Range, mean absolute deviation, variance and standard deviation

What does dispersion measure?

The mean says where returns are centered. Dispersion says how far they stray from it, and for an investor that is the starting point for risk. The measures below are absolute dispersion : they are expressed in the units of the data, either percent or percent squared.

Dispersionhow far returns spread around the center

• Range Use it max − min, units: %
  • Weakness uses only two values
• Mean absolute deviation Use it average of |X − mean|, units: %
  • Weakness absolute values are awkward in later maths
• Variance Memorize Σ(X − mean)² ÷ (n − 1), units: %²
  • Weakness squared units, hard to read
• Standard deviation Memorize √variance, units: %
  • Weakness treats up and down moves alike

How do the range and the mean absolute deviation work?

The range is the largest value minus the smallest. It is quick, but it ignores every observation in between.

$$\text{Range} = X_{\max} - X_{\min}$$

The mean absolute deviation (MAD) is the average distance of the observations from the mean, ignoring direction.

$$MAD = \frac{\sum_{i=1}^{n} |X_i - \bar{X}|}{n}$$

How do you calculate sample variance and standard deviation?

Square each deviation from the mean, add them up and divide by n − 1. That is the sample variance . Its square root is the sample standard deviation .

$$s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n - 1}$$

$$s = \sqrt{s^2}$$

Why n − 1? The deviations are measured from a sample mean that was itself estimated from the same data. That uses up one of the degrees of freedom : once n − 1 deviations are known, the last is fixed. Dividing by n − 1 makes the sample variance an unbiased estimate of the population variance.

Switch between n and n − 1 and watch the standard deviation change. The smaller the sample, the bigger the gap between them.

Exam tip On the BA II Plus, read Sx, not σx. On the HP 12C, [g] [s] is already the sample figure.

Which measure should you use?

Measure	Uses all the data	Units	Weakness
Range	No (two values)	%	Driven by extremes
MAD	Yes	%	Absolute values are awkward in later maths
Variance	Yes	%²	Squared units, hard to read
Standard deviation	Yes	%	Treats up and down moves alike

Squaring gives large deviations more weight, which is why the standard deviation is never smaller than the MAD. The standard deviation is the exam’s default measure of risk, and it feeds portfolio variance and the Sharpe ratio later on.

Go deeper Population versus sample formulas

If you have every member of the group (a “population”), the mean is known rather than estimated and you divide by n. Returns over a few years are nearly always a sample of a longer process, so the exam uses n − 1 unless it says it is working with the whole population. As n grows the two answers converge: with 100 observations the difference is only about 0.5%.

Go deeper Chebyshev's inequality

For any distribution, at least $1 - 1/k^2$ of the observations lie within k standard deviations of the mean (for k above 1). With k = 2 that is at least 75%, and with k = 3 at least 89%. It is a distribution-free bound: it holds for skewed and fat-tailed data alike, but it is loose for bell-shaped data, where about 95% lie within two standard deviations.