When to use geometric vs. arithmetic mean? Why is the former better for percentages?
Simple.  Say in 2012 you were up 50% (brilliant) but then in 2013 you were down 50% (sorry). i.e. if you started with $1000, you were up to $1500, then down to $750. You lost $250 overall. If you were to compute the mean of the percentages using each method, then: Arithmetic mean: The average of +50% and -50% (really 150% and 50% of each period's initial value) is zero, not up or down. Geometric mean: 1.5 * .5 = .75, i.e. you are down 25% over 2 years, or about 13.4% per year. It should be clear the Geometric makes more sense in such a case. "JoeTaxpayer nailed it.  Here's another way to look at it: Generally, we invest in something, then might leave it there for a few years, then take it out, but don't touch it in between.  In that case, to get the final amount X(N), we need to take the initial amount, then multiply by growth in the first year, then multiply by growth in the second year, etc. So, for three years, we have: X(3) = X(0) * G(1) * G(2) * G(3) = X(0) * ""average annual growth"" ^ 3 So, here, we see that we want the average annual growth to the power three equal to the product of the annual growth rates, thus, geometric mean: geometric mean = (G(1) * G(2) * G(3)) ^ (1/3) On the other hand, consider a situation where I have three investments X,Y,Z over one year. Now I have, after one year: X(1)+Y(1)+Z(1) = X(0)*G(1,X) + Y(0)*G(1,Y) + Z(0)*G(1,Z) = ( X(0)+Y(0)+Z(0) ) * ""average annual growth"" Now, in this case, if we assume X(0) = Y(0) = Z(0) = 1, i.e. I put equal amounts in each, we see that the average annual growth rate we want in this case is the arithmetic mean: arithmetic mean = (G(1,X) + G(1,Y) + G(1,Z)) / 3 (if we had unequal amounts at the beginning, it would be a weighted average). TL;DR:"
Degree of Operating Leverage (DOL) measures the percent change in EBIT given a 1% change in Revenue. In other words, if DOL is 1.5, then increasing Revenue by 1% will increase EBIT by 1.5%.  Degree of Financial Leverage (DFL) measures the percent change in NI given a 1% change in EBIT.  Degree of Total Leverage (DTL) cuts EBIT out and measures the percent change in NI given a 1% change in Revenue.   What all three of these numbers measure is “elasticity.” The elasticity between two variables is always the percentage change effect on one due to a 1% change in the other.   Before we talk about why you would multiply elasticities, let’s first just look at absolute changes between a change of three variables, X Y and Z. Lets say X impacts Y (X-&gt;Y) and Y impacts Z (Y-&gt;Z). To make it even more concrete, let’s imagine a classroom where every student is required to have 2 books and every book has 100 pages. So X = # students, Y = # of books, and Z = # of pages. If we add one student to the class, we need to add 2 books. If we add 1 book to the class, we have added 100 pages. With absolute changes like this it is obvious that we don’t add, we multiply. In other words, to directly see the impact of a new student on the number of pages (X -&gt; Z), we would say 1 student -&gt; 2 books -&gt; 200 pages. To say 102 pages would obviously be silly.   For percent changes, this looks pretty similar. To make it easy say the class has 100 students. So adding 1 student is a 1% change. Then we go from 200 books to 202 books, a 1% change. So the degree of Student to Book Leverage is 1 (which make sense there are no “fixed pages”). We go from 20,000 pages to 20,200 pages, also a 1% change so the degree of Book to Page leverage is also 1. Thus, the Student to Page leverage is also just a flat 1.   Let’s add fixed amounts. The teacher has a single 2,000 page book on how to teach. We’ll assume the class has 20 students. So we start with 20*2 + 1 = 41 books and 40*100 + 2,000 = 6,000 pages. Let’s add a student. We have a 21/20 -1 = 4% increase in students. 43/41 -1 = 4.89% increase in books. 6,200/6,000 – 1 = 3.33% increase in pages. So the Degree of Student to Book Leverage is 4.00/4.89 = 0.818 and the Degree of Book to Page Leverage is 4.89/3.33 = 1.47. And the direct Student to Page Leverage is 4.00/3.33 = 1.20. Again, note that here it would be silly to add the degrees of leverage, and when we do multiply them we get 0.82*1.47 = 1.2, the right answer.   From here the application to EBIT [= Sales - Fixed Cost - Variable Cost = (Gross Margin * Sales) - Fixed Cost] and the application to EPS [= EBIT - Interest – Taxes = (1 – tax rate)*(EBIT – Interest)] should be obvious. I can supply the full proof mathematically, however, if you would like. I use StockCharts for spread charting. To take your question as an example, here is the chart of Apple against Nasdaq. "Whatever the percentage divisions are, I'm a big fan of keeping things in separate accounts.  