Question

If $ N = 50 $ , $ \sum {fx} = 300 $ and $ \sum {f{x^2} = 50000} $ , find standard deviation and variance.

Answer 1

Hint: Standard deviation is the square root of the arithmetic mean of the squares of deviations of observations from the mean value. This can be written as,
 $ \sigma = \sqrt {\dfrac{{\sum\limits_{i = 1}^N {{{\left( {{x_i} - \overline x } \right)}^2}} }}{N}} $
Variance is the square of the standard deviation. It represents the expectation of the squared deviation of a variable from the mean. Variance is given as,
 $ {\sigma ^2} = \dfrac{{\sum\limits_{i = 1}^N {{{\left( {{x_i} - \overline x } \right)}^2}} }}{N} $

Complete step by step solution:
We have been given some information about a set of data.
The total number of elements is $ N = 50 $ .
Sum of the values is $ \sum {fx} = 300 $ .
And the sum of the square of the values is $ \sum {f{x^2} = 50000} $ .
We have to find the standard deviation and the variance of this set of data.
The variance is given as,
 $ {\sigma ^2} = \dfrac{{\sum\limits_{i = 1}^N {{{\left( {{x_i} - \overline x } \right)}^2}} }}{N} $
This can be written in terms of frequency of data $ f $ as,
 $ {\sigma ^2} = \dfrac{{\sum\limits_{i = 1}^N {f{{\left( {{x_i} - \overline x } \right)}^2}} }}{N} $
In a more simplified form, this is also written as,
\[{\sigma ^2} = \dfrac{{\sum\limits_{i = 1}^N {f{x_i}^2} }}{N} - {\left( {\overline x } \right)^2}\]
We can find the mean $ \overline x $ as,
 $ \overline x = \dfrac{{\sum {fx} }}{N} = \dfrac{{300}}{{50}} = 6 $
And, \[\sum\limits_{i = 1}^N {f{x_i}^2} = \sum {f{x^2} = 50000} \]
Thus we have,
\[{\sigma ^2} = \dfrac{{50000}}{{50}} - {\left( 6 \right)^2} = 1000 - 36 = 964\]
We can find the standard deviation as the square root of the variance.
Thus,
\[\sigma = \sqrt {{\sigma ^2}} = \sqrt {964} = 31.05\]
Hence, for the given set of data the standard deviation is \[31.05\] and the variance is \[964\].
So, the correct answer is “ \[31.363\] ”.

Note: We did not require the values of the data to find the standard deviation and the variance. Since the sum of the values and the total number of values were given, we used a direct formula to calculate the mean. We used a simplified form of the formula for variance as suitable for the information given in the question. We should be careful that standard deviation is the square root of the variance and not get confused between the two.