An Introduction to Error Analysis

Error Analysis: The Study of Uncertainties in Physical Measurements

Chapter 1: Preliminary Description of Error Analysis

Error Analysis: The study and evaluation of uncertainty in measurement.
Every measurement has some uncertainty.
Error: The inevitable uncertainty that attends all measurements.
Errors are not mistakes; the best you can do is to minimize them and estimate their size.
Problem of definition: The height of the door is not a well-defined quantity.
Uncertainties are always present in measurements.
Knowing the size of uncertainties is crucial.
If uncertainties are too large, measurements are useless.

1.1 Errors as Uncertainties

In science, "error" means inevitable uncertainty, not mistakes.

1.2 Inevitability of Uncertainty

Impossible to know the exact height of a doorway due to limitations of measurement tools and environmental factors.
Problem of definition arises because the height varies in different places and under different conditions.
No physical quantity can be measured with complete certainty.

1.3 Importance of Knowing the Uncertainties

Density of crown example highlights importance of uncertainties. $P{gold} = 15.5 \frac{gram}{cm^3}$ and $P{alloy} = 13.8 \frac{gram}{cm^3}$
Martha's measurement shows crown is not genuine because the density of the suspected alloy, 13.8, lies comfortably inside Martha's estimated range of 13.7 to 14.1, but that of 18-karat gold, 15.5, is far outside it.
Uncertainties have to be reasonably small (perhaps a few percent of the measured value).
Martha must justify her stated range of values.
Measurements would both have been useless if they had not included reliable statements of their uncertainties.

1.4 More Examples

Engineers must know the characteristics of the materials and fuels they plan to use.
Engineers must understand the uncertainties in drivers' reaction times, in braking distances, and in a host of other variables.
The measurement of the bending of light as it passes near the sun. $\alpha = 1.8$ and subsequent ones that give strong support to Einstein's theory of general relativity.
At the time, this result was controversial.

1.5 Estimating Uncertainties When Reading Scales

Measuring length with a ruler, the main problem is to decide where a certain point lies in relation to the scale markings.
'l = 36 mm' means $35.5 mm \le l \le 36.5 mm$ .
The process of estimating positions between the scale markings is called interpolation.

1.6 Estimating Uncertainties in Repeatable Measurements

Example: timing the period of a pendulum.
The spread in your measured values gives a valuable indication of the uncertainty in your measurements.
Best estimate = average.
Cannot always be relied on to reveal the uncertainties.
Even when we can be sure we are measuring the same quantity each time, repeated measurements do not always reveal uncertainties.
Check the clock against a more reliable one.
Reliability of any measuring device is in doubt, it should clearly be checked against a device known to be more reliable.

Chapter 2: How to Report and Use Uncertainties

2.1 Best Estimate ± Uncertainty

The standard form for reporting a measurement of a physical quantity x is (measured value of x) = $x{best} \pm \delta x$ , this statement means that the quantity lies somewhere between $x{best} - \delta x$ and $x_{best} + \delta x$ .
$\delta x$ is called the uncertainty, or error, or margin of error in the measurement of x.

2.2 Significant Figures

Experimental uncertainties should almost always be rounded to one significant figure.
The last significant figure in any stated answer should usually be of the same order of magnitude as the uncertainty.

2.3 Discrepancy

If two measurements of the same quantity disagree, we say there is a discrepancy.
The discrepancy between two measurements is defined as the difference between their best estimates.

$Discrepancy = |Measured Value 1 - Measured Value 2|$

2.4 Comparison of Measured and Accepted Values

The accepted value lies inside her margins of error; her measurement seems satisfactory.

2.5 Comparison of Two Measured Numbers

Uncertainty in a Difference (Provisional Rule): If two quantities x and y are measured with uncertainties $\delta x$ and $\delta y$ , and if the measured values x and y are used to calculate the difference q = x - y, the uncertainty in q is added.

$\delta q \approx \delta x + \delta y$

2.6 Checking Relationships With a Graph

Hooke's law states that the extension of a spring is proportional to the force stretching it, so x = F/k, where k is the “force constant” of the spring.

$x = \frac{mg}{k} = (\frac{g}{k})m$

$\therefore y = Ax^2$ Therefore y/x should be constant

$y = Ae^{bx}$ which represents a exponential relationship

2.7 Fractional Uncertainties

Fractional Uncertainty: $\frac{\delta x}{|x|_{best}}$ , also called relative uncertainty or precision.
Because the fractional uncertainty $\frac{\delta x}{|x|_{best}}$ is therefore usually a small number, multiplying it by 100 and quoting it as the percentage uncertainty is often convenient.

2.8 Significant Figures and Fractional Uncertainties

The number of significant figures in a quantity is an approximate indicator of the fractional uncertainty in that quantity.

2.9 Multiplying Two Measured Numbers

General form is: $(measured value of x) = x{best} (1 \pm \frac{\delta x}{| x{best}|})$
Value of p: $m{best}v{best}(1 \pm \frac{\delta m}{|m{best}|} + \frac{\delta v}{|v{best}|})$
Uncertainty in a product (Provisional Rule): If two quantities x and y have been measured with small fractional uncertainties $\frac{\delta x}{| x{best}|}$ and $\frac{\delta y}{| x{best}|}$ and if the measured values of x and y are used to calculate the product q = xy, then the fractional uncertainty in q is $\frac{\delta q}{|q{best}|} \frac{\delta x}{| x{best}|} + \frac{\delta y}{| y_{best}|}$

Chapter 3: Rules for Propagating Uncertainties

3.1 The Problem

Propagating uncertainties: How to estimate the uncertainty in the final answer.
Example: Volume V=pir2hV=pir2h, V[best} = {pi}r[best]^2h[best].

