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Experiments, samples and populations

Suppose we have a set of N measurements x 1 , x 2 , . . . , x N . Any function of these
measurements (that contains no unknown parameters) is called a sample statistic, or often
simply a statistic. Sample statistics provide a means of characterizing the data. Although the
resulting characterization is inevitably incomplete, it is useful to be able to describe a set of data
in terms of a few pertinent numbers. We now discuss the most commonly used sample statistics.

Averages

The simplest number used to characterize a sample is the mean, which for N values x i , i = 1, 2,
. . . , N, is defined by
N
1 ∑
¯ x = x i . (1.2)
N
i=1
Table 2.1 – Experimental data giving eight measurements of the round trip time in milliseconds for
a computer ’packet’ to travel from Odessa (Ukraine) to Odessa (USA)

188.7 204.7 193.2 169.0
168.1 189.8 166.3 200.0

In words, the sample mean is the sum of the sample values divided by the number of values
in the sample.

Example 1.1. Table 2.1 gives eight values for the round trip time in milliseconds
for a computer ’packet’ to travel from Odessa (Ukraine) to Odessa (USA). Find the
sample mean. ,

Solution. Using (1.2) the sample mean in milliseconds is given by

1
¯ x = (188.7 + 204.7 + 193.2 + 169.0 + 168.1 + 189.8 + 166.3 + 200.0) =
8
1479.8
= = 184.975.
8
Since the sample values in table 2.1 are quoted to an accuracy of one decimal place, it is usual
to quote the mean to the same accuracy, i.e. as ¯x = 185.0.

Strictly speaking the mean given by (1.2) is the arithmetic mean and this is by far the most common
definition used for a mean. Other definitions of the mean are possible, though less common, and
include
1. the geometric mean,
( ) 1/N
N
∏
¯ x g = x i , (1.3)
i=1
2. the harmonic mean,
N
¯ x h = ∑ n , (1.4)
i=1 1/x i
3. the root mean square,

( ) 1/2
N x 2
∑
¯ x rms = i=1 i . (1.5)
N

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