Question

if x bar is the mean of n observations x1, x2......xn, then find the arithmetic mean of x1+a, x2+a, xn+a

Answer 1

Answer:

The arithmetic mean of  x₁ + a , x₂ + a .... x${}_n + a$ is $\bar{x}+a$

Step-by-step explanation:

It is given that $\bar{x}$ is the mean of n observations $x_1 ,x_ 2 ,x_3 ..... x_n$

From the above statement, we can say :

Number of observations = n

Sum of observations = x₁ , x₂ .... x${}_n$

Mean of the observations = $\bar{x}$

From the properties of coordinate geometry, we know  :

Mean = ( sum of observations ) / ( number of observations )

Therefore,

= > $\bar{x}$=( x₁ , x₂ .... x${}_n$ ) / n

= > n . $\bar{x}$ = x₁ , x₂ .... x${}_n$    ...( i )

Hence,

Sum of the total observations is n . $\bar{x}$

_______________________

Now,

Observation = x₁ + a , x₂ + a .... x${}_n + a$

Therefore,

= > Mean = (  ( x₁ + a ) + ( x₂ + a ) .... ( x${}_n + a )$ ) / n

= > Mean = ( x₁ + x₂ + x${}_n$ + a + a + .... n times ) / n

Substituting the value of x₁ + x₂ + x${}_n$ from ( i ) .

= > Mean = ( n . $\bar{x}$ + an ) / n

= > Mean = n( $\bar{x}$+ a ) / n

= > Mean = $\bar{x}$ + a

Hence, the arithmetic mean of  x₁ + a , x₂ + a .... x${}_n + a$ is $\bar{x}+a$

Answer 2

Answer:

Step-by-step explanation:

Given that x bar is the mean of the Observation x1 , x2 , ......... , Xn.

=> Mean ( Average) = Sum of Total Observation/ Number of Observation

=> x bar = ( x1 + x2 + ......, + Xn)/ n

=> n . x bar = ( x1 + x2 + ,........ + Xn)

Sum of the Total Observation = n . x bar.

Now,

Arithmetic Mean of x1+a, x2+a, xn+a.

=> Arithematic Mean = ( x1 + a + x2 + a + ,..... , + Xn)/n

It can be written as:

=> Mean = (x1 + x2 + x3 +,....., + xN , aN)/ n

=> Mean = (n. x bar + aN)/n

=> Mean = n ( x bar + a)/n

=> Mean = x bar + a.

Hence,

The Arithmetic Mean is x bar + a.