Question

How do you find the standard deviation for the numbers $ 4 $ , $ 4 $ , $ 5 $ , $ 5 $ , $ 6 $ and $ 6 $ ?

Answer 1

Hint: In the given question, we are required to find the standard deviation of the six numbers that are given to us in the problem itself. A standard deviation is a statistical measure that measures the dispersion or variation of a dataset relative to its mean. Standard deviation is represented by the symbol $ \sigma $ .

Complete step-by-step answer:
In the problem given to us, we are required to find the standard deviation for the numbers $ 4 $ , $ 4 $ , $ 5 $ , $ 5 $ , $ 6 $ and $ 6 $ .
The formula for finding the standard deviation of the numbers is $ \sigma = \sqrt {\dfrac{{\sum {{\left( {{x_i} - \bar x} \right)}^2}}}{N}} $ where $ \sigma $ is the standard deviation of the observations, N is the number of observations, $ {x_i} $ are the observations, and $ \bar x $ is the mean of the observations.
So, to find the standard deviation of the observations, we have to first calculate the mean of the observations.
Mean of the observations can be calculated as \[\dfrac{{\sum {{x_i}} }}{N}\].
So, we get the mean of the numbers $ 4 $ , $ 4 $ , $ 5 $ , $ 5 $ , $ 6 $ and $ 6 $ as $ \left( {\dfrac{{4 + 4 + 5 + 5 + 6 + 6}}{6}} \right) = \dfrac{{30}}{6} = 5 $
Now, the standard deviation $ = \sqrt {\dfrac{{{{\left( {4 - 5} \right)}^2} + {{\left( {4 - 5} \right)}^2} + {{\left( {5 - 5} \right)}^2} + {{\left( {5 - 5} \right)}^2} + {{\left( {6 - 5} \right)}^2} + {{\left( {6 - 5} \right)}^2}}}{6}} $
 $ \Rightarrow \sqrt {\dfrac{{{{\left( { - 1} \right)}^2} + {{\left( { - 1} \right)}^2} + {0^2} + {0^2} + {1^2} + {1^2}}}{6}} $
Simplifying further, we get,
 $ \Rightarrow \sqrt {\dfrac{{1 + 1 + 0 + 0 + 1 + 1}}{6}} $
 $ \Rightarrow \sqrt {\dfrac{4}{6}} $
Cancelling the common factors in numerator and denominator, we get,
 $ \Rightarrow \sqrt {\dfrac{2}{3}} $
So, the standard deviation of the numbers $ 4 $ , $ 4 $ , $ 5 $ , $ 5 $ , $ 6 $ and $ 6 $ is $ \sqrt {\dfrac{2}{3}} $ .
So, the correct answer is “ $ \sqrt {\dfrac{2}{3}} $ ”.

Note: Standard deviation gives a clear picture of the distribution of observations in real life examples. If the data points are further from the mean, there is a higher deviation within the data set and if the data points are close to the mean of the observations, the standard deviation is less.