The mean of unground data and the mean calculated when the same data is grouped are always the same. Is it true ? Give reasons.

Answer :

The given statement :  The mean of ungrouped data and the mean calculated when the same data is grouped are always the same.

Is wrong , we can justify that by taking a example , As :

In a survey of how many books each of 20 friends owned, and got this:

9, 15 , 11 , 12 , 3 , 5 , 10 , 20 , 14 , 6 , 8 , 8 , 12 , 12 , 18 , 15 , 6 , 9 , 18 , 11

Sum of this data  = 9 + 15 + 11 + 12 + 3 + 5 + 10 + 20 + 14 + 6 + 8 + 8 + 12 + 12 + 18 + 15 + 6 + 9 + 18 + 11  =  222

So mean of this data  is  =  $\frac{222}{20}$  =  11. 1 

Now we make same data as a grouped data
 

Number of Books	Frequency
1 -  5	2
6 - 10	7
11 - 15	8
16 - 20	3

in that case of find mean As :  first we form a different table AS :
 

Number of Books	Frequency ( f )	Midpoint ( x )	fx
1 - 5	2	3	6
6 - 10	7	8	56
11 - 15	8	13	104
16 - 20	3	18	54
	$\underset{}{ \sum }$ f  = 20		$\underset{}{ \sum }$ fx  = 220

SO our mean of grouped data  = $\frac{{\displaystyle \sum _{}} fx}{{\displaystyle \sum _{}}f} = \frac{220}{20}$  =  11 


So we can see that mean obtain by both methods we get different values , As we know from grouped data we can only find estimate mean , So value of mean is different for both methods .