Example 12

EXAMPLE 12 Understanding the Standard Deviation

When you buy stocks or mutual funds, you need to be aware of how to quantify and balance mean gain with the variability or risk of the investment, especially given the volatile years the market has experienced recently. Consider the PIMCO Total Return A (symbol: PTTAX), a fund that invests in intermediate-term, fixed-income securities. Here are its annual total returns for a recent 10-year period:

Table 5.19: Table 5.8 $PTTAX Annual Total Return

Calendar Year	2004	2005	2006	2007	2008	2009	2010	2011	2012	2013
Return (%)	4.65	2.41	3.51	8.57	4.32	13.33	8.36	3.74	9.93	−2.30

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Figure 5.18 shows a dotplot of the data, with their mean (rounded to two decimals) marked by an asterisk (*). The arrows mark two of the deviations from the mean: One is positive and one is negative. We won’t get a useful measure of variability by totaling up all the positive and negative deviations because they will always sum to 0!

Figure 5.18: Figure 5.18 Measuring variability by looking at the deviations of observations from their mean.

Squaring the deviations makes these numbers all positive, and a reasonable measure of variability is the average of the squared deviations. This average is called the variance. The variance is large if the observations are scattered widely around their mean. The variance is small if the observations are fairly close to the mean.

But the variance does not have meaningful units. With the annual return data measured in percentages, the variance of the purchase prices has units of "squared percentages." Taking the square root of the variance yields the standard deviation, which gets us back to the units of the original variable (in this case, percentages).