Example 13

EXAMPLE 13 Calculating the Standard Deviation

To find the standard deviation of the 10 return rates in Example 12, first find the mean.

$\bar{X} = \frac{4.65 + 2.41 + 3.51 + 8.57 + 4.32 + 13.33 + 8.36 + 3.74 + 9.93 + (- 2.30)}{10} = 5.652 % \approx 5.65 %$

For readability of Table 5.9, we have used the mean rounded to two decimals. You will get more accuracy if you include more decimal places for the mean throughout the process and do not round until the end.

Table 5.20: **Table 5.9** Step-by-Step Approach to Calculating Standard Deviation
Observations	Deviations (observation minus mean)	Squared Deviations
$x_{i}$	$x_{i} - \bar{x}$	${(x_{i} - \bar{x})}^{2}$
4.65	$4.65 - 5.65 = - 1.00$	$(- 1.00)^{2} = 1.0000$
2.41	$2.41 - 5.65 = - 3.24$	$(- 3.24)^{2} = 10.4976$
3.51	$3.51 - 5.65 = - 2.14$	$(- 2.14)^{2} = 4.5796$
8.57	$8.57 - 5.65 = 2.92$	$(2.92)^{2} = 8.5264$
4.32	$4.32 - 5.65 = - 1.33$	$(- 1.33)^{2} = 1.7689$
13.33	$13.33 - 5.65 = 7.68$	$(7.68)^{2} = 58.9824$
8.36	$8.36 - 5.65 = 2.71$	$(2.71)^{2} = 7.3441$
3.74	$3.74 - 5.65 = - 1.91$	$(- 1.91)^{2} = 3.681$
9.93	$9.93 - 5.65 = 4.28$	$(4.28)^{2} = 18.3184$
−2.30	$- 2.30 - 5.65 = - 7.95$	$(- 7.95)^{2} = 63.2025$
$Sum = 177.8680$

The variance $s^{2}$ is the sum of the squared deviations divided by 1 less than the number of observations, so it would be $\frac{177.868}{10 - 1} \approx 19.763$ . The standard deviation is the square root of the variance, so we obtain $s = \sqrt{19.763} \approx 4.45 %$ . This value, 4.45%, can be considered small for this context, which suggests that this mutual fund happened to have a great deal of stability during a very turbulent decade.