Page 223 - DMGT404 RESEARCH_METHODOLOGY

Unit 10: Time Series

where Notes
|D| is the absolute deviation,
x is the data element
i
and m(X) is the chosen measure of central tendency of the data set–sometimes the mean (), but
most often the median.
The average absolute deviation or simply average deviation of a data set is the average of the
absolute deviations and is a summary statistic of statistical dispersion or variability. It is also
called the mean absolute deviation, but this is easily confused with the median absolute deviation.
The average absolute deviation of a set {x , x , ..., x } is
1 2 n
S |x x |
-
n
Did u know? What is median absolute deviation?
The median absolute deviation is a measure of statistical dispersion. It is a more robust
estimator of scale than the sample variance or standard deviation; it also exists for some
distributions which may not have a mean or variance.

The MAD is a robust statistic, being more resilient to outliers in a data set than the standard
deviation. In the standard deviation, the distances from the mean are squared, so on average,
large deviations are weighted more heavily, and thus outliers can heavily influence it. In the
MAD, the magnitude of the distances of a small number of outliers is irrelevant.
The MAD can be used to estimate the scale parameter of distributions for which the variance and
standard deviation do not exist, such as the Cauchy distribution.

Self Assessment

Fill in the blanks:
10. Typically the point from which the deviation is measured is the value of either the ………..or
the …….......of the data set.
11. The Mean Absolute Deviation is a robust statistic, being more resilient to outliers in a data
set than the……………………...
12. The Mean Absolute Deviation can be used to estimate the scale parameter of distributions
for which the ………………and standard deviation do not exist.

10.5 Mean Squared Error (MSE)

MSE is the sum of the squared forecast errors for each of the observations divided by the number
of observations. It is an alternative to the mean absolute deviation, except that more weight is
placed on larger errors. While MSE is popular among statisticians, it is unreliable and difficult
to interpret. The mean squared error of an estimator is the expected value of the square of the
“error.” The error is the amount by which the estimator differs from the quantity to be estimated.
The difference occurs because of randomness or because the estimator doesn’t account for
information that could produce a more accurate estimate.

MSE of an estimator is one of many ways to quantify the amount by which an estimator differs
from the true value of the quantity being estimated. As a loss function, MSE is called squared
error loss. MSE measures the average of the square of the “error.” The error is the amount by

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