1
APPENDIX 3.3: CALCULATING SKEW AND KURTOSIS
We mentioned in Chapter 1 that para meters are simply
numbers that characterize the scores of populations.
The mean (µ) and va riance (σ2) are two of the most
importa nt parameters associated with statistical
analysis in psychology. In subsequent chapters, we will
use statistics to estimate these important parameters.
Although µ and σ2 will be our primary concer n, skew
and kur tosis are parameters in the same way as µ
and σ2. Skew and kurtosis can be computed from the
scores in populations in a manner ver y similar to t he
computation of the mean and variance. Therefore,
we will take a moment to comment on how skew and
kurt osis are computed in populations and samples.
Moments
At the beginning of Chapter 3, we noted that the mean
and variance had very similar definitions. The mean
of a population is the sum of all scores d ivided by N.
Statisticians somet imes call this the first raw moment
of the distribution. T he variance in a population is the
sum of squared deviations from the mean, divided by
N. Statisticians call this the second central moment of
a distribution. Central moments are computed by sub-
tracting the mean from all scores and then raising these
deviation scores to some power. (Raw moments do
not subtract the mea n from all scores.) The following
expressions define the second, third, and forth central
moments:
θµ
θµ
θµ
2
2
3
3
4
4
=−
∑
=−
∑
=−
∑
()
()
()
.
y
N
y
N
y
N
The symbol θ is pronounced theta. Therefore, θ1, θ2,
θ3, and θ4 represent the first, second, third, and fourth
central moments of a distribution, respectively. All are
moments computed in the same way, and they differ
only in the power to which the differences between
y and µ are raised. You may not be familiar with
exponents other than 2 (squaring), but they are really
not complicated, as shown in the following examples:
yy
yyy
yyyy
yyyyy
1
2
3
4
=
=
=
=
*
**
***.
So, the exponents simply tell us how many times to
multiply a number by itself.
Skew and Kurtosis
The second, th ird, and fourth cent ral moments are related
to skew and kurtosis. In a population, skew is defined as
skew =
θ
σ
3
3(3.A3.1)
where σ is the population standard deviation (or the square
root of the second central moment, θ2). Skew, as defined
in equation 3.A3.1, can take on positive and negative
values. Symmetr ical distributions (such as in Figures 3.3a
and 3.4a) have zero skew. Distributions that a re skewed
to the right yield positive skew values, and those t hat are
skewed to the left yield negative skew values. The right-
skewed distribution in Figure 3.3 has a skew of about 3.6,
and the left-skewed distr ibution has a skew of about -3.6.
Kurtosis in a population is defined as
kurtosis=
θ
σ
4
4
.(3.A3.2)
Kurtosis, as defined in equation 3.A3.2, can take
on only positive values. The larger the values, the
more leptokur tic the distr ibution. However, when
statisticians t alk about kurtosis, t hey often mean excess
kurtosis. A norm al distribution has a k urtosis of 3, when
defined using equation 3.A3.2. Statisticians define 3
as normal kurtosis. Excess kurtosis is the difference
between kurtosis and normal kur tosis. Therefore,
excess kurtosis is defined as follows:
excess kurtosis=−
θ
σ
4
4
3. (3.A3.3)
A normal distribution has an excess k urtosis of 0.
The leptokur tic distr ibution in Figure 3.4b has an
