Question

The mean of n observations x1 , x2, x3.......xn is x~ (bar).If each observation is decreased by p then show that the mean of the new observation is (x~(bar)-p).

Answer 1

x~=(x1+x2+x3..+xn)/n

Subtracting p from each observation we get mean=

(x1-p+x2-p+...+xn-p)/n=(x1+x2+...+xn-np)/n=(x1+x2+...+xn)/n -p=x~-p

Proved

Answer 2

Answer:

$New \: mean = \overline{x} -p$

Step-by-step explanation:

If mean of

$x_1 ,\: x_2,\: x_3, \: x_4,...x_n \: is \: \: \bar {x}$

Now if each observation is decreased by p, then new mean will be

$\overline{x} -p$

To prove this,apply the basic formula of finding mean

$\overline{x} = \frac{x_1 + x_2 + x_3 + x_4 + ... + x_n}{n} \: \: \: \: eq1$

Now each observation is decreased by p

$New \: mean = \frac{x_1 -p+ x_2 -p + x_3 -p + x_4 -p + ... + x_n -p}{n} \\ \\ New \: mean= \frac{x_1 + x_2 + x_3 + x_4 + ... + x_n -p-p-p-p-... -p}{n} \\\\since\: n\:numbers\:are\:there\:so\:we\: have\:to\:decrease\:p\:n\:times \\ \\New \: mean= \frac{x_1 + x_2 + x_3 + x_4 + ... + x_n}{n} - \frac {p+p+p+p+p+...+p}{n} \\ \\ from \: eq1 \\ New \: mean = \overline{x} - \frac{np}{n} \\ \\ New \: mean= \overline{x} -p$

Hope it helps you.