Suppose that the mean and median of the non-negative numbers 21,8,17, a, 51,103, b, 13,67,…

Question

Suppose that the mean and median of the non-negative numbers $21,8,17, a, 51,103, b, 13,67,(a>b)$, are 40 and 21 , respectively. If the mean deviation about the median is 26 , then $2 a$ is equal to :

Accepted Answer

The given data set has $n=9$ observations: $21,8,17,a,51,103,b,13,67$.

The mean ($\vec{x}$) of these numbers is given as $40$.

The formula for the mean is:

$=\frac{}{n}$

Substitute the given values into the formula:

$40=\frac{21+8+17+a+51+103+b+13+67}{9}$

Multiply both sides by 9:

$360=280+a+b$

$a+b=360-280$

$a+b=80\,$

The median ($M$) of the data is $21$. For $n=9$ (an odd number), the median is the ${(\frac{9+1}{2})}^{}={5}^{}$ observation when aligned in ascending order.

Let's arrange the known numbers in ascending order:

$8,13,17,21,51,67,103$

Here, there are exactly three numbers less than $21$ ($8,13,17$) and three numbers greater than $21$ ($51,67,103$).

For $21$ to remain the middle (5th) observation among all 9 numbers, one of the unknown variables ($a$ or $b$) must be less than or equal to $21$, and the other must be greater than or equal to $21$.

Since it is given that $a>b$, we can conclude:

$a\ge 21$ and $b\le 21$

The formula for mean deviation about the median is:

$=\frac{\sum _{n}^{i=1}|{x}_{i}-M|}{n}$

We are given that $=26$.

Substitute the values:

$26=\frac{|21-21|+|8-21|+|17-21|+|a-21|+|51-21|+|103-21|+|b-21|+|13-21|+|67-21|}{9}$

Multiply by 9 and simplify the absolute values for the known numbers:

$234=0+13+4+|a-21|+30+82+|b-21|+8+46$

$234=183+|a-21|+|b-21|$

$|a-21|+|b-21|=234-183$

$|a-21|+|b-21|=51$

From Step 2, we know that $a\ge 21\, \implies \, |a-21|=a-21$ and $b\le 21\, \implies \, |b-21|=21-b$. Replacing these in the equation:

$(a-21)+(21-b)=51$

$a-b=51\,$

Now we have a system of two linear equations:

1) $a+b=80$

2) $a-b=51$

To find $2a$, simply add Equation 1 and Equation 2:

$(a+b)+(a-b)=80+51$

$2a=131$

Therefore, the value of $2a$ is $131$.

Option D is the correct answer.

Solutions · 1