Standard Deviation of Returns
Easy as 1-2-3!

The standard deviation of returns is important to investors because it is the standard measure of investment risk, and investment risk is the degree of uncertainty of earning the expected rate of return. As you will see, the standard deviation of returns can be used to estimate the probability of the return on an investment falling within a given range relative to the expected rate of return in any given period.

In a nut shell...

1. The mean is what statisticians call the average. If you add up ten numbers and divide by ten, you have the mean of those numbers.
2. Each of those ten numbers is known as a data point.
3. The standard deviation is the average by which the numbers used to calculate the mean vary from the mean.

The normal distribution...

1. Investing returns are known to be approximately normally distributed.
2. The normal probability distribution is a symmetric distribution that is shaped like a bell.
3. The normal distribution is, for all intents and purposes, six standard deviations wide, as six standard deviations bracket 99.74% of the total probability.

Slicing it down the middle...

1. The mean is at the middle of the normal distribution and splits the probability in half.
2. Nearly 50% of the probability is within three standard deviations of each side of the mean.
3. The probabilities are cumulative in both directions.

The Rules of Thumb
The familiar bell-shaped curve is the plot of the normal probability density function and its shape tells us that the probability is more dense near the mean than at the extremes. In other words, the data points are apt to be closer to the mean than the extremes. Which leads us to the rules of thumb.

The approximate probabilities of a data point being within one, two or three standard deviations of the mean are as follows:

+/-2 standard deviations actually spans 95.44% of the probability. But rules of thumb are only approximations and are meant to be convenient and memorable. 96% is close and it is easily divided by two, so we'll use 96%.

If a mutual fund had a mean annual return of 12% and a standard deviation of returns of 15% for the past ten years, each of the ten annual returns used to compute the mean would be a data point and the standard deviation of returns would be computed from the ten data points.

68% of the data points would be expected to be within one standard deviation of the mean, 96% within two standard deviations and virtually 100% within three standard deviations.

There is a 68% probability that a data point selected at random will have a value of 12% +/-15%, which spans the range form -3% and +27%. This also can be interpreted as there being a 34% probability (68% ÷ 2) of the return being between -3% and +12%, and a 34% probability of the return being between +12% and +27%.

Similarly, There is a 96% probability that the return will be between -18% and +42%, and nearly a 100% probability that the return will be between -33% and +57%. And the same logic applies to the probabilities to one side or the other of the mean.

Following this logic, the probabilities can be taken in steps. As the mean splits the total probability, half of each of the ranges, 34%, 48% and 50%, respectively, fall on either side of the mean. Therefore, there is a 48% - 34% = 14% probability that the return will be between one and two standard deviations from the mean in both directions, i.e., a 14% probability of being between -3% and -18%, and a 14% probability of being between +27% and +42%. And so on.

Conclusion...

There's a good reason that they say that past returns are not necessarily an indication of what future returns may be. Statistically, using past returns as a predictor of future returns is unsound due to the randomness of security returns. However, past volatility, as measured by the standard deviation of returns, is considered to be a good predictor of future volatility.

Investment returns must be taken in context and the volatility of returns as measured by the standard deviation of returns provides the means for doing this. Use the rules of thumb to assess the risk of individual investments based on their standard deviation of returns.