When you say that an investment like a stock market index fund has an expected return of 9%, you're saying that in any year there is a chance that your return will be better than 9% and a chance that it will be worse.
To get more specific about your chances, you need to specify the expected volatility of the investment, as well as its expected return.
The volatility of an investment is given by the statistical measure known as the standard deviation of the return rate.
You don't need to know the exact definition of standard deviation to understand this article, although the definition is in the glossary if you really want to know it.
You can just think of standard deviation as being synonymous with volatility.
An S&P 500 index fund has a standard deviation of about 15%;
a standard deviation of zero would mean an investment has a return rate that never varies, like a bank account paying compound interest at a guaranteed rate.
We're initializing the calculators in this article with returns that correspond to a portfolio of 100% stocks during your working years, and a 50/50 mix of stocks and cash during retirement.
This little calculator shows you where the numbers come from (and it also shows you our assumptions about return rates).
The wildcard here is the 9% return for stocks.
Expert opinion is all over the map on expected future returns for the stock market: you'll find estimates that are much lower than that and others that are much higher.
We're going with 9% because it's about equal to the average return of the S&P 500 from 1950 through 2001, which includes some good years and some bad years.