18.10.2015

Root number astrology

Score more in Math with Video Lessons, Sample Papers, Revision Notes & more for Class-IX - CBSESign up now. Hi!Here is the answer to your question.Draw a number line (l) and mark the points O, A and B such that OA = AB = 1. Depending on how accurate a number we need for an answer, 5.65 is a good estimate for the square root of 32. This site is a product of TSI (Technological Solutions, Inc.), Copyright 2016, All Rights Reserved. Perhaps the widest and deepest pitfall lying in wait for any who deal in numerical analyses is forgetting the distinction between precision and accuracy. It is all too easy to be hypnotized by the ability to calculate some metric to a large number of decimal places into believing that such precision equates to accuracy. The idea behind the construction of park factors, stated broadly, is to compare performance in a given park with performance elsewhere. That comes down to average combined (team plus opponents) runs scored per game at home divided by the corresponding figure for away games. But there remain considerable problems, the most obvious being that the pools are still not identical, in that schedules are not perfectly balanced: Teams X and Y can, and probably do, play significantly different numbers of games in each of the other parks. But for our purposes here--getting a grand overview of the plausibility of "park factors"--such niceties, while of interest, can be set aside. But, because we have used a particular park for these figurings, all those numbers are relative to that park. The average is not exactly 1.000 owing to rounding errors, but it's close enough for government work. But before we jump to any conclusions whatever about those results, let's ponder this: they were derived from data for one park, one team. The entire point of this lengthy demonstration has been to lift the lid off those nice, clean-looking, precise park-effect numbers to show the seething boil in the pot. First, yes, the point is that the changes from year to year in parks--structural changes of consequence to performance or, often, entirely new parks--makes multi-year data even worse than one-year data. Second, and related, my belief, which I think the rows and columns of the big table bear out, is that we simply cannot with any shred of honesty apply numerical park corrections to any statistic. I believe--I assure you, with much regret, that we just have to bite the bullet and be more qualitative than quantitative: yes, those are Colorado results, so take them with a handful of salt--that sort of thing.
First of all, all of the issues he speaks of can be handled, statistically, with no problem whatsoever, if one has the know-how and takes the time to do so. That does not mean, however, that we have to use a "bad" park factor as the purveyor of that PF wants us to use it. But he, and other "PF naysayers" don't want us to use those PF's to adjust player and team stats, apparently, at least according to the last part of the statement from him I quoted above. Would it be better to, say, take a run scored in Coors (by some team, player, or whatever) and do nothing or would it be better to say, adjust it by 1% (even though our "bad" PF says to adjust it by 20%)? So, it is not that these "bad" PF's are not useful, it is that it is not necessarily correct to use the exact numerical adjustments that the purveyors of these PF's might want you to use - for example 20% for Coors and 10% for Petco. So the answer is not to "not use them at all" which is (incorrectly) throwing the baby out with the bathwater. Getting back to the Coors and Petco example above and the "bad" (but still reasonable, like the ESPN one) PF system that gives Coors a 120 rating and Petco a 90 rating, so if most of us are in agreement that it would still be correct to adjust Coors and Petco numbers by 1% (in opposite directions of course), then what about 2%?
Where do we cross the line from improving our numbers (by doing SOME adjustments) to doing worse than nothing at all? Let's say that the real numbers are 115 and 95, which would not be unreasonable for a system that gets 120 and 90 (there is always going to be regression towards 100). The claim that "'PF naysayers' don't want us to use those PF's to adjust player and team stats" is a half-truth: I, at least, would love to have park factors usable to adjust obviously biased data, and for years--back when new parks were unusual occurrences and structural changes in older ones not common--I calculated and used them. This is not religion, where you pick a doctrine to believe in because it appeals to you: it is numerical analysis, and if someone knows a method for getting meaningful measures from meaningless sample sizes, I'd be pleased to hear of it.
The real danger is using a one-size-fits-all park impact number, when each player is unique.
The real position to take is that current park impact numbers gets you from step A to some intermediate step, when we still have to get to step Z. Eric is right that they could get pretty large if you are using it on individual players (which is REALLY what we ultimately care about).
No one is going to convince me that Pac Bell has had a depressing effect on Bonds' HR total anywhere close to the 33% drop other LHH have suffered there.


