Median Absolute Deviation Help

By Author: Pierce Brosnan
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Introduction to median absolute deviation help:

Median absolute deviation: Median absolute deviation (MAD) as the name stands is a statistical measure of the Median of the absolute deviation of a Univariate dataset from its Median.

If Univariate Data Sample = X 1, X 2, ..., X n, then MAD is defined as the median of the absolute deviations from the data's median. then MAD = Median ( | Xi - Median(Xi) | )

Uses of MAD:

The median absolute deviation(MAD) is a measure of statistical dispersion. It is often termed as the estimator of scale of data sample than the sample variance or standard deviation itself and is assumed to behaves better with data distributions which do not have a mean or variance, such as the Cauchy distribution.
MAD is believed to be a more accurate and a better statistical measure of a data sample and is more resilient to a distant observation in a data set than the standard deviation. In the standard deviation, the distances from the mean are squared thus resulting in a distant observation heavily influencing the standard deviation by giving a very high value. Whereas in MAD, ...
... there is no square function compared to Variance and is therefore much closer to the median.

Relation to standard deviation: In order to use the MAD as a consistent estimator for the estimation of the standard deviation s or V, one takes V = s = K*MAD where K is a constant scale factor and, which depends on the type of distribution. For a normally distributed data for K is taken to be 1.4826 and for large samples from a uniform continuous distribution, this factor is about 1.1547

Step by Step Help to Solve Problems on Median Standard Deviation

1. Find the MAD For the Univariate Data Sample (1, 1, 2, 3, 4, 6, 9),

Sol:

Step 1:Given Univariate data sample Xi = (1, 1, 2, 3, 4, 6, 9).

Total number of Values = 7

Step 2:Therefore Median Value = Middle Value or 4th Value

Median Value = 3.

Step 3:The absolute deviations about the median 3 for the given univariate data sample Xj = (2, 2, 1 0, 1, 3, 6)

Xj (in ascending order) = (0, 1, 1, 2, 2, 3, 6).

Median of Xj = 2

2. Find the MAD For the Univariate Data Sample (1, 1, 2, 3, 4, 6, 9) and also the standard deviation if it is a normally distributed data

Sol:

Step 1:Given Univariate data sample Xi = (1, 2, 4, 5, 5, 6, 8, 9, 10).

Total number of Values = 9

Step 2:Therefore Median Value = Middle Value or 5th Value

Median Value = 5

The absolute deviations about the median 3 for the given univariate data sample Xj = (4, 3, 1, 0, 0, 1, 3, 4, 5)

Xj (in ascending order) = (0, 0, 1, 1, 3, 3, 4, 4, 5).

Median of Xj = 3

Step 3: Standard Deviation for a normally distributed data V = 1.4826 *3 = 4.4478

3. Find the MAD For the Univariate Data Sample (1, 2, 3, 4, 5) and also the standard deviation if it is a uniformly distributed data

Sol:

Step 1:Given Univariate data sample Xi = (1, 2, 3, 4, 5).

Total number of Values = 5

Step 2:Therefore Median Value = Middle Value or 3rd Value

Median Value = 3

Step 3:The absolute deviations about the median 3 for the given univariate data sample Xj = (2, 1 , 0 , 1, 2)

Xj (in ascending order) = (0, 1, 1, 2, 2)

Median of Xj = 1

Step 4: Standard Deviation for a normally distributed data V = 1.1547 *1 = 1.1547

Practice Problems on Median Absolute Deviation

1. Find the MAD and Standard deviation for the normally distributed data sample ( 3, 5 , 6 , 7 , 9)

Clue: Median absolute deviation is 3rd value in the absolute deviation data and Standar Deviation = 1.4826 * MAD

2. Find the MAD and Standard deviation for the uniformly distributed data sample ( 5, 7, 9 , 15 , 11 , 17, 13, 1 , 3 )

Clue: Median absolute deviation is 5th value in the absolute deviation data and Standar Deviation = 1.1547 * MAD

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