Trouble understanding a step in a proof, algebra with summation

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I have this line in a proof that I do not understand and would like some help understanding please.



$$sum_i=1^nx_i^2-nbarx^2 = sum_i=1^n(x_i^2-2x_ibarx+barx^2)$$



Can anybody enlighten me on this transformation please?



Thanks.










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  • Is there any hypotheses on the $x_i$s?
    – Bernard
    Sep 5 at 11:14










  • what is $barx$?
    – giannispapav
    Sep 5 at 11:16










  • yes, $barx_i$ is the average.
    – Bucephalus
    Sep 5 at 11:17















up vote
0
down vote

favorite












I have this line in a proof that I do not understand and would like some help understanding please.



$$sum_i=1^nx_i^2-nbarx^2 = sum_i=1^n(x_i^2-2x_ibarx+barx^2)$$



Can anybody enlighten me on this transformation please?



Thanks.










share|cite|improve this question





















  • Is there any hypotheses on the $x_i$s?
    – Bernard
    Sep 5 at 11:14










  • what is $barx$?
    – giannispapav
    Sep 5 at 11:16










  • yes, $barx_i$ is the average.
    – Bucephalus
    Sep 5 at 11:17













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have this line in a proof that I do not understand and would like some help understanding please.



$$sum_i=1^nx_i^2-nbarx^2 = sum_i=1^n(x_i^2-2x_ibarx+barx^2)$$



Can anybody enlighten me on this transformation please?



Thanks.










share|cite|improve this question













I have this line in a proof that I do not understand and would like some help understanding please.



$$sum_i=1^nx_i^2-nbarx^2 = sum_i=1^n(x_i^2-2x_ibarx+barx^2)$$



Can anybody enlighten me on this transformation please?



Thanks.







proof-explanation






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asked Sep 5 at 11:08









Bucephalus

455216




455216











  • Is there any hypotheses on the $x_i$s?
    – Bernard
    Sep 5 at 11:14










  • what is $barx$?
    – giannispapav
    Sep 5 at 11:16










  • yes, $barx_i$ is the average.
    – Bucephalus
    Sep 5 at 11:17

















  • Is there any hypotheses on the $x_i$s?
    – Bernard
    Sep 5 at 11:14










  • what is $barx$?
    – giannispapav
    Sep 5 at 11:16










  • yes, $barx_i$ is the average.
    – Bucephalus
    Sep 5 at 11:17
















Is there any hypotheses on the $x_i$s?
– Bernard
Sep 5 at 11:14




Is there any hypotheses on the $x_i$s?
– Bernard
Sep 5 at 11:14












what is $barx$?
– giannispapav
Sep 5 at 11:16




what is $barx$?
– giannispapav
Sep 5 at 11:16












yes, $barx_i$ is the average.
– Bucephalus
Sep 5 at 11:17





yes, $barx_i$ is the average.
– Bucephalus
Sep 5 at 11:17











1 Answer
1






active

oldest

votes

















up vote
1
down vote



accepted










If $bar x$ is the average $bar x = frac1n sum_i=1^n x_i$, then
$$sum_i=1^n 2x_i bar x = 2bar x sum_i=1^n x_i = 2n bar x ^2.
$$
Furthermore,
$$
sum_i=1^n bar x^2 = bar x^2 sum_i=1^n 1 = nbar x ^2
$$






share|cite|improve this answer




















  • Oh, nice, now I get it. Thanks @Kusma
    – Bucephalus
    Sep 5 at 11:21










Your Answer




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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










If $bar x$ is the average $bar x = frac1n sum_i=1^n x_i$, then
$$sum_i=1^n 2x_i bar x = 2bar x sum_i=1^n x_i = 2n bar x ^2.
$$
Furthermore,
$$
sum_i=1^n bar x^2 = bar x^2 sum_i=1^n 1 = nbar x ^2
$$






share|cite|improve this answer




















  • Oh, nice, now I get it. Thanks @Kusma
    – Bucephalus
    Sep 5 at 11:21














up vote
1
down vote



accepted










If $bar x$ is the average $bar x = frac1n sum_i=1^n x_i$, then
$$sum_i=1^n 2x_i bar x = 2bar x sum_i=1^n x_i = 2n bar x ^2.
$$
Furthermore,
$$
sum_i=1^n bar x^2 = bar x^2 sum_i=1^n 1 = nbar x ^2
$$






share|cite|improve this answer




















  • Oh, nice, now I get it. Thanks @Kusma
    – Bucephalus
    Sep 5 at 11:21












up vote
1
down vote



accepted







up vote
1
down vote



accepted






If $bar x$ is the average $bar x = frac1n sum_i=1^n x_i$, then
$$sum_i=1^n 2x_i bar x = 2bar x sum_i=1^n x_i = 2n bar x ^2.
$$
Furthermore,
$$
sum_i=1^n bar x^2 = bar x^2 sum_i=1^n 1 = nbar x ^2
$$






share|cite|improve this answer












If $bar x$ is the average $bar x = frac1n sum_i=1^n x_i$, then
$$sum_i=1^n 2x_i bar x = 2bar x sum_i=1^n x_i = 2n bar x ^2.
$$
Furthermore,
$$
sum_i=1^n bar x^2 = bar x^2 sum_i=1^n 1 = nbar x ^2
$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Sep 5 at 11:16









Kusma

3,355218




3,355218











  • Oh, nice, now I get it. Thanks @Kusma
    – Bucephalus
    Sep 5 at 11:21
















  • Oh, nice, now I get it. Thanks @Kusma
    – Bucephalus
    Sep 5 at 11:21















Oh, nice, now I get it. Thanks @Kusma
– Bucephalus
Sep 5 at 11:21




Oh, nice, now I get it. Thanks @Kusma
– Bucephalus
Sep 5 at 11:21

















 

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