3.2 Addition and Subtraction

If several quantities x,…,zx,…,z are measured with uncertainties deltax,…,deltazdeltax,…,deltaz, and if the measured values are used to compute the sum q=x+…+zq=x+…+z, the uncertainty in q is the sum of the uncertainties in x,…,zx,…,z.

δq≈δx+…+δzδq≈δx+…+δz

If several quantities x,…,zx,…,z are measured with uncertainties deltax,…,deltazdeltax,…,deltaz, and if the measured values are used to compute the quantity q=x+y−zq=x+y−z, the uncertainty in q is the sum of the uncertainties in x,…,zx,…,z.

δq≈δx+δy+δzδq≈δx+δy+δz

3.3 Multiplying by a Constant

If x is measured with uncertainty deltaxdeltax and is used to compute q=Axq=Ax, where A is a known constant, then the uncertainty in q is deltaq=Adeltaxdeltaq=Adeltax

3.4 Products and Quotients

If several quantities x,…,zx,…,z are measured with small fractional uncertainties deltax∣x[best]∣,…,deltaz∣z[best]∣∣x[best]∣deltax,…,∣z[best]∣deltaz, and if the measured values are used to compute the product q=x×…×zq=x×…×z, then the fractional uncertainty in q is deltaq∣q[best]∣=deltax∣x[best]∣+…+deltaz∣z[best]∣∣q[best]∣deltaq=∣x[best]∣deltax+…+∣z[best]∣deltaz
If several quantities x,…,zx,…,z are measured with small fractional uncertainties deltax∣x[best]∣,…,deltaz∣z[best]∣∣x[best]∣deltax,…,∣z[best]∣deltaz, and if the measured values are used to compute the quotient q=x×yzq=zx×y, then the fractional uncertainty in q is deltaq∣q[best]∣=deltax∣x[best]∣+deltay∣y[best]∣+deltaz∣z[best]∣∣q[best]∣deltaq=∣x[best]∣deltax+∣y[best]∣deltay+∣z[best]∣deltaz

3.5 Powers

If a quantity x is measured with a small fractional uncertainty deltax∣x[best]∣∣x[best]∣deltax, and if the measured value is used to compute q=xnq=xn, then the fractional uncertainty in q is deltaq∣q[best]∣=∣n∣deltax∣x[best]∣∣q[best]∣deltaq=∣n∣∣x[best]∣deltax

3.6 Combining Uncertainties in General

If several quantities x,…,zx,…,z are measured with uncertainties deltax,…,deltazdeltax,…,deltaz, and if the measured values are used to compute some quantity q, then the uncertainty in q is

δq=(∂q∂xδx)2+…+(∂q∂zδz)2δq=(∂x∂qδx)2+…+(∂z∂qδz)2

3.7 Averages

If several independent measurements have been made of some quantity x these measurements are x1±δx1,x2±δx2,…,xN±δxNx1±δx1,x2±δx2,…,xN±δxN, then the best estimate for x is

x[best]=x1+x2+…+xNNx[best]=Nx1+x2+…+xN

3.8 Standard Deviation

Standard deviation : If you make N repeated measurements of some quantity x, and if your measurements are x1,x2,…,xNx1,x2,…,xN, then the standard deviation is

σ=1N−1∑i=1N(xi−x[best])2σ=N−11∑i=1N(xi−x[best])2

3.9 Standard Deviation of the Mean

Standard deviation of the mean: If you make N repeated measurements of some quantity x, and if your measurements are x1,x2,…,xNx1,x2,…,xN, then the standard deviation of the mean is

σm=σNσm=Nσ

Chapter 4: Normal Distribution

4.1 The Normal Distribution

The Normal Distribution : The probability density function is given by

P(x)=1σ2πe−12(x−μσ)2P(x)=σ2π1e−21(σx−μ)2

4.2 Using the Normal Distribution

68%confidence : x=x[best]±σ68%confidence : x=x[best]±σ
95%confidence : x=x[best]±2σ95%confidence : x=x[best]±2σ
99.7%confidence : x=x[best]±3σ99.7%confidence : x=x[best]±3σ

4.3 Why the Normal Distribution Works

Central Limit Theorem: The sum of N random numbers (no matter what their individual distributions) is distributed according to the normal distribution, provided N is large enough.

4.4 Standard Deviation of the Mean (Again)

Standard deviation of the mean: If you make N repeated measurements of some quantity x, and if your measurements are x1,x2,…,xNx1,x2,…,xN, then the standard deviation of the mean is

σm=σNσm=Nσ

4.5 Caveats Concerning the Use of Standard Deviations

Standard deviations give no indication of the possible presence of systematic errors.

4.6 Averages (Again)

If several independent measurements have been made of some quantity x, these measurements are x1±δx1,x2±δx2,…,xN±δxNx1±δx1,x2±δx2,…,xN±δxN, then the best estimate for x is the weighted average

x[best]=∑i=1Nwixi∑i=1Nwix[best]=∑i=1Nwi∑i=1Nwixi