Range, mean absolute deviation, standard deviation, and variance

CFA level I / Quantitative Methods: Basic Concepts / Statistical Concepts and Market Returns / Range, mean absolute deviation, standard deviation, and variance

Dispersion is the variability around the central tendency, and it signifies the risk. The most common measures of dispersion are range, mean absolute deviation, variance, and standard deviation.

The range is the difference between the maximum and minimum values in a sample data. It is easy to compute, but it doesn't tell about the distribution as it uses only two pieces of information.

Range = Maximum value - Minimum value

The inter-quartile range is the difference between the third and first quartiles of a data set. It contains the middle 50 percent of the data.

Inter-quartile range = Q₃ - Q₁ = 75th percentile - 25th percentile

The mean absolute deviation is the arithmetic average of the absolute deviation of all sample data around the mean.

MAD = (∑|X_i-(X-bar)|/n

The variance is defined as the average of the squared deviations from the mean. The standard deviation is the positive square root of the variance.

The symbol σ² denotes the population variance, and the symbol σ denotes the population standard deviation. We calculate population parameters for the population.

σ² = ∑(X_i-X-bar)²/N

σ = √∑(X_i-X-bar)²/N

Example 9: Calculating range, inter-quartile range, mean absolute deviation and standard deviation

The annual returns (in percentage) of a stock since its IPO has been given below:

14.50, 4.20, 23.20, -12.30, 6.70, 8.20, 1.50, -10.50, 18.70. 9.10, 3.10

Compute the range, interquartile range, mean absolute deviation and standard deviation for the population of returns.

Solution:

Arranging the data in the ascending order:

-12.30,-10.50,1.50,3.10,4.20,6.70,8.20,9.10,14.50,18.70,23.20

Range = Maximum value - Minimum value = 23.20 - (-12.30) = 35.50 percent.

75th percentile = (1+11)*0.75 = 9th place = 14.50 percent
25th percentile = (1+11)*0.25 = 3rd place = 1.50 percent
Inter-quartile range = 75th percentile - 25th percentile = 14.50 - 1.50 = 13.00 percent.

Observation	Annual return	X_i - X ̅	\|X_i - X ̅\|	(X_i - X ̅)²
1	14.50	8.464	8.464	71.633
2	4.20	-1.836	1.836	3.372
3	23.20	17.164	17.164	294.590
4	-12.30	-18.336	18.336	336.222
5	6.70	0.664	0.664	0.440
6	8.20	2.164	2.164	4.681
7	1.50	-4.536	4.536	20.579
8	-10.50	-16.536	16.536	273.451
9	18.70	12.664	12.664	160.368
10	9.10	3.064	3.064	9.386
11	3.10	-2.936	2.936	8.622
Mean	6.036		8.033	107.577

Mean absolute deviation = 8.033 percent
Variance = 107.577 percent square
Standard deviation = √107.577 = 10.372 percent

Please note that the standard deviation will always be higher than or equal to the mean absolute deviation because the standard deviation gives more weight to the large deviations as compared to the small ones as the deviations are squared.

The symbol s² denotes the sample variance, and the sample standard deviation is denoted by the symbol 's'. We calculate sample statistics for the sample of a population. s² = ∑(X_i-X-bar)²/(n-1)

s = √∑(X_i-X-bar)²/(n-1)

We use n-1 in the denominator for calculating the sample variance and sample standard deviation. The quantity n-1 is known as degrees of freedom in estimating the population variance. The statistical properties of the sample variance are improved by using n-1, and it becomes an unbiased estimator of the population variance.

There are other measures of measuring deviation as well like semivariance and semi-deviation. Semivariance is defined as the average squared deviations below the mean. Semideviation is the positive square root of semivariance. These are helpful in measuring the downside risk. Similarly, we have a concept of target semivariance and target semi-deviation where we use the average squared deviations below a target return.

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