Question

Find the mean of \[8,12,16,10,22,4\] when each number is multiplied by $3$ .

Answer 1

Hint: At first, we arrange the numbers in the ascending order. After that, we find the order by multiplying each number by $3$ . We then find their mean according to the formula \[Mean\text{ }=\text{ }\dfrac{Sum\text{ }of\text{ }observations}{Number\text{ }of\text{ }observations}\] and finally get the mean.

Complete step by step answer:
Mean is the average of observations. The discrete data given has six observations. Other measures of finding the average of the middle value of data are median and mode. Median is calculated by arranging the discrete data in discrete order and then applying the formula of finding the mean of even number of observations. For an even number of observations, the median is calculated as the average of the third and fourth observations. Mode is calculated as the highest number of times a value appears in the data available.
Mean is a measure of analysis as it helps in finding the average value in data. The observations are arranged in ascending order to determine the value of mean. Ascending order indicates arranging the numbers from smallest to largest. The numbers are arranged as,
\[4,8,10,12,16,22\]
Each of the numbers multiplied by $3$ gives the order as,
$\Rightarrow 12,24,30,36,48,66$
Now,
\[\begin{align}
  & Mean\text{ }=\text{ }\dfrac{Sum\text{ }of\text{ }observations}{Number\text{ }of\text{ }observations} \\
 & \Rightarrow Mean=\dfrac{12+24+30+36+48+66}{6}=\dfrac{216}{6}=36 \\
\end{align}\]
Thus, we can conclude that the mean of the numbers \[8,12,16,10,22,4\] when each number is multiplied by $3$ is $36$ .

Note: We can also solve the problem using a different approach. After arranging the given data set in ascending order, we find the mean of them using the known formula of mean. The mean comes out as $\dfrac{4+8+10+12+16+22}{6}=\dfrac{72}{6}=12$ . Now, multiplying or dividing a data set by a common number changes their mean by the same multiplying or dividing factor accordingly. So, the mean of the multiplied numbers becomes $12\times 3=36$ .