What is the difference between mean absolute deviation and standard deviation?

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 = (∑|Xi-(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 = ∑(Xi-X-bar)2/N

σ = √∑(Xi-X-bar)2/N

Inhaltsverzeichnis Show

Is standard deviation same as mean absolute deviation?
What is the difference between absolute deviation and deviation?
What is the difference between mean and mean standard deviation?
What does absolute mean deviation mean?

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

Xi - X ̅

|Xi - X ̅|

(Xi - X ̅)2

14.50

8.464

71.633

4.20

-1.836

1.836

3.372

23.20

17.164

294.590

-12.30

-18.336

18.336

336.222

6.70

0.664

0.440

8.20

2.164

4.681

1.50

-4.536

4.536

20.579

-10.50

-16.536

16.536

273.451

18.70

12.664

160.368

9.10

3.064

9.386

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 = ∑(Xi-X-bar)2/(n-1)

s = √∑(Xi-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.

Previous LOS: Quartiles, quintiles, deciles, and percentiles

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View the discussion thread.

Consider three set of data having same mean and MD but their ranges are changing. It is interesting to see how SD changes with change in the range of the data.

SET 1: 1, 3,5,7,9,11,13,15,17,19 Range:1-19 Mean=10, MD=5 SD= 6.05

SET 2: 2,3,5,7,7,9,13,15,14,23 Range: 1-23 Mean=10 MD=5 SD=6.28

SET 3: 3,5,5,7,7,8,10,12,13,30 Range: 1-30 Mean =10 MD=5 SD=7.70

It can be observed that all the three sets have same mean and MD. It is to be highlighted that while MD do not change with change in range, SD show changes with every change in ranges. This clearly establishes the supremacy of SD as compared to MD in dealing with variation in the data.

Is standard deviation same as mean absolute deviation?

While both measures rely on the deviations from the mean (x - \bar{x}), the MAD uses the absolute values of the deviations and the standard deviation uses the squares of the deviations. Both methods result in non-negative differences. The MAD is simply the mean of these nonnegative (absolute) deviations.

What is the difference between absolute deviation and deviation?

The average deviation, or mean absolute deviation, is calculated similarly to standard deviation, but it uses absolute values instead of squares to circumvent the issue of negative differences between the data points and their means.

What is the difference between mean and mean standard deviation?

Thus, the mean tells us what the average value is and the SD tells us what the average scatter of values is, around the mean. Taken together, especially along with the range, these statistics give us a good mental picture of the sample.

What does absolute mean deviation mean?

Mean absolute deviation (MAD) of a data set is the average distance between each data value and the mean. Mean absolute deviation is a way to describe variation in a data set.