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:"
The S&P 500 represents a broadly diversified basket of stocks.  Silver is a single metal.  If all else is equal, more diversification means less volatility.  A better comparison would be the S&P 500 vs. a commodities index, or silver vs. some individual stock. If you're getting the same total amount of money every year, then the main issue is psychological. I mean, you may find it easier to manage your money if you get it on one schedule rather than another. It's generally better to get money sooner rather than later. If you can deposit it into an account that pays interest or invest it between now and when you need it, then you'll come out ahead. But realistically, if we're talking about getting money a few days or a week or two sooner, that's not going to make much difference. If you get a paycheck just before the end of the year versus just after the end of the year, there will be tax implications. If the paycheck is delayed until January, then you don't have to pay taxes on it this year. Of course you'll have to pay the taxes next year, so that could be another case of sooner vs later. But it can also change your total taxes, because, in the US and I think many other countries, taxes are not a flat percentage, but the more you make, the higher the tax rate. So if you can move income to a year when you have less total income, that can lower your total taxes. But really, the main issue would be how it affects your budgeting. Others have discussed this so I won't repeat. "Debt increases your exposure to risk.  What happens if you lose your job, or a major expense comes up and you have to make a hard decision about skipping a loan payment?  Being debt free means you aren't paying money to the bank in interest, and that's money that can go into your pocket. Debt can be a useful tool, however.  It's all about what you do with the money you borrow.  Will you be able to get something back that is worth more than the interest of the loan? A good example is your education.  How much more money will you make with a college degree?  Is it more than you will be paying in interest over the life of the loan?  Then it was probably worth it. Instead of paying down your loans, can you invest that money into something with a better rate of rate of return than the interest rate of the loan?  For example, why pay off your 3% student loan if you can invest in a stock with a 6% return?  The money goes to better use if it is invested.  (Note that most investments count as taxable income, so you have to factor taxes into your effective rate of return.) The caveat to this is that most investments have at least some risk associated with them.  (Stocks don't always go up.)  You have to weigh this when deciding to invest vs pay down debts.  Paying down the debt is more of a ""sure thing"". Another thing to consider: If you have a long-term loan (several years), paying extra principal on a loan early on can turn into a huge savings over the life of the loan, due to power of compound interest.  Extra payments on a mortgage or student loan can be a wise move.  Just make sure you are paying down the principal, not the interest!  (And check for early repayment penalties.)" "Like a lot of businesses, they win on the averages, which means lucrative customers subsidize the money-losers.  This is par for the course.  It's the health club model.  The people who show up everyday are subsidized by the people who never show but are too guilty to cancel. When I sent 2 DVDs a day to Netflix, they lost their shirt on me, and made it up on the customers who don't. In those ""free to play"" MMOs, actually 95-99% of the players never pay and are carried by the 1-5% who spend significantly.  In business thinking, the overall marketing cost of acquiring a new customer is pretty big - $50 to $500. On the other side of the credit card swiper, they pay $600 bounty for new merchant customers - there are salesmen who live on converting 2-3 merchants a month.   That's because as a rule, customers tend to lock-in.  That's why dot-coms lose millions for years giving you a free service.  Eventually they figure out a revenue model, and you stay with it despite the new ads, because changing is inconvenient. When you want to do a banking transaction, they must provide the means to do that.  Normal banks have the staggering cost of a huge network of branch offices where you can walk in and hand a check to a teller.  The whole point of an ATM is to reduce the cost of that.  Chase has 3 staffed locations in my zipcode and 6 ATMs.  Schwab has 3 locations in my greater metro, which contains over 400 zipcodes. If you're in a one-horse town like French Lick, Bandera or Detroit, no Schwab for miles.  So for Schwab, a $3 ATM fee isn't expensive, it's cheap - compared to the cost of serving you any other way. There may also be behind-the-scenes agreements where the bank that charged you $3 refunds some of it to Schwab after they refund you.  It doesn't really cost $3 to do a foreign ATM transaction.  Most debit cards have a Visa or Mastercard logo.  Many places will let you run it as an ATM card with a PIN entry.  However everyone who takes Visa/MC must take it as a credit card using a signature. In that case, the merchant pays 2-10% depending on several factors.** Of this, about 1.4% goes to the issuing bank.  This is meant to cover the bank's risk of credit card defaults. But drawing from a bank account where they can decline if the money isn't there, that risk is low so it's mostly gravy. You may find Schwab is doing OK on that alone.  Also, don't use debit cards at any but the most trusted shops -- unless you fully understand how, in fraud situations, credit cards and debit cards compare -- and are comfortable with the increased risks.   **  there are literally dozens of micro-fees depending on their volume, swipe vs chip, ATM vs credit, rewards cards, fixed vs online vs mobile, etc. (Home Depot does OK, the food vendor at the Renaissance Faire gets slaughtered). This kind of horsepuckey is why small-vendor services like Square are becoming hugely popular; they flat-rate everything at around 2.7%. Yay!" You've flipped the numerator and denominator around, and need to multiply by 100 to get percentage rather than 10: I like to use a simple example to assess reasonableness of an approach, if you had invested $100 and after 1 year had $150, your approach would yield: But since $50 is half of $100, we know the rate of return should be 50%, so we know that approach is off. But, flipping the numerator and denominator and multiplying by 100 gets us the 50% we expected: Edit: Good catch by @DJohnM you've called it 9 years, but it's actually 11, so you'd want to adjust accordingly. id like to start by saying youre still doing this yourself, and i dont actually have all the info required anyway, dont send it but  &gt;[3] Descriptive Statistical Measures: Provide a thorough discussion of the meaning and interpretation of the four descriptive &gt;statistical measures required in your analysis: (1) Arithmetic Mean, (2) Variance, (3) Standard Deviation and (4) Coefficient of &gt;Variation. For example, how are these measures related to each other? In order to develop this discussion, you may want to &gt;consult chapters 2 and 3 of your textbook. This topic is an important part of your report. can be easily interpreted, im guessing the mean is simply just the observed (and then projected stock price for future models) the standard deviation determines the interval in which the stock price fluctuates. so you have like a curve, and then on this curve theirs a bunch of normal distributions modeling the variance of the price plotted against the month  also the coefficient of variation is just r^2 so just read up on that and relate it to the meaning of it to the numbers you have  actually my stats are pretty rusty so make sure you really check into these things  but otherwise the formulas for part 4 is simple too. you can compare means of a certain month using certain equations, but there are different ones for certain situations  you can test for significance by comparing the differences of the means and if its outside of your alpha level then it probably means your company is significantly different from the SP index. (take mu of SP - mu of callaway)  you can also find more info on interpreting the two different coefficients your given if you look up comparing means of linear regression models or something So what? Everybody knows that labor statistics go well beyond a single metric (funny how the article ends up focusing only on one). The US Bureau of Labor statistics already provides different measurements of unemployment.  The 23.3% figure comes from shadowstats, which is well know to be entirely bullshit. Actually, total return is the most important which isn't necessarily just price change as this doesn't account for dividends that may be re-invested.  Thus, the price change isn't necessarily that useful in terms of knowing what you end up with as an ending balance for an investment. Secondly, the price change itself may be deceptively large as if the stock initial price was low, e.g. a few dollars or less adjusting for stock splits as most big companies will split the stock once the price is high enough, then the percentages can be quite large years later. Something else to consider is the percentage change would be based on what as the initial base.  The price at the start of the chart or something else?  Carefully consider what you want the initial starting point to be in determining price shifts here as one could take either end and claim a rationale for using it. Most people want to look at the price to get an idea of what would X shares cost to purchase rather than look at the percentage change from day to day.