Biomedical Engineering Reference
In-Depth Information
quantity
W
. You might have come across Gibbs in a different context as the pioneer of
modern vector calculus.
The link between microscopic and macroscopic behaviour is provided by
statistics
, and
in this case a special branch of mechanics called
statistical mechanics
. Statistics deals with
the collection, organization and analysis of experimental data and it is important that you
should have the basic concepts at your fingertips before progressing.
The idea of a random number will appear again and again through the text; a sequence
of random numbers is one where there is no discernible trend from one number to another.
Your pocket calculator has a key labelled RAN#; press the key 10 times and make a note
of the numbers generated. My results are given in Table 7.1; yours will be different. This
is a series of random numbers; there is no discernible trend over the series and in this case
they are equally distributed over the range of values 0 to 1. If you switch off your pocket
calculator, switch it back on again and repeat the experiment, you will almost certainly
find exactly the same sequence because they are generated numerically from a
seed
. For
this reason (amongst others), mathematicians refer to such series as
pseudo-random
. Such
numbers are good enough for experiments where random numbers are needed, and the
sequence can be reproduced time and time again. If you choose a different seed, you will
get a different sequence.
Table 7.1
Ten pseudo-random numbers from a calculator
Key press
1
2
3
4
5
6
7
8
9
10
Number
0.861
0.478
0.127
0.427
0.648
0.275
0.571
0.813
0.471
0.063
The first thing to do is to calculate the mean, which is given by the sum of the set of
numbers divided by the number of values. If I write
x
1
,
x
2
, ...,
x
n
for the numbers then we
usually write the mean in brackets <
x
>, given by
n
1
n
x
=
x
i
(7.2)
i
=
1
The next thing we might want to know is the spread of points about the average; some-
times the points will be widely spread about the average, sometimes they will be closely
clustered. For this experiment, the points are chosen at random with values between 0 and
1, and so the spread is wide. To get a measure of the spread, we calculate the standard
deviation
n
1
n
)
2
σ
n
=
(
x
i
−
x
(7.3)
i
=
1
For reasons based on statistical theory, you will often find the standard deviation calculated
with a factor of
n
−
1 instead of
n
n
1
σ
n
−
1
=
(
x
i
−
)
2
x
(7.4)
n
−
1
i
=
1
