Sharp JX-9400 Informations techniques Page 177

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Variance and standard deviation
A figure representing the importance of the scattering around the average value
is the mean square deviation or variance:
S
x
¼
P
i
ðx
2
i
hx
2
ðN 1Þ
¼
P
i
ðx
2
i
ÞNhxi
2
ðN 1Þ
ð7:51Þ
The square root of S
x
is the estimate, s
x
, of the standard deviation,
x
:
s
x
¼
ffiffiffiffiffi
S
x
p
x
ð7:52Þ
The larger the number of measurements, the better the estimate.
Covariance
An estimate s
xy
of the covariance
xy
of two random variables x and y , of which
N measurements x
i
and y
i
were done, is calculated by:
S
x
¼
P
i
ðx
i
hxiÞðy
i
hy
ðN 1Þ
¼
P
i
ðx
i
y
i
ÞNhxihyi
ðN 1Þ
ð7:53Þ
This figure gives the tendency of two quantities to vary together. If these two
variables are totally independent, the covariance will be zero. The covariance of
a quantity with itself is the variance, already defined in Equation 7.51.
Statistical distributions
There are numerous probability distributions with a mathematical model. It is
not the place here to present all of them. They can be found in the specialized
handbooks such as Bevinton (1969), Diem and Lentner (1970) and Box et al.
(1978). The two most used distributions, which are also used afterwards to
estimate the confidence intervals, are presented below.
The probability density function of the normal or Gaussian distribution (see
Figure 7.3, left) is:
fðcÞ¼
1
ffiffiffiffiffi
2
p
exp
c
2
2
where c ¼
x
ð7:54Þ
where is the average and the standard deviation of the variable x.
The probability of the normal distribution (see Figure 7.3, right) is:
FðcÞ¼
1
2
1 þ erf
c
ffiffi
2
p

ð7:55Þ
where the error function erf(x) is:
erfðxÞ¼
2
ffiffi
p
ð
x
0
expð
2
Þd ð7:56Þ
with erfðxÞ¼erfðxÞ.
156 Ventilation and Airflow in Buildings
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