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August 14th, 2011, 07:02 AM   #21
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Re: Arithmetic mean: Why use its computation as its definiti

Quote:
 Originally Posted by Envroe â is a single value meant to characterize the entire set of a values collectively.
How does one go about choosing a value for â? August 14th, 2011, 01:20 PM   #22
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Re: Arithmetic mean: Why use its computation as its definiti

Quote:
 Originally Posted by CherryPi Since we're talking about things that have been forgotten by the populous: The method of squaring, cubing, and 'n'ing--finding roots of equations manually--has been forgotten. I recall there being a method similar to division, but I also recall a common sensical method. So here is my challenge to you: What are those two methods? (Bonus points if applied uniquely and specially to fractional powers!)
Yes, not totally forgotten, but buried away in the dusty halls of yesteryear (along with slide rules, calculating devices such as "Consul" The Educated Monkey), nomographs, etc. Probably better discussed in a new thread. I know that finding the n-th root begins by segmenting the number in groups of n digits, usually with apostrophes, and then the division-like process begins. Then there's Newton's method of approximation.

There's also methods for multiplying (yes, multiplying) on your fingers, and probably lots of other cool math methods. I have my own method of multiplying longhand (especially squaring) that's more efficient and involves much fewer "carries".

Quote:
Originally Posted by greg1313
Quote:
 Originally Posted by Envroe â is a single value meant to characterize the entire set of a values collectively.
How does one go about choosing a value for â?
That's what the derivation determines by setting the sum of the residuals to zero.

However, you could calculate it step-wise, beginning with some nominal value (eg, zero seems like a good value to begin for the estimate â). Then, calculating the sum of the residuals, and using the result to determine which way to move the estimate along the number line.
For example, using the set of {4, 5, 9}, the sum of residuals is as follows:

â ? ?
0 ? 18
1 ? 15
2 ? 12
3 ? 9
4 ? 6
5 ? 3
6 ? 0
7 ? -3
8 ? -6
9 ? -9
etc

The estimate â can be adjusted until the sum of the residuals equals zero (and then the mean would equal the value of â at that point), or the sum of residuals can be set to zero and the mean determined from it. Tags arithmetic, computation, definition Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ZardoZ Complex Analysis 0 May 4th, 2013 01:12 PM unm Number Theory 1 November 17th, 2012 09:58 AM mathbalarka Calculus 9 August 31st, 2012 05:40 AM Chengpu Real Analysis 3 June 2nd, 2010 11:37 AM Chengpu Linear Algebra 0 May 23rd, 2010 04:55 PM

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