Operating instructions

The “Statistics” application
73
6.4.4 Formulas used for calculating statistical values
Calculating the mean value and standard deviation
Terminology
i
x
= Individual measurement values of a measurement series of
n
measurement values
ni ...1=
x
= Mean value and
s
standard deviation of these measurement values
The formula for calculating the mean value is:
(1)
The usual formula for calculating standard deviation, as seen i
s
the literature
(2)
is not suitable for numerical calculation, since the variance (individual value-mean value) can result in deletion in measurement series
that have very small deviations. Moreover, when this formula is used, each individual measurement value must be stored before the
standard deviation can be determined at the end.
The following formula is mathematically equivalent but significantly more stable numerically. It can be derived from (1) and (2) through
appropriate recasting.
To use this formula for calculating the mean value and the standard deviation, you just need to store
n
,
∑
i
x
and
2
∑
i
x
.
Standard deviation
Numerical stability can be improved even more by scaling the measurement value:
With
0
Xxx
ii
−=∆
, where
0
X
(depending on the application) is either the first measurement value of a measurement series or the
nominal value of a measurement series, the result is:
Mean value
The mean value is then calculated as follows:
Relative standard deviation
The relative standard deviation is calculated by means of the following formula:
percent
Number of digits in the results
Mean value and standard deviation are always expressed and displayed to one more decimal place than the corresponding individual
measurement values. When interpreting the results, keep in mind that the additional decimal place is not meaningful when it comes
to small measurement series (less than approx. 10 measurement values).
The same is also true of percentages (such as those used in expressing relative standard deviations), which are always shown to two
decimal places (for example, 13.45 percent). Here, too, the significance of the decimal places depends on the size of the background
data.
∑
=
=
n
i
i
x
n
x
1
1
( )
2
1
1
∑
−
−
= xx
n
s
i
==
















−
−
=
∑∑
n
i
n
i
ii
x
n
x
n
s
1
2
1
2
1
1
1
( )
==
















∆−∆
−
=
∑∑
n
i
n
i
ii
x
n
x
n
s
1
2
1
2
1
1
1
∑
=
∆+=
n
i
i
x
n
X
x
1
0
1
100
x
s
s
rel
=