This works especially well for fixed bills.  It is essentially a bank account version of the envelope method.  If I take out everything from your paycheck I know is reserved (including reserving savings), for me it is a lot easier to not even miss the money when I'm going towards ""fun"" expenditures - which come from the checking account." A shorter term MA would be used for short term changes in price whilst a long term MA would be used for longer term movements in price.  A 200 day SMA is widely used to determine the trend of the stock, simply a cross above the 200 day SMA would mean the stock may be entering an uptrend and a cross below that the price may be entering a downtrend. If the price is continuosly going above and below in a short period of time it is usually range trading. Then there are EMAs (Expodential Moving Averages) and WMAs (weighted moving averages) which give more emphasis to the latest price data than the earlier price data in the period chosen compared to a SMA. MAs can be used in many different ways, too many to list all here. The best way to learn about them is to read some TA books and articles about them, then choose a couple of strategies where you can use them in combination with a couple of other indicators that are complimentary with each other. As has been said before, going from nothing to something is an infinite percent increase! It is not 100%. Maybe you had a dollar and now have $101 that is a 10000% increase! Quite remarkable. I often work with symmetrized percent changes like: spc = 100 * (y2-y1)/(0.5 * (y1+y2)) Where I compute the percent with respect to the average. First this is more stable as often measurements can have noise, the average is more reliable.  Second advantage is also that this is symmetric. So going from 95 to 105 is a 10 % increase while going from 105 to 95 a 10% decrease. Of course you need to explain what you show. I like to look at Alpha, Beta, St. Dev., Sharpe Ratio, and R-Squared. It's also good to know how they work together. i.e.: Say you're comparing a fund to an index and the fund has a low beta, but the r-squared is low (&lt;70 is low for my usage). The beta loses some significance in that instance.   You want to be able to look at these 5 metrics, know what they mean on their own, and what they say about each other.  Sorry if that was poorly worded, Mondays... The percentage of gross income is a quick and easy way to arrive at the rough ballpark figure of the mortgage one can afford. The net income is something which one does not know offhand. If you want to seriously evaluate then most of the standard mortgage calculators will ask you to enter your gross income, your other liabilities like Auto Loan, Student Loan, personal Loan, Card Payments, Any other regular payments like Child Support etc. After factoring all these one arrives at the actual mortgage one can afford. As is evident, the above requires a good amount of calculation and hence the preffered method becomes on the basis of gross income. If you mean the percentages of long/short positions within a mutual fund or ETF, then it's a percentage of the total value of the fund portfolio. In that case, positions of 50% in X, -50% in Y are not the same as 100% in X, -100% in Y. If the long and short positions are both for the same asset, then, as D Stanley mentions, all that matters is the net position. If you're equally long and short X, then the net position is always 0%. If held in one savings account, how can I easily manage what percentage is planned for which   purpose? I used a spreadsheet for some years, but found it clumsy for everyday use. Thus I wrote some software which my wife and I use for our short-term as well as long-term planning, available at http://budgeter.sourceforge.net. It specifically helps with splitting the money in one or several accounts into logical categories. (The software is not the most user-friendly ever, so there may be better suggestions that follow, but it works well for us. Please feel free to suggest improvements to it as well.) Volatility is a shitty metric and is sample dependent, What is more interesting is point recurrence, i.e. how many times has a certain point been touched in x time, you can make good day trading strategies off point recurrence (relative that is). This was an interesting little article, thanks for sharing it.  However I'm a bit perplexed by part of the analysis.    The authors give some fairly substantial hints that the value they are providing is *not* the standard deviation, but most of the respondents miss these hints.  