Question

⭑$\huge{\mathfrak{Bonjour!♡}}$⭑•°⭑

→ The mean and standard deviation of a group of 100 observation were found to be 20 and 3 respectively. Later on it was found that three observation were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.
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Answer 1

$\Large\underline\mathfrak{Question}$

The mean and standard deviation of a group of 100 observation were found to be 20 and 3 respectively. Later on it was found that three observation were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted. ?

$\Large\bold\star\underline{\underline\textbf{Formula\:used}}$

• Mean = (sum of observation) / (Total No. of observation)
• Standard Deviation = [ ⅀ ( x - xi)² / n ]

$\huge\overbrace{\underbrace{\pink{\boxed{\bf{\red{A}\green{n}s\blue{w}\orange{e} r:}}}}}$

Given , that, Mean of 100 observations is 20 ,, So, we can say that,,

→Mean = (sum of observation) / (Total No. of observation)

→ 20 = ( sum ) / 100

→ sum of observations = 2000 = Incorrect Sum.

$\rule{200}{4}$

Now, it has been said that , three observation were incorrect, which were recorded as 21, 21 and 18 .. So,

→ Sum of new observation = 2000 - 21 - 21 - 18 = 1940.

→ And, New Number of observation now = 100 - 3 = 97 .

So,

→ New Mean = sum / No. of observation .

→ New Mean = 1940/97 = $\pink{\large\boxed{\boxed{\bold{20}}}}$

$\rule{200}{4}$

Now, Given that, standard deviation of 100 observations is 3.

So, we can say that,

→ (3)² = [ (⅀ xi² /n) - ( ⅀ xi /n)²

→ 9 = [ (⅀ xi² / 100) - (20)² ]

→ 9 = ( ⅀ xi² / 100) - 400

→ 9 + 400 = ( ⅀ xi² / 100)

→ ⅀ xi² = 40900

Now,

→ Correct ⅀ xi² = 40900 - (21)² - (21)² - (18)²

→ Correct ⅀ xi² = 40900 - 441 - 441 - 324

→ Correct ⅀ xi² = 39694 .

$\rule{200}{4}$

So, Correct Standard Deviation :-

→ SD = √ [ ( 39694/97) - (20)² ]

→ SD = √ [ ( 39694 - 400*97) / 97 ]

→ SD = √ [ (894/97)

→$\bold{\boxed{\large{\boxed{\orange{\small{\boxed{\large{\red{\bold{\:SD=3.03}}}}}}}}}}$

Hence, the mean and standard deviation if the incorrect observations are omitted are 20 and 3.03 Respectively.

$\huge\bold{\red{\ddot{\smile}}}$

Answer 2

Answer:

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