Find the values of $n$ and $X$ in each of the following cases:
$\sum\limits _{i=1}^{n}( x_{i} -10) =30$ and $\sum\limits _{i=1}^{n}( x_{i} -6) =150$.

Given:

$\sum\limits _{i=1}^{n}( x_{i} -10) =30$ and $\sum\limits _{i=1}^{n}( x_{i} -6) =150$.

To do:

We have to find the values of $n$ and $X$.

Solution:

We know that,

Mean $\overline{X}=\frac{Sum\ of\ the\ observations}{Number\ of\ observations}$

Therefore,

$\sum_{i=1}^{n}(x_{i}-10)=30$...........(i)

$\sum_{i=1}^{n}(x_{i}-6)=150$...........(ii)

From (i) and (ii), we get,

$n \bar{x}-10 n=30$........(iii)

$n \bar{x}-6 n=150$.........(iv)

Subtracting (iv) from (iii), we get,

$-4 n=-120$

$n=\frac{-120}{-4}$

$n=30$

From (iii),

$n \bar{x}-10 \times 30=30$

$30 \bar{x}=30+300$

$30 \bar{x}=330$

$\bar{x}=\frac{330}{30}$

$=11$

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Updated on: 10-Oct-2022

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