Question

The mean of 10 obseravtion is 200. If one observation is included, the new mean became 205. Find the included observation.

Answer 1

Given

Mean of 10 observation= 200

To find :-

we have to find the included observation by which the mean become 205

Solution

Let the included observation be x

Now the total number of observation = 11

A/Q

Firstly find the sum of 10 observation :-

$\boxed{\sf\ Mean =\dfrac{sum\ of\ observation}{Number\ of\ observation}}$

Let the number of observation be

$\sf\ x_1,\ x_2\ ,\ x_3........,\ x_{10}$

Now

→ $\sf\ 200= \dfrac{x_1+x_2+x_3......+x_{10}}{10}$

→ 200×10=x1+x2+x3+x4........+x10

→ 2000= x1+x2+x3+x4.....x10. -------eq.(i)

If one more observation included

● mean = 205

● number of observation= 11

● Included observation= x

then ,

→ $\sf\ 205= \dfrac{x_1+x_2+x_3+x_4.......+x_{10}+x_{11}}{11}$

→ 205×11= 2000+x ------- from eq.(i)

→2255= 2000+x

→ 2255-2000=x

→255= x

•°• Included observation is 255.

Answer 2

$\Large{\underline{\bf{\color{indigo} GiVeN,}}}$

• The mean of 10 observation is 200.

$\longmapsto\:\:\bf\blue{Mean_{(x_1\:+\:x_2\:+\:...\:+\:x_{10})}\:=\:200\:}$

• If one observation is included, i.e.

$\longmapsto\:\:\bf{No.\:of\:total\:observation\:=\:10\:+\:1\:}$

$\longmapsto\:\:\bf\purple{No.\:of\:total\:observation\:=\:11\:}$

• After including 11th observation, the new mean becomes 205.

$\longmapsto\:\:\bf\pink{Mean_{(x_1\:+\:x_2\:+\:...\:+\:x_{11})}\:=\:205\:}$

$\bf\red{We\:know\:that,}$

$\red\bigstar\:\:{\underline{\green{\boxed{\bf{\color{peru}Mean\:=\:\dfrac{\Sigma_{(observation)}}{Total\:no.\:of\:observation}\:}}}}}$

$\bf\pink{Let,}$

• Included observation is X₁₁.

Cᴀsᴇ - 1 ;-

• Mean = 200

• $\bf{\Sigma_{(observation)}}$ = X₁ + X₂ + ..... + X ₁₀

• Total no. of observation = 10

$:\implies\:\:\bf{Mean\:=\:\dfrac{(X_1\:+\:X_2\:+\:...\:+\:X_{10})}{10}\:}$

$:\implies\:\:\bf{(X_1\:+\:X_2\:+\:...\:+\:X_{10})\:=\:200\times{10}\:}$

$:\implies\:\:\bf\orange{(X_1\:+\:X_2\:+\:...\:+\:X_{10})\:=\:2000\:}--(i)$

Cᴀsᴇ - 2 ;-

• Mean = 205

• $\bf{\Sigma_{(observation)}}$ = X₁ + X₂ + ..... + X₁₁

• Total no. of observation = 11

$:\implies\:\:\bf{Mean\:=\:\dfrac{(X_1\:+\:X_2\:+\:...\:+\:X_{11})}{11}\:}$

$:\implies\:\:\bf{205\:=\:\dfrac{(X_1\:+\:X_2\:+\:...\:+\:X_{11})}{11}\:}$

$:\implies\:\:\bf{(X_1\:+\:X_2\:+\:...\:+\:X_{11})\:=\:205\times{11}\:}$

$:\implies\:\:\bf{(X_1\:+\:X_2\:+\:...\:+\:X_{10})\:+\:X_{11}\:=\:2255\:}$

$:\implies\:\:\bf{2000\:+\:X_{11}\:=\:2255\:}$

$:\implies\:\:\bf{X_{11}\:=\:2255\:-\:2000}$

$:\implies\:\:\bf\green{X_{11}\:=\:255}$

$\Large\bold\therefore$ The included observation is 255.