Question

What is the standard deviation of \[1,2,3,4\text{ }and\text{ }5\] ?

Answer 1

Hint: For solving this problem we need to have a clear understanding of what does variance and standard deviation signify. By calculating the mean of the five numbers and using the standard formula, we get the standard deviation of the five numbers.

Complete step-by-step answer:
Unlike range and quartiles, the variance combines all the values in a data set to produce a measure of spread. The variance and standard deviation (the square root of the variance) are the most commonly used measures of spread. We know that variance is a measure of how spread out a data set is. It is calculated as the average squared deviation of each number from the mean of a data set. Standard deviation is the measure of spread most commonly used in statistical practice when the mean is used to calculate central tendency. Thus, it measures spread around the mean. Because of its close links with the mean, standard deviation can be greatly affected if the mean gives a poor measure of central tendency.
According to the given problem, we need to find the standard deviation of \[1,2,3,4\text{ }and\text{ }5\] . For solving this, we first need to find the mean of the five numbers. We denote the numbers as X and the total number of numbers (which is five) as N.
\[Mean~X=\dfrac{\sum X}{N}=\text{ }\dfrac{\left[ 1+2+3+4+5 \right]}{5}=\dfrac{15}{5}=3\]
Thus, we get that the mean of the five numbers is three. Now employing the standard formula for calculating the variance and standard deviation, we get that,
\[\begin{align}
  & Variance~=\dfrac{\sum {{X}^{2}}}{N}-{{\left( \dfrac{\sum X}{N} \right)}^{2}}=\text{ }\dfrac{\left[ {{1}^{2}}+{{2}^{2}}+{{3}^{2}}+{{4}^{2}}+{{5}^{2}} \right]}{5}-{{3}^{2}}=\dfrac{55}{5}-9=2 \\
 & \therefore ~Variance~=2 \\
 & And,~Standard\text{ }Deviation=\sqrt{2} \\
\end{align}\]

Note: These types of problems may seem simple but there are high chances of miscalculations due to the lengthy calculations for variance and standard deviation. We need to carefully perform the calculations after using the correct formula.