{}$ + $\left(\vphantom{ .5d-.5b}\right.$.5d - .5b$\left.\vphantom{ .5d-.5b}\right)^{{2}}_{}$$\left.\vphantom{ \left( .5a-.5c\right) ^{2}+\left(
.5c-.5a\right) ^{2},\left( .5b-.5d\right) ^{2}+\left( .5d-.5b\right)
^{2}}\right]$

The square root of the variance is called the standard deviation. It is the most commonly used measure of dispersion.

$\displaystyle \sqrt{{\frac{\sum_{i=1}^{n}\left( x_{i}-\frac{\sum_{j=1}^{n}x_{j}}{\stackrel{%
\vspace{4pt}}{n}}\right) ^{2}}{n-1}}}$

$\blacktriangleright$ Statistics + Standard Deviation

$\left[\vphantom{ 5,1,89,4,29,47,18}\right.$5, 1, 89, 4, 29, 47, 18$\left.\vphantom{ 5,1,89,4,29,47,18}\right]$, Standard deviation(s): 31.664  

$\left(\vphantom{
\begin{array}{r}
18.1 \\
5.3 \\
7.6
\end{array}
}\right.$$\begin{array}{r}
18.1 \\
5.3 \\
7.6
\end{array}$$\left.\vphantom{
\begin{array}{r}
18.1 \\
5.3 \\
7.6
\end{array}
}\right)$, Standard deviation(s): 6.8237

$\left(\vphantom{
\begin{array}{cc}
x & y \\
a & b \\
c & d
\end{array}
}\right.$$\begin{array}{cc}
x & y \\
a & b \\
c & d
\end{array}$$\left.\vphantom{
\begin{array}{cc}
x & y \\
a & b \\
c & d
\end{array}
}\right)$, Standard deviation(s): $\left[\vphantom{ .70711\sqrt{\left( \left(
-1.0a+c\right) ^{2}\right) },.70711\sqrt{\left( \left( -1.0b+d\right)
^{2}\right) }}\right.$.70711$\sqrt{{\left( \left(
-1.0a+c\right) ^{2}\right) }}$,.70711$\sqrt{{\left( \left( -1.0b+d\right)
^{2}\right) }}$$\left.\vphantom{ .70711\sqrt{\left( \left(
-1.0a+c\right) ^{2}\right) },.70711\sqrt{\left( \left( -1.0b+d\right)
^{2}\right) }}\right]$

Note that the preceding matrix was treated as a labeled matrix, and the first row was ignored.

$\blacktriangleright$ Statistics + Standard Deviation

$\left[\vphantom{
\begin{array}{rrr}
23 & 5 & -6 \\
18 & 23 & -22 \\
5 & 0 & 0
\end{array}
}\right.$$\begin{array}{rrr}
23 & 5 & -6 \\
18 & 23 & -22 \\
5 & 0 & 0
\end{array}$$\left.\vphantom{
\begin{array}{rrr}
23 & 5 & -6 \\
18 & 23 & -22 \\
5 & 0 & 0
\end{array}
}\right]$, Standard deviation(s): $\left[\vphantom{ 9.2916,12.097,11.372}\right.$9.2916, 12.097, 11.372$\left.\vphantom{ 9.2916,12.097,11.372}\right]$

$\left[\vphantom{
\begin{array}{rrr}
-8.5 & 5.0 & 5.7 \\
-5.5 & 7.9 & -5.9...
...5.6 & 4.5 \\
-3.5 & 4.9 & -8.0 \\
9.7 & 6.3 & -9.3
\end{array}
}\right.$$\begin{array}{rrr}
-8.5 & 5.0 & 5.7 \\
-5.5 & 7.9 & -5.9 \\
-3.7 & 5.6 & 4.5 \\
-3.5 & 4.9 & -8.0 \\
9.7 & 6.3 & -9.3
\end{array}$$\left.\vphantom{
\begin{array}{rrr}
-8.5 & 5.0 & 5.7 \\
-5.5 & 7.9 & -5.9...
...5.6 & 4.5 \\
-3.5 & 4.9 & -8.0 \\
9.7 & 6.3 & -9.3
\end{array}
}\right]$, Standard deviation(s): $\left[\vphantom{ 7.0014,1.23,7.1456}\right.$7.0014, 1.23, 7.1456$\left.\vphantom{ 7.0014,1.23,7.1456}\right]$