There are two types of dispersions:

**Absolute dispersion**is the amount of variability without comparison to any benchmark. Measures of absolute dispersion include range, mean absolute deviation, variance, and standard deviation.**Relative dispersion**is the amount of variability in comparison to a benchmark. Measures of relative dispersion include the coefficient of variance.

The

The range of the numbers 1, 2, 4, 6,12,15,19, 26 = 26 - 1 = 25

Recall that the deviation from the arithmetic mean is the distance between the mean and an observation in the data set. The

In calculating the MAD, we ignore the signs of deviations around the mean. Remember that the sum of all the deviations from the mean is equal to zero. To get around this zeroing-out problem, the mean deviation uses the absolute values of each deviation. MAD is superior to the range as a measure of dispersion because it uses all the observations in the sample. However, the absolute value is difficult to work with mathematically.

The

where:

- μ = the mean
- N = the number of scores

When the variance is computed in a sample, the statistic (m = the mean of the sample) can be used. However, s

This gives an unbiased estimate of σ

The formula for the sample variance is nearly the same as that for the population variance except for the use of the sample mean, X, and the denominator. In the case of the population variance, we divide by the size of the population, N.

The major problem with using the variance is the difficulty interpreting it. Why? The variance, unlike the mean, is in terms of units squared. How does one interpret squared percentages or squared dollars? The solution to this problem is to use the standard deviation. The formula for the

An important attribute of the standard deviation as a measure of spread is that if the mean and standard deviation of a normal distribution are known, it is possible to compute the percentile rank associated with any given score. In a normal distribution, about 68% of the scores are within one standard deviation of the mean and about 95% of the scores are within two standards deviations of the mean.

The standard deviation has proven to be an extremely useful measure of spread in part because it is mathematically tractable. Many formulas in inferential statistics use the standard deviation.

A direct comparison of two or more measures of dispersion may be difficult. For example, the difference between the dispersion for monthly returns on T-bills and the dispersion for a portfolio of small stocks is not meaningful because the means of the distributions are far apart. In order to make a meaningful comparison, we need a relative measure, to standardize the measures of absolute dispersion.

It is often useful to compare the relative variation in data sets that have different means and standard deviations, or that are measured in different units.

It gives a measure of risk per unit of return, and an idea of the magnitude of variation in percentage terms. It allows us direct comparison of dispersion across data sets. The lower the CV, the better; investments with low CV numbers offer less risk per unit of return. This measurement is also called

Note that because s and X-bar have the same units associated with them, the units effectively cancel each other out, leaving a unitless measure which allows for direct comparison of dispersions, regardless of the means of the data sets.

The CV is not an ideal measure of dispersion. What if the expected return is zero!? Generally, the standard deviation is the measure of choice for overall risk (and beta for individual assets).

The mean monthly return on T-bills is 0.25% with a standard deviation of 0.36%. For the S&P 500, the mean is 1.09% with a standard deviation of 7.30%. Calculate the coefficient of variation for T-bills and the S&P 500 and interpret your results.

T-bills: CV = 0.36/0.25 = 1.44

S&P 500: CV = 7.30/1.09 = 6.70

Interpretation: There is less dispersion relative to the mean in the distribution of monthly T-bill returns when compared to the distribution of monthly returns for the S&P 500 (1.44 < 6.70).

achu: RSD (rel std dev) = CV. |

Gooner7: What does variance even tell you? |

alester83: think of variance as vary. it speaks to the likelihood of experiencing returns that are different that average. that is the most simplistic way of looking at it, but to interpret you really need the standard deviation. hope that provides some level of insight |

LoveIvie: Good overview |

chipster: what is beta? |

sgossett86: hope you're kidding. beta is the first thing you learn in finance class. if you aren't familiar why would you be interested in the cfa.the scarier thing is that you'd be relying on someone to answer this in the thread instead of taking it upon yourself to research. keep checking back. someone will answer it eventually. |

sgossett86: without referencing anything, basically beta is a correlation coefficient to a security relative to a major index like the s&p or dow. it's supplied for you from a multitude of online financial sources, and is used in many valuation calculations. it is derived using a regression analysis of returns correlated with the index returns to get ur correlation coefficient |

sgossett86: in theory there are leverage calculating formulas, basically finance sourcing, that can solve for a beta without regression |

fobucina: Beta - essentially, measures the sensitivity of the stock's returns to that of the market portfolio (i.e., we use a major index to proxy for this theoretical portfolio). The levered beta (i.e., the beta incorporating the capital structure of the firm) can be derived using regression analysis... As you will see later on in the curriculum, there are formulas that take the leverage of the firm (D/E), marginal tax rate, and levered beta to determine the firm's unlevered beta (i.e., the beta reflecting the firm's operating risk)... |