Error Analysis: Understanding Uncertainties in Measurements, Study notes of Physics

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APPENDIX A

MEASUREMENT AND UNCERTAINTY

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Experimental Error

When performing an experiment, one needs to keep in mind that a measurement that is precise is not necessarily accurate, and vice versa. For example, a vernier caliper is a precision instrument used to measure length (the resolution is 0.005cm). However, if you used a damaged caliper that reads 0.25cm too short, all measurements taken would be incorrect. The values are precise, but they are not accurate.

EXPERIMENTAL ERRORS

Experimental errors are generally classified into three types: systematic, personal, and random. Systematic errors and personal errors are seldom valid sources of uncertainty when performing an experiment as simple steps can be taken to reduce or prevent them. The specifics follow.

Systematic Error Systematic errors are such that measurements are pushed in one direction. Examples include a clock that runs slow, debris in a caliper (increases measurements), or a ruler with a rounded end that goes unnoticed (decreases measurements). To reduce this type of error all equipment should be inspected and calibrated before use.

Personal Error Carelessness, personal bias, and technique are sources of personal error. Care should be taken when entering values in your calculator and during each step of the procedure. Personal bias might include an assumption that the first measurement taken is the “right” one. Attention to detail will reduce errors due to technique.

Parallax, the apparent change in position of a distant object due to the position of an observer, could introduce personal error. To see a marked example of parallax, close your right eye and hold a finger several inches from your face. Align your finger with a distant object. Now close your left eye and open your right eye. Notice that your finger appears to have jumped to a different position. To prevent errors due to parallax, always take readings from an eye-level, head-on, perspective.

Experimental Error

Random Error Random errors are unpredictable and unknown variations in experimental data. Given the randomness of the errors, we assume that if enough measurements are made, the low values will cancel the high values. Although statistical analysis requires a large number of values, for our purposes we will make a minimum of six measurements for those experiments that focus on random error.

Error Analysis

ERROR ANALYSIS

Random Error When analyzing random error, we will make a minimum of six measurements (N=6). From these measurements, we calculate an average value (mean value):

x = x^1 +^ x^2 +^ Nx^3^ +^ ...^ +^ xi = x^1 +^ x^2 +^ x^3 + 6 x^4 +^ x^5 +^ x^6 = N^1 xi

i = 1

N Â =^

6 i = 1 xi

6 Â Eq.^ A-3.

To determine the uncertainty in this average, we first compute the deviation, di:

di = xi - x Eq. A-4.

The average of di will equal zero. Therefore, we compute the standard deviation, s (Greek letter, sigma). Analysis shows that approximately 68% of the measurements made will fall within one standard deviation, while approximately 95% of the measurements made will fall within two standard deviations.

s = N^1 - 1 ( d i )^2

i = 1

N Â =^

5 (^ di )

2 i = 1

6 Â =^

5 (^ xi^ -^ x )

2 i = 1

6 Â Eq.^ A-

or

s = (^ x^1 -^ x )

2 + x

( 2 -^ x )

2 + x

( 3 -^ x )

2 + x

( 4 -^ x )

2 + x

( 5 -^ x )

2 + x

( 6 -^ x )

2

5 Eq.^ A-6.

Propagation of Error

Uncertainty of Measurements: d In this lab the uncertainty, d (Greek letter, delta), of a measurement is usually 1/2 of the smallest division of the measuring device. The smallest division of a 30-cm ruler is one

millimeter, thus the uncertainty of the ruler is d x = 0.5 mm = 0.05 cm.

For example, an object is measured to be x ± d x = (23.25 ± 0.05) cm. The 5 in 23.25 is

estimated. The measurement, (23.25 ± 0.05) cm, means that the true measurement is most likely between 23.20 cm and 23.30 cm.

If you have multiple measurements of the same quantity, dx can be represented by the standard deviation, s.

To calculate

d% x ,^ the^ percent^ uncertainty^ (which^ has^ no^ units),^ of^ a measured^ value:

d% x =^ d xx ¥ 100 Eq. A-7.

Propagation of Error Once the uncertainties for your measurements are known they can be propagated in all mathematical manipulations that use the quantities measured. This allows a reader to know the precision of your work. When we propagate error the following rules apply.

Added and Subtracted Quantities When measured values are added (or subtracted) the uncertainties are added in quadrature (i.e., combining two numbers by squaring them, adding the squares, then taking the square root). The uncertainty of A, dA, is added in quadrature in Eq. A-8.

(x±dx) + (y±dy) => A = x + y

dA =

(^ d x )^2 +(^ d y )^2 Eq.^ A-8.

Express this result as A ± dA.

Propagation of Error

Multiplied and Divided Quantities

To determine the uncertainty in a product (or quotient):

x ± d x , y ± d y => A = xy and d A = A ¥^ d xx

Ê

Ë^ Á^

¯^ ˜

2

+^ d yy

Ê

Ë

ÁÁ

2 Eq. A-9.

Express this result as A ± d A.

d A can be expressed as a percent uncertainty,

d% A , by using Eq. A-7.

The result would be expressed as

A ± d% A.

When a measured quantity is raised to a power the uncertainty is determined by:

V = xn d V = V ¥ n^ d xx Eq. A-10.

Uncertainty is rounded to one or two significant figures, depending on the case. Keep only one significant figure unless the uncertainty begins with the number one. Keep two significant figures if the uncertainty begins with the number one.

Uncertainty Example: Multiplied and Divided Quantities

An area, A, and its uncertainty, dA, are to be determined from the following measurements:

Length = x ± d x = (2.40 ± 0.05) cm

Width = y ± d y = (1.35 ± 0.05) cm.

Area is length times width:

A = xy = 2.40 cm ¥ 1.35 cm = 3.24 cm^2.

The uncertainty in the area is given by

d A = A ¥^ d x

x

Ê

Ë^ Á^

¯^ ˜

2

+^ d y

y

Ê

Ë^ Á^

¯^ ˜

2

= 3.24 cm^2 ¥ Ê Ë Á 0.05 2.40 cmcm ˆ ¯ ˜

2

+ Ê Ë Á 0.05 1.35 cmcm ˆ ¯ ˜

2

= 0.13768cm 2 => 0.14cm^2.

Propagation of Error

The area and its uncertainty is expressed as

A ± d A = (3.24 ± 0.14) cm^2.

d % A can be determined using Eq. A-7. For this example

d% A =^ d AA ¥ 100 = 0.14 cm

2

3.24 cm^2 ¥^100 =^ 4.3%^ fi^ 4%^.

A ± d A , in terms of percent uncertainty, would be expressed as

A ± d% A = 3.24 cm^2 ± 4%

A simpler method to determine uncertainties for products (quotients) follows. This method yields slightly higher results than the quadrature method as well, but it is quite useful for most of our experiments.

Simple Uncertainty Example: Multiplied and Divided Quantities

First calculate the percent uncertainty of each quantity:

Length =

x ± d x = (^) (2.40 ± 0.05) cm = 2.40 cm ± 0.

¥ 100 = 2.40 cm ± 2%

Width =

y ± d y = (^) (1.35 ± 0.05) cm = 1.35 cm ± 0.05 1.35 ¥ 100 = 1.35 cm ± 4%.

Second simply add the percent uncertainties. Thus

d% A is^ 2%^ + 4%^ = 6%^ and

A ± d% A = 3.24 cm^2 ± 6%

Comparing the simple method (6%) to the quadrature method (4 %) for this example shows that the simple method yields an uncertainty about 30% too large.