025. Variance and the Standard Deviation for a Population

The two most important measures of dispersion of the data: the Variance and the Standard deviation.

The Variance is the mean of the squared deviations from the mean. It means (1) finding the amount by which each observation deviates from the mean, (2) squaring those deviations, and (3) finding the average of those squared deviations.

The Variance for a population is

, (3.12)

Or , (3.12a)

Where X1, X2,.., XN are the individual observations; μ is the population mean; N is the number of observations.

The Standard deviation is the square root of the variance.

. (3.13)

The concept of the standard deviation is often quite important in business and economics. For example, in finance the standard deviation in used as a measure of the risk associated with various investment opportunities. Generally, the higher the standard deviation of the rate of return of a particular investment, the greater the degree of risk.

Example 3.8. Markus Boggs is manager of Nest Egg Investments, a financial planning firm that assists individuals in setting up their personal portfolios. Recently, Markus was interested in the rates of return over the past five years of two different mutual funds. Megabucks, Inc. showed rates of return over the five-year period of 12, 10, 13, 9 and 11 percent, while Dynamic Corporation yielded 13, 12, 14, 10, and 6 percent, a client approached Boggs and expressed an interest in one of these mutual funds. Which one should Boggs choose for his client?

Solution. The both funds offer an average return of 11 percent. Since that, the safer investment is the one with the smaller degree of risk as measured by the standard deviation. Boggs calculates the variance and takes square root to get the standard deviation for each stock. For Megabucks, it becomes

.

The standard deviation is

.

For Dynamics,

.

The standard deviation is therefore

.

Interpretation. Since Megabucks exhibits less variability in its returns and offers the same rate of return in average as doe Dynamics, Megabucks represents the safer of the two investments and is therefore the preferred investment opportunity.

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