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

Question

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

Filo · Accepted Answer

Given, number of observation n=100 Incorrect mean (xˉ)=20We know thatxˉ=n∑xi​​⇒20=1001​i=1∑100​xi​ ⇒i=l∑100​xi​=20×100=2000 So, the incorrect sum of observations =2000.Given incorrect observations are 21,21,18 and these has to be omitted.So, correct sum of observations =2000−21−21−18=2000−60=1940 Correct mean =100−3Correctsum​=971940​=20 Given, incorrect standard deviation σ=3σ=n∑i=1n​xi2​​−(n∑i=1n​xi​​ )2​σ2=n∑i=1n​xi2​​−(xˉ)2 ⇒32=100∑ xi2​​−(20)2 ⇒Incorrect∑ xi2​=100(9+400)=40900Correct  i=l∑n​xi2​=Incorrecti=l∑n​xi2​−(21)2−(21)2−(18)2=40900−441−441−324=39694∴ Correct standard deviation = ncorrect∑xi2​​−(Correct mean)2​  =9739694​−(20)2​  =409.216−400​ =9.216​ =3.036

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

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