{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 3. Propagation of Uncertainty" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3.1 Introduction" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Suppose that you've measured quantities $a$, $b$, $c$, … with uncertainties $U_a$, $U_b$, $U_c$, …, and you are calculating a function $f(a,b,c,\\ldots)$. For example, you might calculate the average speed of an object by dividing the distance covered by the time elapsed. How would you find the uncertainty of the average speed from the uncertainties of the distance and time? The general process of finding the uncertainty of a calculated quantity is called the propagation of uncertainty." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3.2 General Method" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If quantity $a$ varies by $U_a$ while $b$, $c$, … are held constant, then the function $f(a,b,c,\\ldots)$ will vary by $(\\partial f/\\partial a)U_a$. There are similar variations in $f$ when the other variables are varied individually. If variations in $f$ due to each of the quantitiess are independent of each other, the uncertainty of $f$ due to the uncertainties of all of the quantities is\n", "\n", "\\begin{equation}\n", "U_f = \\sqrt{\\left(\\frac{\\partial f}{\\partial a} U_a\\right)^2 + \\left(\\frac{\\partial f}{\\partial b} U_b\\right)^2 +\\ldots}. \\tag{3.1}\n", "\\end{equation}\n", "\n", "In other words, the variations due to each variable combine like the components of a vector do to give the length of the vector. Note that this is based on the assumption that the variations in $a$, $b$, ... are independent of each other." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3.3 Two Frequently-Used Examples" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 3.3.1 Addition or Subtraction" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If the function is a sum or difference, $f = a \\pm b$, then the partial derivatives are $\\partial f/\\partial a = 1$ and $\\partial f/\\partial a = \\pm 1$. Using equation 3.1, the uncertainty in $f$ is\n", "\n", "\\begin{equation}\n", "U_f = \\sqrt{ U_a^2 + U_b^2}, \\tag{3.2}\n", "\\end{equation}\n", "\n", "for both addition and subtraction.\n", "\n", "If $f = a + b + \\ldots - c - d - \\ldots$, then equation 3.2 generalizes to\n", "\n", "\\begin{equation}\n", "U_f = \\sqrt{ U_a^2 + U_b^2 + U_c^2 + \\ldots}. \\tag{3.3}\n", "\\end{equation}\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 3.3.2 Multiplication or Division" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If the function is $f = ab$, then $\\partial f/\\partial a = b$ and $\\partial f/\\partial b = a$. Using equation 3.1, the uncertainty in $f$ is\n", "\n", "\\begin{equation}\n", "U_f = \\sqrt{ \\left(b U_a\\right)^2 + \\left(a U_b\\right)^2} = \\sqrt{(ab)^2 \\left[\\left(\\frac{U_a}{a}\\right)^2 + \\left(\\frac{U_b}{b}\\right)^2\\right]}. \\tag{3.4}\n", "\\end{equation}\n", "\n", "This can be rewritten as\n", "\n", "\\begin{equation}\n", "U_f = f\\sqrt{ \\left(\\frac{U_a}{a}\\right)^2 + \\left(\\frac{U_b}{b}\\right)^2}, \\tag{3.5}\n", "\\end{equation}\n", "\n", "or \n", "\n", "\\begin{equation}\n", "\\frac{U_f}{f} = \\sqrt{ \\left(\\frac{U_a}{a}\\right)^2 + \\left(\\frac{U_b}{b}\\right)^2}. \\tag{3.6}\n", "\\end{equation}\n", "\n", "The same result holds for division (see problem 3.1). The uncertainty of a quantity divided by that quantity ($U_a/a$, for example) is called a fractional uncertainty. For multiplication or division, the fractional uncertainties combine the same way as uncertainties commbine for addition or subtraction. Note that equations 3.5 and 3.6 cannot be used to calculate the uncertainty of $a^2$ by setting $a = b$ (see problem 3.2). \n", "\n", "If $f = (ab\\cdots)/(cd\\cdots)$, then equation 3.6 generalizes to\n", "\n", "\\begin{equation}\n", "\\frac{U_f}{f} = \\sqrt{ \\left(\\frac{U_a}{a}\\right)^2 + \\left(\\frac{U_b}{b}\\right)^2 + \\left(\\frac{U_c}{c}\\right)^2 + \\ldots}. \\tag{3.7}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3.4 Examples" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example 3.1**: Suppose that you measure the dimensions of the object below as $D = 4.24 \\pm 0.03 \\ \\rm{cm}$ and $H = 6.07 \\pm 0.04 \\ \\rm{cm}$. \n", "\n", "