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?
Answers
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.
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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