Question

How do you calculate the sample range, sample mean, sample median, sample variance and sample standard deviation of $18,19,34,38,24,18,22,51,44,14,29$?

Answer 1

Hint: We first express the given sample inputs in ascending order. Then we express the individual process and formulas to find sample range, sample mean, sample median, sample variance and sample standard deviation.

Complete step by step solution:
We have been given a sample of 11 discrete data $18,19,34,38,24,18,22,51,44,14,29$ and we need to find their sample range, sample mean, sample median, sample variance and sample standard deviation. In this sample $n=11$.
Arranging them in ascending order we get $14,18,18,19,22,24,29,34,38,44,51$. We consider the inputs as ${{x}_{i}},i=1(1)11$.
We express the mathematical formulas to find those attributes and put the values.
Sample range is equal to the difference between the highest and lowest value of the whole sample which is $51-14=37$.
Sample mean can be expressed as $\overline{x}=\dfrac{\sum{{{x}_{i}}}}{n}$. Putting the values, we get
\[\overline{x}=\dfrac{14+18+18+19+22+24+29+34+38+44+51}{11}=\dfrac{311}{11}=28.28\].
Sample median is expressed as the middle point of the ordered sample either in ascending or descending order. In this case of $14,18,18,19,22,24,29,34,38,44,51$, the median is 24.
Sample variance can be expressed as ${{\sigma }^{2}}=\dfrac{1}{n}\sum{{{x}_{i}}^{2}}-{{\left( \overline{x} \right)}^{2}}$. Now we find
\[\begin{align}
  & \sum{{{x}_{i}}^{2}}={{14}^{2}}+{{18}^{2}}+{{18}^{2}}+{{19}^{2}}+{{22}^{2}}+{{24}^{2}}+{{29}^{2}}+{{34}^{2}}+{{38}^{2}}+{{44}^{2}}+{{51}^{2}} \\
 & \Rightarrow \sum{{{x}_{i}}^{2}}=10243 \\
\end{align}\]
Putting the values, we get ${{\sigma }^{2}}=\dfrac{1}{11}\left( 10243 \right)-{{\left( 28.28 \right)}^{2}}=\dfrac{15952}{121}=131.83$.
Sample standard deviation can be expressed as \[\sigma =\sqrt{{{\sigma }^{2}}}=\sqrt{\dfrac{1}{n}\sum{{{x}_{i}}^{2}}-{{\left( \overline{x} \right)}^{2}}}\]. Putting the values, we get \[\sigma =\sqrt{131.83}=11.48\].

Note:
These all-required attributes are the measurement of central tendency and dispersion. The sample range, sample mean, sample median is part of central tendency and sample variance and sample standard deviation are part of dispersion.