Titel_SS06

2.5 Numerical Summaries

Central Measures

One of the most useful numerical summaries is the sample mean. If the data set is collected in
the vector x( , ,..,x 12 xxn)Tthe sample mean x is simply given as:

1

1 n

i

i

x x

n 

  (2.12)

The sample mean may be interpreted as a central value of the data set. If, on the basis of the
data set, one should give only one value characterising the data, one would normally use the
sample mean. Another central measure is the mode of the data set i.e. the most frequently
occurring value in the data set. When data samples are real values, the mode in general cannot
be assessed numerically, but may be assessed from graphical representations of the data as
will be illustrated in Section 2.6.

As it will be seen repeatedly in the present lecture notes it is often convenient to work with an
ordered data set which is readily established by rearranging the original data set
( , ,.., 12 )
T
x x xxn
12 .. ..

OO O

such that the data are arranged in increasing order as

1

O

in

O

x Oxx n

i

x x. In the subsequent the value of an ordered data set is denoted
by

ith

x.

The median of the data set is defined as the middle value in the ordered list of data if n is odd.
If n is even the median is taken as the average value of the two middle values (see also the
examples below).

Example 2.3 - Concrete compressive strength data

Consider the data set given in Table 2.2 corresponding to concrete cube compressive strength
measurements. In the table the data are listed both unordered, e.g. in the order they were
observed and ordered according to increasing values.

1 2 3 4 5 6 7 8 9

10

11

12

13

14

15

16

17

18

19

20

35.8

39.2

34.6

27.6

37.1

33.3

32.8

34.1

27.9

24.4

27.8

33.5

35.9

39.7

28.5

30.3

31.7

32.2

36.8

30.1

24.4

27.6

27.8

27.9

28.5

30.1

30.3

31.7

32.2

32.8

33.3

33.5

34.1

34.6

35.8

35.9

36.8

37.1

39.2

39.7

i xi xOi

Unordered Ordered