Square Root   The square root of a number is a number that must be squared (multiplied by itself) to give the original number.
Cube Root The cube root of a number is a number that must be cubed (multiplied by itself 3 times) to give the original number. The Index Laws An 'index' (plural 'indices') is a number written in small case to the upper right of a number to indicate the number's size. The Index Law of Raised Powers - Indices of terms in brackets, raised to another power have their indices multiplied by the index outside the brackets.
The Index Law of Reciprocal Powers - The index of a reciprocal indexed quantity is multiplied by '-1' when turned upside down(inverted). Numbers to the power of zero - All number and quantities to the power of zero have a value of '1'. The square root of a non-negative number, is a non-negative number that when multiplied by itself results in the original number. Example: Since 25 is a non-negative number, there is a non-negative number 5, such that 52 = 25.
When trying to estimate the square root of a number, benchmark values of whole numbers can be used. When solving a simple equation such as x2 = 25, it must be observed that there are two solutions.
When someone is proficient with square roots, usually only the steps shown below are written out. When working with equations involving the Pythagorean Theorem, it is important to remember that the lengths of the sides of the triangle are positive numbers.
The number to the "power of 2" is the same as the number "squared" or the "square" of the number. You can think of it as the "root" of the square or the number that was used to make the square. Check out tons of baseball gloves, youth baseball gloves and catchers gear from Rawlings, Wilson, Nike & Under Armour. As an example, a widely used method for educing park factors for a simple but basic metric, run scoring, is the one used by (but not original to) ESPN. Let us see what some of the things wrong with that basic approach are, and if we can improve on it. Perhaps the most obvious failing of the ESPN method is made manifest by the simple question compared to what?
Even if we throw out inter-league data, which is especially corrupt owing to the variable use of the DH Rule, we still have differing pools for differing teams, at least by division (and possibly even within divisions, owing to rainouts never made up). We have been using the simple--or rather, simplistic--idea of "games" as the basis for comparing parks' effects on run scoring. Yet, if the methodology is sound, we ought to get at least roughly the same results no matter which park we initially use.
The end results are not totally meaningless: we can say with fair credibility that San Diego's is a considerably more pitcher-friendly park than Colorado's, and that the Mets and the Marlins were playing in parks without gross distorting effects. In 2007, Greg Rybarczyk at The Hardball Times noted that the home-run "factor" for the park in Arizona was 48 in one season and 116 in the next. Given that the error in single season park factors is so large, wouldn't a 10 year sample be more accurate than a 1 year sample? But if the increase in sample-size introduces systematic error, resulting error can be larger than 1-year sample. Park factors have bothered me for a long time, and you articulating what I did not have the patience nor the math skills to do.
As you suggest, we know directionally from different park factor methods that Petco supresses runs and Coors increases them.
Subtle distinction - our paper was published in The Journal of Quantitative Analysis in Sports and presented at The New England Symposium on Statistics in Sports. It is undoubtable that ballparks influence the results that players achieve playing in them, and in many cases--"many" both as to particular parks and as to particular statistics--those influences are substantial.
The elements that go into it are team runs scored (R) and opponents' runs scored (OR) at home and away, and total games played at home and away. In the calculation above, run scoring at Park X is being compared to run scoring at all parks except X.
Well, one thinks, we can see how to deal with that: we would normalize away data park by park, then combine the results, so the "away" pool would, finally, represent the imaginary "league-average park" against which we would ideally like to compare any particular park's effects.


But even at that level, there are inequalities needing adjustment, in that the numbers of innings are not going to be equally apportioned among home batters, away batters, home pitchers, and away pitchers, in that a winning team at home does not bat in the bottom of the last inning. For example, if we had chosen the stingiest park in the league, all the factors would be greater than 1; had we chosen the most generous, all the factors would be under 1. Imagine a Twilight-Zone universe in which the 2008 season was played out in some timeless place where each team played ten thousand games with each other team, yet still at their natural and normal performance levels as they were in 2008. The "average" column is a little better than the ESPN column because it allows better for the differing numbers of games on the schedule, but by using the average for each park of the values derived from all the other parks we are approximating the ESPN method. But to try to numerically correct any team's results--much less any particular player's results--by means of "park factors" is very, very wrong. Back in 2001, Rich Rifkin at Baseball Prospectus remarked that "Unfortunately, it is problematic to average out a park factor over more than a few years because the conditions of one or more of the ballparks in a league change. For example, when Royals pulled fences in, they made their park more hitter-friendly, and every other park in the league more pitcher-friendly. The question becoems, in my mind, is how precise do we need to be for the information to be useful? I cant find it online, but here is another chart worth remembering that can help with square roots and such. Park factors are intended as correctives, numbers that ideally allow inflating or deflating actual player or team results in a way that neutralizes park effects and give us a more nearly unbiased look at those players' and teams' abilities and achievements. Thus, each park for which we calculate such a factor is being compared to some different basis: the pool of "away" parks for Park X is obviously different from the pool of "away" parks for Park Y (in that X's pool includes Y but excludes X itself, while Y's includes X but excludes Y itself). We have seen that we need to use normalized runs per game on a park-by-park basis if we are to avoid gross distortions from schedule imbalances and related factors. But all we have to do is average the various factors--in which process we assign the park itself, here San Francisco (I refuse to use the corporate-name-of-the-day for that or any park), a value of 1, since it is necessarily identical to itself--and then normalize the factors relative to that average. Surely it is clear that we then could indeed use any one park as a basis for deriving "park factors" since, in the end, we normalize away that park to reach an all-league basis. That would be nice, and useful, were no park changed structurally over a period of some years.
Any data set that has data from both old and new configuration will introduce systematic error in the calculation and for AL Central stadiums that error may be larger than random error of 1-year sample. Now that rather basic folly can be fairly easily corrected for; let's call the average combined runs per game at home and away RPGh and RPGa, respectively.
For most stats, the wanted basis for comparison is batter-pitcher confrontations, whether styled PA or BFP. In that Twilight Zone world, any variations from using this or that particular park can only be relatively minor random statistical noise. No, what we are dealing here, plain and simple, is the traditional statistical bugaboo--an inadequate sample size.
But consider: not even counting structural changes, in the last ten seasons (counting 2009), a full dozen totally new ballparks have come on line. But, justifiably proud as they are of their improved methodology, even they concluded that "Unfortunately, the lack of longer-term data in Major League Baseball . San Francisco is to Los Angeles thus, and San Francisco is to San Diego so, hence Los Angeles is to San Diego thus-and-so (in a manner of speaking). When one considers that pace, plus the changes (some even to a few of those new parks), it becomes painfully obvious that trying multi-year data is as bad or worse. Even for a particular park that might itself not have been at all changed for many years, there remains the issue that the standard of comparison--that imaginary league-average park--will have changed, probably quite a lot, over that time, owing to changes in the other real parks. That will increase run scoring in a manner that a metric measured against PAs will not fully capture. So we can't use multi-season values, and single-season values are comically insufficient for anything beyond broad-brush estimations, estimations more qualitative than quantitative. And there are yet other questions, such as whether strikeouts should be normalized to plate appearances or to at-bats. Shown are the "park runs factors" for each park as calculated from each of the other parks as a basis.



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