Lets say:

$X = \{x_1, x_2, x_3, ... \} $ be a set of Real numbers in range $(R_1, R_2)$ and $m =$ mean of $x$

If I have to increase mean of set $X$ by $3$, each number in the set has to be increase by $3$. But how to increase mean of set $X$ by $3$, by only changing a subset of X. Is there any mathematical relations as such?

2 Answers

Let $x_1, \dots, x_k$ be the subset that does not change and let $x_{k+1},\dots, x_N$ be the subset that does. We have by hypothesis that

$$ {1 \over N} (x_1 + \cdots + x_k) + {1 \over N} (x_{k+1} + \cdots + x_N) = m. $$

And you want to find some function $f$ such that

$$ {1 \over N} (x_1 + \cdots + x_k) + {1 \over N} f(x_{k+1},\dots,x_N) = m + \sigma. $$

Using the expression for $m$ above, this becomes

$$ {1 \over N} (x_1 + \cdots + x_k) + {1 \over N} f(x_{k+1},\dots,x_N) = {1 \over N} (x_1 + \cdots + x_k) + {1 \over N} (x_{k+1} + \cdots + x_N) + \sigma. $$

$$ \Rightarrow f(x_{k+1},\dots,x_N) = (x_{k+1} + \cdots + x_N) + N \sigma. $$

As long that is satisfied, you will change the mean by a value of $\sigma$.

Now the sum of the $n$elements is $s$ and so $$s=nm$$ Now we want a new mean $M=m+3$ and the new sum $S$ should be $$S=nM=nm+3n=s+3n$$ So you have to add $y_i$ to every element $x_i$ such that $$\sum y_i=3n$$

ِAdded

• you can construct another set $y_i$ of elements with mean $3$

• you can add $3$ to every element

• you can add $3n$ to one element