If 5 is added to each and every item of a data, then the arithmetic mean is A) 5 times to the first arithmetic mean 

Correct option is: B) increased by 5 to the first arithmetic mean

Let \(x_i\) be the original observations of the data.

Let there are total n number of observations in the data.

\(\therefore\) Sum of original observation = n \(\times\) original mean

\(\Rightarrow\) \(\sum x_i = n\overline x\) , where \(\overline x\) is original mean of data.

Since, 5 is added to each and every item of the data.

\(\therefore\) Each observations of new data is of type (\(x_i+5\)).

\(\therefore\) Sum of new observations = \(\sum x_i+5\)

\(\therefore\) New mean = \(\frac {\sum x_i+5}n = \frac {\sum x_i+\sum5}n\)

= \(\frac {n\overline x + 5n}{n} = \frac {n(\overline x+5)}n = \overline x+5\)

Hence, the new arithmetic mean is increased by 5 to the first arithmetic mean.

Correct option is: B) increased by 5 to the first arithmetic mean