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: it puts a number on how uncertain a return is around its average. 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 math
• Median absolute deviation Use it median of |X − median|, robust
• Variance Memorize population: ÷ N; sample: ÷ (n − 1)
  • 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. It is also called the mean deviation or the average absolute deviation. It beats the range because it uses every observation, but it is seldom used: absolute values are awkward to work with in later formulas.

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

The median absolute deviation takes the median of the distances from the median. Like the median, it is robust: one wild return barely moves it. On this site MAD always means the mean absolute deviation; the median version is written out in full.

$$\text{Median absolute deviation} = \text{Median}\left(\,|X_i - \text{Median}(X)|\,\right)$$

For returns of 1%, 2%, 4%, 9% and 30%, the median is 4. The distances from it are 3, 2, 0, 5 and 26; sorted, 0, 2, 3, 5, 26. Their median is 3%, and the 30% barely matters.

How do you calculate variance and standard deviation?

Square each deviation from the mean, add them up and divide. For a whole group, a population, you divide by N and get the population variance . Its square root is the population standard deviation .

$$\sigma^2 = \frac{\sum_{i=1}^{N} (X_i - \mu)^2}{N}$$

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

With sample data you divide by n − 1 instead. That is the sample variance , and its square root is the sample standard deviation .

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

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

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.

Why divide a sample by n − 1?

Memorize

An unbiased estimate is one whose average across many samples equals the true value. The sample mean is unbiased. But deviations measured from a sample mean are slightly too small, so dividing their squares by n systematically underestimates the population variance. Dividing by n − 1 instead, known as Bessel’s correction , removes that bias from the sample variance. n − 1 is also the number of degrees of freedom : 100 observations leave 99.

Two cautions. First, the correction fixes the variance, not the standard deviation: because a square root bends the scale, s still tends to understate σ a little. Second, it matters most in small samples with widely spread values; above roughly 100 observations the difference is small (with n = 100, √(99 ÷ 100) = 0.995, so about 0.5%).

If a figure was computed by dividing by n and you need the sample version, multiply by n ÷ (n − 1).

$$s^2 = \sigma^2 \times \frac{n}{n - 1}$$

How do you get these in a spreadsheet?

Know the idea

Measure	Excel	Google Sheets
Population variance	VAR.P	VARP
Sample variance	VAR.S	VAR
Population standard deviation	STDEV.P	STDEVP
Sample standard deviation	STDEV.S	STDEV

The .S versions apply Bessel’s correction; the BA II Plus shows both (Sx is the sample figure, σx the population one).

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 math
Median absolute deviation	Yes, through the median	%	Ignores how far the extremes are
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 (Learning Module 8) and the Sharpe ratio later on.

A standard deviation also depends on how often you measure: daily, monthly and annual returns give very different figures. Lesson 7 shows how to convert between them.

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 questions use n − 1 unless they say the data are the whole population. As n grows the two converge: with 100 observations the standard deviations differ by only about 0.5%.