Question

23. The mean of 100 observations is 50. If one of the observation which was 50 is replaced by 150 then find the new mean.
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Answer 1

Given

Mean of 100 observations is 50.

To find

The new mean if one of the observation, 50, is replaced by 150

$\rule{200}2$

Answer

51

Solution

We know that,

sum of observations/total number of observations = mean

→ sum of observations = mean × total number of observations

Here, mean = 50, total number of observations = 100

→ sum of observations = 100 × 50

→ sum of observations = 5000

Now, one of the observation, that's 50 is replaced by 150.

This means that 150 is added when 50 is subtracted

→ new sum of observations = 5000 - 50 + 150

→ 5100

So, new mean = 5100/100

→ 51.

Answer 2

$\bold{\underline{\underline{\huge{\sf{AnsWer:}}}}}$

$\sf{\pink{New\:Mean\:=\:51}}$

$\bold{\underline{\large{\underline{\sf{StEp\:by\:stEp\:explanation:}}}}}$

$\bold{\underline{\underline{\red{\tt{Given:}}}}}$

• The mean of 100 observations is 50.
• One of the observation,50 is replaced by 150.

$\bold{\underline{\underline{\blue{\tt{To\:find:}}}}}$

• The new mean

$\bold{\underline{\underline{\red{\tt{Solution:}}}}}$

We know the formula for mean.

$\bold{\large{\boxed{\tt{\green{Mean\:=\:{\dfrac{Sum\:of\:observations\:}{Number\:of\:observations}}}}}}}$

• Mean = 50
• Sum of observations (unknown)
• Number of observation = 100

$\leadsto$ $\tt{50\:=\:{\dfrac{Sum\:of\:observation\:}{100}}}$

$\leadsto$ $\tt{50\:\times\:100\:=\:Sum\:of\:observations}$

$\leadsto$ $\tt{5000\:=\:Sum\:of\:observations}$

$\tt{\therefore{Sum\:of\:100\:observations\:having\:mean\:50\:is\:5000}}$

One of the observation is replaced :

•°• New sum of the 100 observation will be,

$\leadsto$ $\tt{5000\:-50\:+\:150}$

$\leadsto$ $\tt{4950\:+\:150}$

$\leadsto$ $\tt{5100}$

$\tt{\therefore{\underline{\blue{New\:Sum\:of\:100\:observations\:\:is\:5100}}}}$

Now, the new mean will be, new sum of 100 observations (5100) over number of observation (100)

New Mean :

$\leadsto$ $\tt{\dfrac{5100}{100}}$

$\leadsto$$\tt{51}$

$\sf{\underline{\therefore{\purple{New\:mean\:of\:100\:observations\:is\:51}}}}$