mathematics-physics-wiki/docs/en/physics/mathematical-physics/error-analysis/standard-error.md

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 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}.

??? 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 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.

??? 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 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.

??? 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 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.

??? 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.