Question

Find the mean of the following data:

Class interval	$ 10 - 25 $	$ 25 - 40 $	$ 40 - 55 $	$ 55 - 70 $	$ 70 - 85 $	$ 85 - 100 $
Frequency( \[{f_i}\] )	$ 2 $	$ 3 $	$ 7 $	$ 6 $	$ 6 $	$ 6 $

Answer 1

Hint: Use formula for mean of group data and find required values for the formula.
First, we know that frequency is denoted as $ {f_i} $ and for finding the mean of grouped data. The formula is \[\dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}\] , so we have to find \[{x_i}\] , which is the midpoint of the respective class intervals.Then we find the required values to be substituted in the formula and find mean.

Complete step by step solution:
 We have to find the midpoint of all the given class intervals.

Class interval	$ 10 - 25 $	$ 25 - 40 $	$ 40 - 55 $	$ 55 - 70 $	$ 70 - 85 $	$ 85 - 100 $
Frequency ( \[{f_i}\] )	$ 2 $	$ 3 $	$ 7 $	$ 6 $	$ 6 $	$ 6 $
\[{x_i}\]	$ 17.5\, $	$ \,32.5\,\, $	$ 47.5\, $	$ \,62.5\,\, $	$ 77.5\, $	$ \,92.5 $

Now, we have found the midpoint of all the given class intervals, now we should find \[\sum {{f_i}} \] and \[\sum {{f_i}{x_i}} \] , So let’s find them
To find \[\sum {{f_i}} \] , it is the total of all the frequency. So, it is
 \[\sum {{f_i}} = 2 + 3 + 7 + 6 + 6 + 6 = 30\]
Next, we have to find \[\sum {{f_i}{x_i}} \] , so for that first we have to find \[{f_i}{x_i}\] and then we have to total it up, then we get \[\sum {{f_i}{x_i}} \] .
So

Class interval	$ 10 - 25 $	$ 25 - 40 $	$ 40 - 55 $	$ 55 - 70 $	$ 70 - 85 $	$ 85 - 100 $
Frequency ( \[{f_i}\] )	$ 2 $	$ 3 $	$ 7 $	$ 6 $	$ 6 $	$ 6 $
\[{x_i}\]	$ 17.5\, $	$ \,32.5\,\, $	$ 47.5\, $	$ \,62.5\,\, $	$ 77.5\, $	$ \,92.5 $
\[{x_i}{f_i}\]	$ 35\,\,\, $	$ 97.5\, $	$ \,332.5\, $	$ \,375\, $	$ \,465\, $	$ \,555 $

Now, we have found the \[{x_i}{f_i}\] , we add them up and find the total \[\sum {{f_i}{x_i}} \] . Now, we add them
 \[\sum {{f_i}{x_i}} = 35 + 97.5 + 332.5 + 375 + 465 + 555 = 1860\]
Now, we have found all the necessary values for finding the mean of the given data which is grouped data
So, we substitute them into the formula of mean
Mean= \[\dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}\]
We get;
 $ mean = \dfrac{{1860}}{{30}} = 62 $
Therefore, we have found the mean of the given data which is classified as grouped data
So, the correct answer is “ $ mean = \dfrac{{1860}}{{30}} = 62 $ ”.

Note: We can also find the mean of the grouped data with another formula which is called step-deviation. Which has the formula of \[\mathop {x{\text{ }}}\limits^ - = {\text{ a + h}}\left( {\dfrac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }}} \right)\] . The mean obtained by any method, the value of the mean is going to be the same and exact with no approximation and when we find the summations, we have to be careful and make no calculation mistakes.