Question

Suppose a constant c is added to each value in a set of data. Prove that the mean increases by c by comparing the formula for the mean of the original data with the formula for the mean of the recentered data.

Answer 1

Let x ₁, x ₂, x ₃ , ..., x _nbe the given set of data having  n  number of observations.

So, mean = sum of the observations / number of observations

  

("x+'"' + Er + II + )

$=\overline{x}$

Suppose a constant c is added to each value in a set of data.

Then the new observations becomes,

  x ₁+ c , x ₂+ c , x ₃ + c , ..., x _n+ c

and the number of observations will remain the same i.e., n

So, new mean =   sum of the observations / number of observations

(21+ c) + (22+c) + (23 +c) + ... + (n + c)

(21 + x2 +13 + ... + n) + nc

  

(21 +22 +13 + ... +un)

$=\overline{x}+\frac{nc}{n}$

= +c

i.e., New mean = old mean + c

Hence, the mean increases by c by comparing the formula for the mean of the original data with the formula for the mean of the recentered data.