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 \(x_i \in \mathbb{R}\).

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 spread \(S\) in the measurements is defined as

\[ S = \sqrt{\frac{1}{N - 1} \sum_{i=1}^N (\bar x - x_i)^2}, \]

for all \(x_i \in \mathbb{R}\).

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 mean \(S_{\bar x}\) 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.

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 spread \(S_{\bar y}\) for \(n \in \mathbb{N}\).

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 spread \(S_{\bar y}\) 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 \((\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n\).

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 \((\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n\).

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 spread \(S_{\bar y}\) may be given by

\[ S_{\bar y} = |\lambda| (S_{\bar x_1} + \dots + S_{\bar x_n}), \]

for all \((\bar x_1 \pm S_{\bar x_1}, \dots, \bar x_n \pm S_{\bar x_n}) \in \mathbb{R}^n\).

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 \((\bar x_1 \pm S_{\bar x_1}, x_2 \pm S_{\bar x_2}) \in \mathbb{R}^2\).

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 \(\{\bar y_1 \pm S_{\bar y_1}, \dots, \bar y_m \pm S_{\bar y_m}\} \in \mathbb{R}^m\).

Proof:

Will be added later.