Question

1. The numbers 7, 6, 10, 13, 7, 2, 5, 6 and x have only one mode. And the mean, median and mode are all equal. What is the value of x?

Answer 1

Solution:

Numbers given => 7, 6, 10, 13, 7, 2, 5, 6, x

And also its is given that mean, median and mode are all equal.

Mode => It is the value which occurs the most times in the list of observations.

Median => It is the value of the middle most item when all the items are in the order of magnitude.

Mean => In mean we add all the observations and divide with the number of observations given.

Now, if we see in the given numbers 6 and 7 occurs two times.

But, mode can only be one.

Let's assume x be 6 and 7 once and check which satisfies this condition that mean, median and mode all are equal.

Condition 1 => Let x be 6.

Arranging numbers in ascending order => 2, 5, 6, 6, x i.e. 6, 7, 7, 10, 13

Mode = 6 (Occurring most times)

Mean = (2 + 5 + 6 + 6 + 6 + 7 + 7 + 10 + 13)/9 = 62/9 = 6.8

Median =  (Number of observations + 1)/2 = (9+1)/2 = 5th term i.e. 6

So, in this Mode = Median ≠ Mean i.e. 6 = 6 ≠ 6.8

Condition 2 => Let x be 7.

Arranging numbers in ascending order => 2, 5, 6, 6, 7, 7, x i.e. 7, 10, 13

Mode = 7 (Occurring most times)

Mean = (2 + 5 + 6 + 6 + 7 + 7 + 7 + 10 + 13)/9 = 63/9 = 7

Median =  (Number of observations + 1)/2 = (9+1)/2 = 5th term i.e. 7

So, in this Mode = Median = Mean i.e. 7 = 7 = 7

We can see that in Condition 2 which says to take x = 7 is satisfying the condition of mean, median and mode to be equal.

The value of x is 7.

Answer 2

Given :  The numbers 7, 6, 10, 13, 7, 2, 5, 6 and x have only one mode .  mean, median and mode are all equal

To find : Value of x

Solution:

7, 6, 10, 13, 7, 2, 5, 6 and x have only one mode

arranging data in order

2 , 5 , 6  , 6  , 7  ,  7  , 10 , 13

and position of x is unknown

there is only one mode

hence  x would  be either 6 or  7

Mode would be x

so data in order would be

2 , 5 , 6  , 6  , x  ,  7  ,  7  , 10 , 13

x is middle term  hence median = x

mean, median and mode are all equal

=> Mean = x

=> x  =  ( 2 + 5 + 6 + 6 + x + 7 + 7 + 10 + 13)/9

=> 9x  = 56 + x

=> 8x = 56

=> x = 7

Value of x  = 7

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