Titel_SS06

distribution, which is equal to 3.0. The kurtosis for the concrete cube compressive strength
(Table 2.2) is evaluated as equal to 2.23, i.e. the considered data set is less peaked than the
Normal distribution. For the traffic flow data (Table 2.3) it is equal to 5.48 and 7.44 for
direction 1 and 2 respectively.

Measures of Correlation

Observations are often made of two characteristics simultaneously as shown in Figure 2.2
where pairs of data observed simultaneously are plotted jointly along the x-axis and the y-axis
(this representation is also called a two-dimensional scatter diagram as outlined in Section
2.6.).

Figure 2.2: Two examples of paired data sets.

As a characteristic indicating the tendency toward high-high pairings and low-low pairings, i.e.
a measure of the correlation between the observed data sets the sample covariance is
useful, and is defined as:

sXY

1

1

()(

n

XY i i

i

sxxy

n 

 y) (2.17)

The sample covariance has the property that, if there is a tendency in the data set that the
values of xiand yi are both higher than xand y at the same time, and the trend is linear, then
most of the terms in the sum will be positive and the sample covariance will be positive. The
other way around will result in a negative sample covariance. Such behaviours are referred to
as correlation.

In the scatter diagram to the left in Figure 2.2 there appears to be only little correlation
between the observed data pairs whereas the opposite is evident in the example to the right.

The sample covariance may be normalised in respect to the sample standard deviations of the
individual data sets and and the result is called the sample correlation coefficient
defined as:

sX sY rXY

1

(-)(-)

1

n

ii

i

XY

XY

x xy y

r

nss

 



(2.18)

The sample correlation coefficient has the property that it is limited to the interval   11 rXY