# Mastering Measures of Dispersion | CFA Level I Quantitative Methods

Welcome back to your study notes on measures of dispersion! The mean tells us where returns are centered, but to better understand an investment, we need to study the variability of returns. That’s why we’re covering the measures of dispersion like range, mean absolute deviation, variance, standard deviation, downside deviation, and the coefficient of variation. So, buckle up and let’s dive right in!

## Choosing an Investment: Asset A vs. Asset B

Imagine two investments:

• Investment asset A has a mean return of 7% based on returns from the past 10 years.
• Investment asset B also has a mean return of 7% based on returns from the past 10 years.

If you had to choose between the two, which would you pick? It really depends on your investment objective. If you’re an average investor who’s satisfied with a 7% return, you might go for asset B. Why? Investors want to minimize risk, and variability in returns is a risk.

## Measuring Dispersion: Range, Mean Absolute Deviation, and Variance

Let’s explore some measures of dispersion:

1. Range: The difference between the maximum and minimum value. It’s a simple measure, but limited in the information it provides.
2. Mean Absolute Deviation: The average of the absolute deviations of each observation from the mean. The formula is:
(Sum of Absolute Deviations) / Number of Observations
3. Variance: The average of the squared deviations of each observation from the mean. It’s denoted by sigma square (σ²). The formula is:
(Sum of Squared Deviations) / Number of Observations

To get the same dimension as the observations, we take the square root of variance, which gives us the standard deviation (σ). Standard deviation is always greater or equal to the mean absolute deviation because it gives more weight to larger deviations when the deviations are squared.

## Population vs. Sample Variance and Standard Deviation

If we can’t collect data about the entire population, we can use a sample to estimate the population parameters. The sample variance and sample standard deviation are denoted as “s square” and “s” respectively. The main difference is that the denominator is n-1, instead of N (the entire population size).

EXAMPLE

Calculate the sample standard deviation of the past 5 years return for a mutual fund.

Sample mean = 4%

Sum of squared deviations = 131.6

Sample variance = 32.9 (131.6 / 4)

Sample standard deviation = 5.7% (square root of 32.9)

## Downside Deviation and Coefficient of Variation

Downside deviation is a measure of downside risk. Instead of the mean, we choose a target value to measure each outcome. We only include negative deviations from the target, and the denominator remains the sample size (n-1).

Coefficient of Variation (CV) is a dimensionless measure of relative dispersion that helps compare investments with different means and standard deviations. It is calculated as the ratio of the standard deviation to the mean. A smaller CV indicates less risk for a given return.

CV = Standard Deviation / Mean

## Application: Asset A vs. Asset B Revisited

Let’s now analyze Asset A and Asset B using measures of dispersion:

• Asset A: Mean return = 7%, Standard Deviation = 5%
• Asset B: Mean return = 7%, Standard Deviation = 8%

Using the coefficient of variation:

• Asset A: CV = 5% / 7% = 0.714
• Asset B: CV = 8% / 7% = 1.143

Asset A has a lower coefficient of variation, which means it has a lower relative risk compared to Asset B. Therefore, a risk-averse investor would choose Asset A over Asset B.

## Conclusion

Measures of dispersion such as range, mean absolute deviation, variance, standard deviation, downside deviation, and coefficient of variation help investors analyze the risk associated with investment returns. By understanding these measures, you can make informed decisions about which investments are most suitable for your risk tolerance and objectives.

Now that you’ve got a strong foundation in measures of dispersion, you’re one step closer to mastering CFA Level I material. Keep studying, and best of luck in your preparation!

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