5.5 KiB
Standard error
The spread in the mean
Definition 1: for a series of
N \in \mathbb{N}
independent measurements\{x_1, \dots, x_N\}
of the same quantity, the mean\bar x
of the measurements is defined as
\bar x = \frac{1}{N} \sum_{i=1}^N x_i,
for all
x_i \in \mathbb{R}
.
??? note "Derivation from the expectation value:"
Will be added later.
Which is closely related to the expectation value defined in probability theory, the difference is the experimental notion of a finite amount of measurements. Similarly, the mean should be an approximation of the true value.
Definition 2: for a series of
N \in \mathbb{N}
independent measurements\{x_1, \dots, x_N\}
of the same quantity, the spreadS
in the measurements is defined as
S = \sqrt{\frac{1}{N - 1} \sum_{i=1}^N (\bar x - x_i)^2},
for all
x_i \in \mathbb{R}
.
??? note "Derivation from the variance:"
Will be added later.
Which is closely related to the variance defined in probability theory, the difference is once again the experimental notion of a finite amount of measurements.
With the spread S
in the measurements the spread in the mean S_{\bar x}
may be determined.
Theorem 1: for a series of
N \in \mathbb{N}
independent measurements\{x_1, \dots, x_N\}
of the same quantity, the spread in the meanS_{\bar x}
is given by
S_{\bar x} = \sqrt{\frac{1}{N(N-1)} \sum_{i=1}^N (\bar x - x_i)^2},
for all
x_i \in \mathbb{R}
with\bar x
the mean.
??? note "Proof:"
Will be added later.
Determining the transformed spread
In this section a method will be postulated and derived under certain assumptions to determine the spread in the transformed means with a map f
.
Definition 3: let
f: \mathbb{R}^n \to \mathbb{R} :(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}
be a function that maps the mean for each independent measurement series with a corresponding spread to a new mean quantity\bar y
with a spreadS_{\bar y}
forn \in \mathbb{N}
.
In assumption that the spread in the mean for each independent measurement series is small, the following may be posed.
Postulate 1: let
f: (\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}
, the spreadS_{\bar y}
may be given by
S_{\bar y} = \sqrt{\sum_{i=1}^n \Big(\partial_i f(\bar x_1, \dots, \bar x_n) S_{\bar x_i} \Big)^2},
and
\bar y = f(\bar x_1, \dots, \bar x_n)
correspondingly for all(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n
.
??? note "Derivation:"
Will be added later.
With this general expression the following properties may be derived.
Properties
The sum of the independently measured quantities is posed in the following corollary.
Corollary 1: let
f: (\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}
with\bar y
given by
\bar y = f(\bar x_1, \dots, \bar x_n) = \bar x_1 + \dots \bar x_n,
then the spread
S_{\bar y}
may be given by
S_{\bar y} = \sqrt{S_{\bar x_1}^2 + \dots + S_{\bar x_n}^2},
for all
(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n
.
??? note "Proof:"
Will be added later.
The multiplication of a constant with the independently measured quantities is posed in the following corollary.
Corollary 2: let
f: (\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \overset{.} = \bar y \pm S_{\bar y}
with\bar y
given by
\bar y = f(\bar x_1, \dots, \bar x_n) = \lambda(\bar x_1 + \dots \bar x_n),
for
\lambda \in \mathbb{R}
then the spreadS_{\bar y}
may be given by
S_{\bar y} = |\lambda| (S_{\bar x_1} + \dots + S_{\bar x_n}),
for all
(\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n
.
??? note "Proof:"
Will be added later.
The product of two independently measured quantities is posed in the following corollary.
Corollary 3: let
f: (\bar x_1 \pm S_{\bar x_1}, \bar x_2 \pm S_{\bar x_2}) \mapsto f(\bar x_1 \pm S_{\bar x_1}, \bar x_2 \pm S_{\bar x_2}) \overset{.} = \bar y \pm S_{\bar y}
with\bar y
given by
\bar y = f(\bar x_1, \bar x_2) = \bar x_1 \bar x_2,
then the spread
S_{\bar y}
may be given by
S_{\bar y} = \sqrt{\bigg(\frac{S_{\bar x_1}}{\bar x_1}\bigg)^2 + \bigg(\frac{S_{\bar x_2}}{\bar x_2} \bigg)^2},
for all
(\bar x_1 \pm S_{\bar x_1}, x_2 \pm S_{\bar x_2}) \in \mathbb{R}^2
.
??? note "Proof:"
Will be added later.
Combining measurements
If by a measurement series m \in \mathbb{N}
values \{\bar y_1 \pm S_{\bar y_1}, \dots, \bar y_m \pm S_{\bar y_m}\}
have been found for the same quantity then \bar y
is given by
\bar y = \frac{\sum_{i=1}^m (1 / S_{\bar y_i})^2 \bar y_i}{\sum_{i=1}^m (1 / S_{\bar y_i})^2},
with its corresponding spread S_{\bar y}
given by
S_{\bar y} = \frac{1}{\sqrt{\sum_{i=1}^m (1 / S_{\bar y_i})^2}},
for all \{\bar y_1 \pm S_{\bar y_1}, \dots, \bar y_m \pm S_{\bar y_m}\} \in \mathbb{R}^m
.
??? note "Proof:"
Will be added later.