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The variance is the ______ of the standard deviation.
A. Square
B. Cube
C. Square root
D. Cube root

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Last updated date: 23rd May 2024
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Answer
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Hint: As we know that the formula of standard deviation is \[\sigma =\sqrt{\dfrac{\underset{i=1}{\overset{n}{\mathop \Sigma }}\,{{\left( {{x}_{i}}-\overline{x} \right)}^{2}}}{n}}\] . It will tell how the data will be deviated from its mean and the formula of the variance is \[\dfrac{\underset{i=1}{\overset{n}{\mathop \Sigma }}\,{{\left( {{x}_{i}}-\overline{x} \right)}^{2}}}{n}\]. Hence, we can find the solution from both of these.

Complete step-by-step solution:
As standard deviation tells us, the measure of how spread out numbers are is equal to the square root of the arithmetic mean of the squares of the deviations measured form the arithmetic mean of the data.
The standard deviation formula is given as follows:
\[\sigma =\sqrt{\dfrac{\underset{i=1}{\overset{n}{\mathop \Sigma }}\,{{\left( {{x}_{i}}-\overline{x} \right)}^{2}}}{n}}\ldots \ldots \ldots \left( i \right)\]
${{x}_{i}}$ is the ${{i}^{th}}$ data point.
$\overline{x}$ is the mean of all the data points.
$n$ is the number of data points.
And the variance mean static is a measurement of the spread between the numbers in a data set. That is, it measures how far each number in the data set is from the mean and therefore from every other number in the set.
The average of the squared differences from the mean.
Variance is equal to,
\[\dfrac{\underset{i=1}{\overset{n}{\mathop \Sigma }}\,{{\left( {{x}_{i}}-\overline{x} \right)}^{2}}}{n}\ldots \ldots \ldots \left( ii \right)\]
Where, ${{x}_{i}}$ is the ${{i}^{th}}$ data point.
$\overline{x}$ is the mean of all the data points.
$n$ is the number of data points.
Hence from formula (i) and (ii) we can say that the square of the standard deviation is equal to variance.
Hence the correct option is A.

Note: Standard deviation along median is being calculated as the formula is,
\[\sigma =\sqrt{\dfrac{\underset{i=1}{\overset{n}{\mathop \Sigma }}\,{{\left( {{x}_{i}}-M \right)}^{2}}}{n}}\]
Where, ${{x}_{i}}$ is the ${{i}^{th}}$ data point.
$M$ is the median of all the data points.
$n$ is the number of data points.
It will show how much the data is deviated from the median.