Question

if the mean of first 10 observations is 15and the mean of 15 observations is 10 find the mean of all observation​

Answer 1

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Question :-

If the mean of first 10 observations is 15 and the mean of 15 observations is 10.

Find the mean of all observations.

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Answer :-

Given :-

Mean of first 10 observations is 15 .

No. of observations = 10

Mean of next 15 observations is 10 .

No. of observations = 15

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Required to find :-

• Mean of all observations ?

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Formulae used :-

$\large{\rightarrow{\boxed{\mathsf{Mean \;=\; \dfrac{Sum\;of\;the\; observations}{No. \: of \; observations}}}}}$

$\large{\rightarrow{\boxed{\mathsf{Sum \; of \; the \; observations \; =\; Mean \times No. \; of \; observations }}}}$

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Solution :-

Given that ;

Mean of first 10 observations is 15.

Since ,we are given with mean

Let's find the sum of the observations

For this we have to use a formula

The formula is ,

Sum of the observations = Mean x No. of observations

Here, No. of observations are 10

So,

$\rightarrow{\textsf{Sum of the observations}}{\tt{\;= 15 \times 10}}$

$\rightarrow{\textsf{Sum of the observations}}{\tt{ \;= 150 }}$

Similarly,

It is also given that ,

Mean of next 15 observations is 10 .

However,

Here mean is given so , let's find the sum using the same formula .

No. of observations = 15

So,

$\rightarrow{\textsf{Sum of the observations}}{\tt{\;= 10 \times 15}}$

$\rightarrow{\textsf{Sum of the observations}}{\tt{ \;= 150 }}$

Now we have to find the mean of all observations .

In order to find the mean of all observations ,

we should add the sum of the observations which we got in the first case with the sum of the observations which we got in the second case .

Similarly,

we should add the no. of observations in both cases in order to find the Total no. of observations .

So,

Total No. of observations = 10 + 15

Total No. of observations = 25

Similarly,

Total sum of observations = sum of observations in 1st case + 2nd case

Hence,

Total sum of the observations = 150 + 150

Total sum of the observations = 300

Hence,

Now we have to use a formula to find the mean of all observations .

So,

The formula is ,

$\large{\rightarrow{\boxed{\mathsf{Mean \;=\; \dfrac{Sum\;of\;the\; observations}{No. \: of \; observations}}}}}$

Using this formula we can find the mean of all observations .

Hence,

$\tt{Mean = \dfrac{ 300}{25}}$

$\red{\implies{\tt{ Mean \; of \; all \; observations\; = 12 }}}$

Therefore,

$\green{\large{\boxed{\textsf{Mean of all observations = 12}}}}$

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Point to remember :-

If mean is giving we should find the sum .

Similarly,

When sum is given we should find the mean .

$\mathsf{Mean \;=\; \dfrac{Sum\;of\;the\; observations}{No. \: of \; observations}}$

$\textsf{ Sum of the observations \: = }{\tt{\: Mean \times No. of observations}}$

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Answer 2

Answer:

• Mean of First 10 = 15
• Mean of Next 15 = 10

$\bigstar\:\boxed{\sf Mean=\dfrac{Sum\:of\: Observations}{Number\:of\: Observations}}$

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$\underline{\bigstar\:\textbf{According to the Question :}}$

$\bf{\dag}\:\underline{\textsf{Mean of First 10 Observations}}\\\\:\implies\sf Mean_{10}=\dfrac{Sum\:of\: Observations}{10}\\\\\\:\implies\sf15 = \dfrac{Sum\:of\: Observations}{10}\\\\\\:\implies\sf 15 \times 10 = Sum\:of\: Observations\\\\\\:\implies\sf Sum\:of\: Observations = 150$

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$\bf{\dag}\:\underline{\textsf{Mean of Next 15 Observations}}\\\\:\implies\sf Mean_{15}=\dfrac{Sum\:of\: Observations}{15}\\\\\\:\implies\sf10 = \dfrac{Sum\:of\: Observations}{10}\\\\\\:\implies\sf 10 \times 15 = Sum\:of\: Observations\\\\\\:\implies\sf Sum\:of\: Observations = 150$

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$\underline{\bigstar\:\textsf{Mean of all 25 Observations :}}$

$\dashrightarrow\sf\:\:Mean=\dfrac{Sum\:of\: Observations}{Number\:of\: Observations}\\\\\\\dashrightarrow\sf\:\:Mean_{25}=\dfrac{150 + 150}{10 + 15}\\\\\\\dashrightarrow\sf\:\:Mean_{25} = \dfrac{300}{25}\\\\\\\dashrightarrow\:\:\underline{\boxed{\sf Mean_{25} = 12}}$

⠀

$\therefore\:\underline{\textsf{Mean of all 25 Observations will be \textbf{12}}}.$