Square sum and Arithmetic mean

I came across this coding question where one is given as input a sequence of n numbers, say  a_{1}, a_{2},...., a_{n}   and one has to output the number (need not be one of the input values) which would minimize the difference from that number squared and summed over all numbers in the input sequence i.e. if that number is K, then this sum

\displaystyle \sum_{i=1}^{n} (a_{i} - X)^{2} calculated over \forall X \in R is minimum when X=K.

My first guess was that number would be either mean or median. After taking a few small examples, I could see that median wouldn't be the desired answer. So, I submitted my solution by calculating the mean of the numbers and it was accepted by the system.

But, I had to convince myself that this squared sum of differences of a sequence is indeed the minimum only about the arithmetic mean. Hence, I did the following elementary mathematical proof to convince myself.

Let's calculate the square sums across, a number X, which is  \delta away from M. So, X = M + \delta  where \delta \neq 0 . Let's see that difference of square sum calculated about mean, S_{M}, is always lesser than square sum calculated, S_{X} about any other number X.

Arithmetic Mean, M = \frac {\displaystyle \sum_{i=1}^{n} a_{i}}{n} 
S_{M} = \displaystyle \sum_{i=1}^{n} (a_{i} - M)^{2} 
S_{X} = \displaystyle \sum_{i=1}^{n} (a_{i} - X)^{2}

S_{X} - S_{M}  
= \displaystyle \sum_{i=1}^{n} (a_{i} - X)^{2} - \sum_{i=1}^{n} (a_{i} - M)^{2}
= \displaystyle \sum_{i=1}^{n} [(a_{i} - X)^{2} - (a_{i} - M)^{2}]
= \displaystyle \sum_{i=1}^{n} [X^{2} - M^{2} - 2 a_{i} (X - M)]
= \displaystyle \sum_{i=1}^{n} [(M + \delta)^{2} - M^{2} - 2 a_{i} (M + \delta - M)] [Since, X = M + \delta]
= \displaystyle \sum_{i=1}^{n} [\delta^{2} + 2 M \delta - 2 a_{i} (\delta)]
= n (\delta^{2} + 2 M \delta)  - 2 \delta \displaystyle \sum_{i=1}^{n}  a_{i}  
= n \delta^{2} + 2 n M \delta - 2 \delta (n M)  [From definition of mean]
= n \delta^{2}
> 0 [Since \delta \neq 0 ]

Hence proved, S_{X} - S_{M} > 0 .

Hence proved that arithmetic mean is the right choice for minimizing the square difference sums of a given sequence of numbers.

Dear Facebook

Dear Facebook,

I was away for 2 days, trying to wear off the hallucinating effects of that wonderful thingie. As the effects gradually toned down, I started feeling more in control of myself. So, I thought of visiting you, go liking around every goddamn update that appeared, commenting LOLs on the not-so-amusing updates of my girlfriend, poking around strangers and sending friend requests to them . Yes, don't be surprised, that's what would have made my day complete.

But then as soon as I visited you, I felt as if I was still hallucinating. It appeared to me that the spirit of my deleted Orkut account has hijacked my Facebook account and is running amok, taking it's revenge, hurting my eyes and stressing my brain cells. My impulsive reaction was to close my browser and then close my eyes, hoping that I was just imagining things and all shall be well when I open it.

I opened my eyes and then visited you again, but I was dismayed to find out that it wasn't my illusion. I saw the home page covered with hate messages about new-looking interface. I tried hard to get used to the new interface but you made me feel so dumb. The 'smart-lists' were way too smart for me, the 'ticker' ticked me off and the relevance of  'top stories' went over my top.  Sorry, but I feel like I have no option left but to 'deactivate' you from my life.

Yours sincerely,
one of your ticked off users

Flirting in a different sense …