Statistics-1 (XAT-2011) (5-Marks)

In a list of seven integers, one integer denoted as x is unknown. The other six integers are 20,4,10,4,8 and 4. If the mean, median and mode of these 7 integers are arranged in increasing order, they form an arithmetic progression. The sum of all possible values of x is:

(1)26   (2) 32   (3) 34   (4) 38   (5) 40

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Solution:

Arranging the given six integers in increasing order:

4,4,4,8,10,20

Irrespective of the value of x, the number of occurrences being three, 4 is the mode.

Mode = 4; Mean = (4+4+4+8+10+20+x)/7 = (50+x)/7

Here three cases arise:

Case-I                 If x≤4, median = 4

Case-II                If 4

Case-III              If x>8, median = 8

Case-I:

x≤4;       Median = 4;

Given that mean, median and mode arranged in ascending are in AP

Since mode=median=4, ascending order is not possible here and hence this case is excluded.

Case-II:

4

4 possible values for x are 5,6,7 and 8

But, Mean which is (50+x)/7 is integer only for x=6

 => x = 6

Case-III:

x>8;      Median = 8;

x>8 => (50+x)/7 > 8

Hence 4, 8, (50+x)/7 are in AP => 8 - 4 = (50+x)/7 - 8 => 4 = (50+x-56)/7

=> 28 = x-6 => x = 34

In all the cases, the possible values of x are 6 and 34

Sum of possible values of x = 6+34 = 40

Answer (5)