Question

What is the mean, median and mode of 72, 75, 78, 72, 73?

Answer 1

Hint: Use the formula \[\bar{x}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}\] to calculate the mean of the given numbers, where \[\bar{x}\] denotes the mean and n is the number of observations. Substitute the value of n equal to 5. Now, for odd number of observations provided apply the formula Median = \[{{\left( \dfrac{n+1}{2} \right)}^{th}}\] term to calculate the median after arranging the observations in ascending order of their numerical value. Finally, to calculate the mode of the data set given, observe the data which is appearing most number of times.

Complete step by step solution:
Here we have been provided with the numbers 72, 75, 78, 72, 73 and we are asked to calculate the mean, median and mode of these numbers.
(1) Now, we know that the mean of n observations is given by the formula \[\bar{x}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}\], i.e. the ratio of sum of all the observations to the number of observations. In the above formula \[\bar{x}\] denotes the mean. Clearly we can see that the number of observations is 5 so we have the value of n equal to 5. Substituting n = 5 in the formula for mean we get,
\[\begin{align}
  & \Rightarrow \bar{x}=\dfrac{\sum\limits_{i=1}^{5}{{{x}_{i}}}}{5} \\
 & \Rightarrow \bar{x}=\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}}{5} \\
\end{align}\]
Substituting the values of given observations we get,
\[\begin{align}
  & \Rightarrow \bar{x}=\dfrac{72+75+78+72+73}{5} \\
 & \Rightarrow \bar{x}=\dfrac{370}{5} \\
 & \therefore \bar{x}=74 \\
\end{align}\]
Therefore, the mean of the given numbers is 74.

(2) Now, to calculate the median first we have to arrange the given numbers in ascending order of their numerical value. So on arrangement we get 72, 72, 738, 75, 78. Therefore applying the formula of median for n = 5 terms (which is odd) we get,
$\Rightarrow $ Median = \[{{\left( \dfrac{n+1}{2} \right)}^{th}}\] term
$\Rightarrow $ Median = \[{{\left( \dfrac{5+1}{2} \right)}^{th}}\] term
$\Rightarrow $ Median = ${{3}^{rd}}$ term
In the above arrangement we see that the ${{3}^{rd}}$ term is 73, so we get,
$\therefore $ Median = 73
Therefore, the median of the numbers is 73.

(3) Now, Mode of a given data set is defined as the data which appears the most number of times. On observing the given data set we can conclude that the number 72 is appearing two times and the other three numbers are appearing only one time each, so we have,
$\therefore $ Mode = 72
Therefore, the mode of the numbers is 72.

Note: Note that in case we have even number of terms then the median of the data set is given by the formula Median = \[\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}\]. One important thing you must note is that there is a relation between mean, median and mode given as: - Mode = 3(median) – 2(Mean). This relation is known as the empirical relation. However, we haven’t used this relation to solve the above question because it is used for moderately asymmetrical distribution of data but in the above question we haven’t been provided with the distribution of data but simply a data set of numbers is given.