Question

Find mean median of all positive factors of 72.

Answer 1

Hint:
We will find the positive factors of 72. We will arrange them in ascending order. If the number of factors is odd, we will locate the \[{\left( {\dfrac{{n + 1}}{2}} \right)^{th}}\] factor. If the number of factors is even, we will add the \[{\left( {\dfrac{n}{2}} \right)^{th}}\] and \[{\left( {\dfrac{{n + 2}}{2}} \right)^{th}}\]factor and divide the sum by 2. This will give us the median of the factors. Then, we will add all the factors and divide the sum by \[n\]. Here, \[n\] is the total number of factors.
Formula used: \[\bar x = \dfrac{{\sum {{x_i}} }}{n}\] where \[\bar x,\sum {{x_i}} \] and \[n\] are the mean, sum of observations and total number of observations respectively.

Complete step by step solution:
We will list out the factors of 72. We will find all numbers whose product with another natural number is 72. We will start with 1 and move further.
\[\begin{array}{l}1 \times 72 = 72\\2 \times 36 = 72\\3 \times 24 = 72\\4 \times 18 = 72\\6 \times 12 = 72\\8 \times 9 = 72\end{array}\]
We can see that the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
The factors are already arranged in ascending order. The total number of factors is 12. So, \[n = 12\]and \[n\] is even.
So \[{\left( {\dfrac{n}{2}} \right)^{th}}\] factor \[ = \dfrac{{12}}{2} = 6\]
\[{\left( {\dfrac{{n + 2}}{2}} \right)^{th}}\] factor \[ = \dfrac{{12 + 2}}{2} = \dfrac{{14}}{2} = 7\]
We will locate the \[{\left( {\dfrac{n}{2}} \right)^{th}}\] and \[{\left( {\dfrac{{n + 2}}{2}} \right)^{th}}\]factor i.e. we will locate the 6th and 7th factor. They are 8 and 9.
We will find the sum of 8 and 9.
\[8 + 9 = 17\]
We will now divide the sum by 2.
\[\dfrac{{17}}{2} = 8.5\]
\[\therefore\] The median of positive factors of 72 is 8.5.
We will add all the factors.
\[1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 + 72 = 195\]
We will substitute 195 for \[\sum {{x_i}} \] and 12 for \[n\] in the formula for the mean.
\[\begin{array}{l}\bar x = \dfrac{{195}}{{12}}\\\bar x = 16.25\end{array}\]

\[\therefore\] The mean of positive factors of 72 is \[16.25\].

Note:
Median is the middle observation of data when all the observations are arranged in \[{\left( {\dfrac{n}{2}} \right)^{th}}\] ascending/ descending order. So, we can also arrange the observations in descending order. This will not hamper the solution. It is easy to find the median if the number of observations is odd but we might get confused in case of an even number of observations. An even number of observations will have 2 observations that form the middle of data. We must remember to find the average of these 2 middle observations to get the median.