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STS2320 Ali S. H.
violations of the expected norm can have a severe influence on the computed
index numbers as they maylead to biased index values.
The rest of this paper is organized as follows: Sections 2 and 3 discuss ways
of identifying and treating the above mentioned violations. Section 4 discusses
what to do with the outliers once they are identified.

2. The Univariate Approach
Each variable in a composite index data should be examined individually
for the presence of skewness, kurtosis, and/or the presence of univariate
outliers before the composite index is computed.

2.1 Skewness and Kurtosis
The skewness coefficient is a measure of the lack of symmetry in data
distribution about the mean. Let x1, x2, ..., denote the n observation of a

variable X. A common definition of the skewness coefficient is given by, see,
e.g., Groeneveld and Meeden, (1984) and Johnson, Kotz, and Balakrishnan
(1994),

∑ ( − ̅) 3
= =1 , ℎ ̅ = −1 ∑ = √ −1 ∑ ( − ̅) 2

3 =1 =1

Values of 0, negative, positive SC indicates that the distribution of the
variable is symmetric, negatively skewed, and positively skewed distribution.
The Kurtosis is a measure of the thickness of the tail of the data
distribution relative to the Normal distribution. A definition of the Kurtosis
coefficient is given
bytrialalallalalllalaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaalalallal
∑ ( − ̅) 4
= =1 − 3
4

A zero value of KC indicates that the distribution has a tail similar in
thickness as that of a normal distribution. A positive or negative value indicates
that the distribution has a heavy-tails or light-tails relative to the tails of the
Normal distribution, respectively.
Other measures of skewness and kurtosis are discussed in Joanes and Gill
(1998). See also, Hair et al. (2015).
When making statistical inference based of index data, the data are
assumed to be Normally-distributed. If the Skewness and/or Kurtosis
coefficients are far from zero, they indicate departure from the Normality
assumptions. Variables with significant Skewness and/or Kurtosis coefficients
may require special treatment before the index is computed. As an

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