The answers for the annualized rate also suggest that many of the respondents knew how to compute this, given the standard deviation.  The authors then go on to say that basically all of the respondents gave the correct mathematical definition of the standard deviation.  Then they go on to say that,  &gt; Whatever reason there was for their error, it did not result from ignorance of the concept.  But this strikes me as both disingenuous and incorrect.  On the contrary the most likely explanation would seem to be that most of them were not aware of the concept of 'mean absolute deviation', probably because they had either never learned it, or never had occasion to apply it.  I agree that people - even highly trained people - tend to be very bad at intuiting probability - just look up the Monty Hall Problem for a primer on that.  But I strongly disagree with the notion that this paper provides any evidence of this tendency, as there seems to be a much stronger case for ignorance.  This alternative is further supported by the authors' assertion that,  &gt; debriefings with respondents revealed that they rarely had an immediate understanding of the error when it was pointed out to them.  again, suggesting ignorance as the most - not least - likely culprit.  I also found it odd that they asked the respondents specifically about the standard deviation definition, but not about the mean absolute deviation, despite the fact that the latter is almost certainly more obscure.  To that last bit, I just went through the textbooks I have on hand, and this concept is not mentioned in either of the undergraduate statistics/probability textbooks I own, nor in the one undergraduate finance textbook I have.  I'm in machine learning and compsci, not finance, but it is mentioned in just one of the four graduate-level texts I have on applied statistics and probability - once on one page in a textbook on Bayesian statistics.  And the wikipedia page is terrible.  In short, while this is an interesting little experiment and write-up, I think the analysis and conclusions are inappropriate and not supported by the evidence the authors provide.  That doesn't mean that their conclusions are incorrect, just that I disagree with the notion that this paper corroborates them.  ...This response ended up being way long... "You may as well ask why a piece of wood is 25 centimeters long but only 10 inches. Most units of measure are very arbitrary. Somebody decides that this amount of heat or distance or money is a convenient unit, and so that's what they use. Suppose that tomorrow the government issued a whole new currency that had 10 times the value of the old currency. So if you used to make 10,000 foobars a year, now you make 1,000 new foobars. And likewise the price of everything you buy is divided by 10. If a certain model car used to cost 2,000 foobars, now it costs 200 new foobars. Are you better or worse off? Clearly if ALL prices change by the same percentage, then it makes absolutely no difference. (Aside from the hassle of making the switch and getting used to the new numbers.) A currency where 1 unit of money buys more is not necessarily a ""stronger currency"". Any more than inches are ""better"" than centimeters because you get more wood for an inch than you get for a centimeter. A currency is said to be ""strong"" when it's value is stable or increasing relative to other currencies. If yesterday I could trade 10 foobars for 1 plugh, but today I only need 9 foobars to buy 1 plugh, then foobars are stronger than plughs. Even though I still need more foobars than plughs to buy the same item." 'In demand', partly due to the larger supply available.    Employers might prefer a math major, but when they are 5-10x more finance majors, they have to settle for the finance skillset + excel, rather than a proverbially higher IQ math major with greater modeling and math skills. R would work, and would be what I'd use, but I only make scripts for myself, so sometimes they're dirty and I just call them from my R environment.    I guess it depends if it's a tool for individual use, or an app you want to sell or have others use.  If it's a standalone app others have to use, I'd probably use Java. "I'd personally display ""n/a"" The only other answer that makes sense to me other is ""infinity"" (phone keyboard doesn't allow me to input the symbol). This would at least allow you to show direction by using positive and negative  infinity and mathematical as the the initial value approaches zero the percentage change approaches infinity which is the closet you can get to a meaningful value"