Finding the (X-overlineX)^2 of first 5 data of dataset given mean and population variance

Simone Watts

Simone Watts

Answered question

2022-11-04

Finding the ( X X ¯ ) 2 of first 5 data of dataset given mean and population variance
Mean and population variance of the dataset x 1 , x 2 . . x 10 are 19 and 49 respectively. If the value i = 6 10 x i 2 = 1900, what is the value of i = 1 5 x i 2 = ?.
I've solved it as following and it is wrong:
Population variance:
S 2 = ( x i x ¯ ) 2 n 49 = i = 1 5 x i 2 10 + 1900 10 49 = i = 1 5 x i 2 10 + 190 190 + 49 = i = 1 5 x i 2 10 i = 1 5 x i 2 = 141 10 = 1410
And this solution is wrong. How to solve this problem?

Answer & Explanation

Deanna Sweeney

Deanna Sweeney

Beginner2022-11-05Added 14 answers

Step 1
You don't use the whole definition of the population variance which is
s 2 = i = 1 10 ( x i x ¯ ) 2 10 = 1 10 ( i = 1 10 x i 2 2 x i x ¯ + x ¯ 2 )
1 10 ( i = 1 10 x i 2 i = 1 10 2 x i x ¯ + i = 1 10 x ¯ 2 )
2 , x ¯ and x ¯ 2 are constants. They can be put in front the sigma signs.
1 10 ( i = 1 10 x i 2 2 x ¯ i = 1 10 x i + x ¯ 2 i = 1 10 1 )
We have 1 10 i = 1 10 x i = x ¯ i = 1 10 x i = 10 x ¯
1 10 ( i = 1 10 x i 2 2 x ¯ 10 x ¯ + x ¯ 2 10 ) = 1 10 ( i = 1 10 x i 2 10 x ¯ 2 )
1 10 i = 1 10 x i 2 x ¯ 2 = 1 10 i = 1 10 x i 2 ( 1 10 i = 1 10 x i ) 2
= 1 10 ( i = 1 5 x i 2 + i = 6 10 x i 2 ) ( 1 10 i = 1 10 x i population mean ) 2 = s 2
Step 2
Now you can insert the values: 1 10 i = 1 10 x i = 19 , s 2 = 49 and i = 6 10 x i 2 = 1900. Finally solve for i = 1 5 x i 2 .

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