Question

What are the variance and the standard deviation \[\left\{ 18,-9,-57,30,18,5,700,7,2,1 \right\}\]?

Answer 1

Hint: Firstly, we have to determine the sum of the given data and then calculate the mean of the data. And then we have to calculate the deviation from mean from each observation. Then we have to calculate the square of the deviations. Then we have to divide the sum of the squared deviations by the original data values. Then in order to find the standard deviation, we have to find the square root of the variance.

Complete step-by-step solution:
Now let us learn about the variance and standard deviation. Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Both measures reflect variability in a distribution, but their units differ. Standard deviation is expressed in the same units as the original values.
Now let us find out the variance and the standard deviation \[\left\{ 18,-9,-57,30,18,5,700,7,2,1 \right\}\].

DATA	DEVIATION	\[{{\left( \text{DEVIATION} \right)}^{\text{2}}}\]
\[18\]	\[53.5\]	\[2862.25\]
\[-9\]	\[80.5\]	\[6480.25\]
\[-57\]	\[128.5\]	\[16512.25\]
\[30\]	\[41.5\]	\[1722.25\]
\[18\]	\[53.5\]	\[2862.25\]
\[5\]	\[66.5\]	\[4422.25\]
\[700\]	\[-628.5\]	\[395012.3\]
\[7\]	\[64.5\]	\[4160.25\]
\[2\]	\[69.5\]	\[4830.25\]
\[1\]	\[70.5\]	\[4970.25\]

Let us calculate the sum and mean of the observations.
Sum of the observations= \[18-9-57+30+18+5+700+7+2+1\] \[=715\]
Mean of the observations= \[\dfrac{18-9-57+30+18+5+700+7+2+1}{10}\]
Upon solving, we get
\[=\dfrac{715}{10}=71.5\]
The deviations can be calculated by subtracting the data values from the mean.
Now let us calculate the variance of the data.
Variance calculated by \[{{\sigma }^{2}}=\dfrac{1}{N}(\sum {{X}^{2}}-\dfrac{{{(\sum X)}^{2}}}{N})\]
\[\begin{align}
&\sum{{X}^{2}}={{18}^{2}}+{{9}^{2}}+{{57}^{2}}+{{30}^{2}}+{{18}^{2}}+{{5}^{2}}+{{700}^{2}}+{{7}^{2}}+{{2}^{2}}+{{1}^{2}} \\
 & =494957 \\
\end{align}\]
\[\begin{align}
  & {{\left( \sum\limits_{{}}^{{}}{X} \right)}^{2}}={{\left( 715 \right)}^{2}} \\
 & = 511225 \\
\end{align}\]
Upon substituting the values in the formula, we get
\[\begin{align}
  & = \dfrac{1}{10}\left( 494957-\frac{511225}{10} \right) \\
 & =\dfrac{1}{10}\left( 494957-51122.5 \right) \\
 & =\dfrac{443834.5}{10} \\
 & =44383.45 \\
\end{align}\]
\[\therefore \] The variance of the data is \[44383.45\]
Now let us calculate the standard deviation of the data.
The standard deviation is nothing but the square root of the variance.
\[\sqrt{{\sigma}^2 }=\sqrt{44383.45}=210.67\]
\[\therefore \] The variance and standard deviation are \[44383.45\] and \[210.67\] respectively.

Note: The value of variance is equal to the square of standard deviation, which is another central tool. Variance is symbolically represented by \[{{\sigma }^{2}},\text{ }{{s}^{2}},\text{ }or\text{ }Var\left( X \right)\]. The standard deviation is the average amount of variability in your data set. It tells you, on average, how far each score lies from the mean.