Handling Measurement Data #

This section discusses how to estimate mean values and uncertainties after conducting repeated experiments or measurements.

Rounding Error

1. Number of Operations and Total Rounding Error # $$ \begin{align} X’ &= X \pm d \\ d &= \varepsilon \sqrt{N} \end{align} $$ Machine precision $\varepsilon$ Number of operations $N$ True value $X$ Computed result $X'$ Uncertainty in calculation $d$ 2. Explanation # Consider performing repeated operations on a computer. Here, “operations” refer to basic arithmetic calculations such as addition, subtraction, multiplication, and division in decimal. Since all variables in a computer are represented in bits of 0 and 1, any value used by the computer is rounded to a finite representation.

Uncertainty

1. Summary # To determine the true value $X$ from multiple measurements $x_1, x_2, \cdots, x_N$, the true value can be estimated with uncertainty as follows: $$ \begin{align} \label{eq1} X \to \bar{x} \pm \Delta\bar{x} \end{align} $$ Here, $\bar{x}$ is the most probable value (or mean) of $x_1, x_2, \cdots, x_N$, and $\Delta\bar{x}$ is the uncertainty in the mean, given by: $$ \begin{align} \bar{x} &= \frac{1}{N}\sum_{i=1}^N x_i \label{eq2}\\ \Delta\bar{x} &= \frac{\Delta x}{\sqrt{N}}=\frac{1}{{\sqrt{N}}}\sqrt{\frac{1}{N-1}\sum_{i=1}^N (x_i-\bar{x})^2} \label{eq3} \end{align} $$