Question

How do I calculate the variance and standard deviation of $102,104,106,108,110$?

Answer 1

Hint: To find the variance, you must find “the average of the squared distances from the mean”. You individually subtract the mean from each data point, squaring all the numbers and then add them and finally, we have to take the average of all of them. Also, to find the standard deviation we have to take the square root of variance.

Complete step-by-step solution:
The measurement of the distance between the mean or average value of a data set and how far a data point has dispersed is called the variance. It is represented as.
Now to find the variance first we have to find the mean. We do this by adding all the numbers of the data set together, then dividing them by how many numbers there were, so:
Therefore,
$\text{Mean} = \dfrac{{102 + 104 + 106 + 108 + 110}}{5}$
$Mean = \dfrac{{530}}{5}$
$Mean = 106$
Therefore, our required mean is $106$.
Next, we find the variance. The process is as follows: select a value from the data set $\left( {102} \right)$. Subtract the mean from it $\left( {102 - 106 = - 4} \right)$. Then square the result $\left( {{{\left( { - 4} \right)}^2} = 16} \right)$. We do this for the rest of the numbers, to get
${\left( {104 - 106} \right)^2} = {\left( { - 2} \right)^2} = 4$
${\left( {106 - 106} \right)^2} = {0^2} = 0$
${\left( {108 - 106} \right)^2} = {\left( 2 \right)^2} = 4$
${\left( {110 - 106} \right)^2} = {\left( 4 \right)^2} = 16$
The last thing to do is to add these results together, then divide them by how many there are like so:
$ \Rightarrow \dfrac{{16 + 4 + 0 + 4 + 16}}{5}$
$ \Rightarrow \dfrac{{40}}{5}$
$ \Rightarrow 8$
Therefore, the variance of given data is $8$.
To find the standard deviation, we simply take the square root of the variance, so:
$ \Rightarrow \sqrt 8 = 2.828$
Therefore, the standard deviation is roughly equal to $2.828$.

Note: There is also a direct formula to find the variance of the given data. But here we have used the most basic method to find the variance and standard deviation. Variance tells us how far the random numbers are spread out from their mean value.