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Chapter 3: Numbers and Physical Reality

74 Version 4: Last updated: 1/26/2013, copyright Pearson, for personal use only, not for reproduction.

CHAPTER 3 NUMBERS AND PHYSICAL REALITY

How big are things and are they really changing?

Science depends on the comparison of theory with experiments; experiments depend on

measurements and measurements involve numbers. Likewise, theories are most reliably

expressed with numbers. Ultimately scientific decisions are made when someone compares the

numbers predicted by a scientific theory with the numbers produced in an experiment. Because

science often involves numbers and mathematics, we will develop enough mathematics to give

students an idea of how scientists view and use mathematics. We emphasize two things: 1)

numbers associated with experimental data (this chapter), and 2) the vocabulary and grammar of

the mathematical language used in theory (next chapter). These are central tools for science

literacy.

Expectations

When dealing with experimental data or theoretical calculations associated with physical

phenomena, we are always dealing with physical quantities that are expressed via numbers and

units. After completing this chapter, successful students are expected:

To recognize that there are two distinct ways in which numbers are used: discrete

counting and continuous measurements.

To develop basic skills as estimating quantities and understanding the role of units and

simple formulas in estimations.

To understand that numbers in science not only include the value and units, but also

an estimate of the error. To recognize the difference between precision and accuracy, and

why each is important. The successful student will also develop basic skills in estimating

errors.

To understand different sources of fluctuations in experiments and how scientists use

basic statistical quantities, such as mean and standard deviation, to quantify fluctuations.

To have a basic understanding of the importance of probability in describing physicalphenomena.


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3-A Numbers

Numbers appear in experimental measurements and in theoretical equations throughout the field

of physics, chemistry, biology, and engineering. Scientists generally work hard to see to it that

the numbers they use are as precise as is possible. William Thomson (later Lord Kelvin) is often

quoted as follows: When you can measure what you are speaking about, and express it innumbers, you know something about it; but when you cannot express it in numbers, your

knowledge is of a meager and unsatisfactory kind. Many scientists might believe that

Thomsons quote was an overstatement, but all would agree that numbers play a crucial role in

the success of science. As a science spectator, there are some basic features of numbers you need

to review, or recognize for the first time! The first issue is that numbers come in different types.

In everyday life, we dont spend a lot of time worrying about different types of numbers and

their accuracy explicitly. And yet, implicitly it shows up everywhere. When you arrange to meet

someone, because most peoples watches do not agree exactly, there is generally a 5 minute

rule, or some other range of allowed values that people could show up within and not be

late. However, as everyone switches to cell phone time that is tied to the same GPS system, the

range of allowed values may shrink! This is a great example of how the improvement in

technology (our tools) changes how we measure things. In contrast, when you are getting your

prescription filled, you expect the number of pills to be exact! There should be no error in this

case, because one pill too few is not acceptable and pills come in discrete units (each pill is its own

countable object). Just thinking about these everyday situations a little carefully suggests there is

even more to numbers than we usually think.

Therefore, it should be no surprise that scientists and mathematicians use a wide variety of types

of numbers. You probably learned about integers, whole numbers, rational numbers, real

numbers, and maybe even imaginary and complex numbers. For our purposes, it is useful to

categorize numbers found when describing nature as either countable or measurable. This is the

distinction we made between meeting at a certain time and counting out pills. It is a distinctionthat focuses on the degree of error associated with a particular quantity.

Countable quantities are generally expressed as integers (such as 1, 2, 3, ), and another word

that is often used for such quantities is discrete such as the number of pills you need. There are

cases where you can use fractions for discrete entities. For example, when using money, a half

dollar is an exact quantity and is considered countable. In physics, we find this in music, where

we can make waves that are exact fractions of a wavelength, or more esoterically, in quantum

mechanics where objects can possess a half integer spin value.

Measurable quantities are expressed as decimals and are generally associated with continuous

processes or situations such as telling time. Measurable numbers can never be exact

(completely free of error and uncertainty). Though we mentioned some special examples where

fractions were countable, they are more generally measurable quantities. Think of the classic

challenge of splitting a cookie between two people. Since it is almost impossible to split it exactly

in half, one person breaks and the other person chooses! This is an example of one half being

measurable and having an error associated with it.

It is worth mentioning a third category of numbers: defined numbers. These are quantities with a

very precise definition, such as the number which relates the radius and circumference of a


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circle. Most physical constants are in this category. Generally, these numbers have an exact value

inprinciple that is given by their definition, but in practice, we have to use an approximation for

these numbers.

Examples: Here are a few examples of different types of numbers.Measurable: 14.2 meters

Defined and exact: 3.14159 ....

(the famous Pi that relates the circumference and radius of circles)

Defined and countable: 5! (= 5 4 3 2 1 = 120)

(a mathematical operation calledfactorial).

Measurable: 3 x 102 kilograms

Countable: 47 eggs (or any other item you might count)

Why do we care about countable versus measurable numbers? The main reason is our interest inthe error in any given measurements. We will return to this concept in Section 3-C.

3-B. Estimations: how big are things?

A central skill for any science spectator is developing a feel for how big things are. For example, if

you were asked how big a typical sports stadium is, you should know that it is probably bigger

than 100 yards (the size of a football or soccer field, depending on the sport central to your

culture!) if it is going to fit a sports field. Also, it is probably not 100 times bigger than that. So,

adding some room for stands, you might guess 200 300 yards for the diameter. (Or, you might

guess something similar in meters if you prefer the metric system.) A quick search of some

typical stadiums finds dimensions in exactly this range. For example, the famous RomanCollosseum is basically 200 yard across. Notice, this estimate involves two key pieces: units and a

basic feel for the size of things.

Figure: When making an estimate, it can help to draw a quick

sketch. In this case, by drawing the field (rectangle in the center)

and the stands (gray area) roughly to scale, we see that our

estimate of 200 300 yards is quite reasonable.


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3-B.1. Units

In discussing numbers related to science, most people are aware that most numbers in science

come with units. People often complain about units and why do we even bother. But, as we

already pointed out, one of the most important science spectator skills is having an

understanding for the size of things that matters most! And this is a fundamental role of units they are associated with the size of something. Think of all the medical horror stories of people

receiving the wrong dosage because the units were wrong. After all, there is a big difference

between 2 mg and 2 g of the typical medicine. Another common example is a persons height. It

makes no sense to say that you are 6 tall. Is this 6 feet (which seems reasonable) or 6 inches

(which does not)? If you are in charge of a company and an employee predicts there will be a $2

million loss, you want to know is this expected in the next few days, years, or decade? Or, as

scientists would put it what is the time scale? Your response to the expected loss clearly depends

on how much time you have to deal with it. Finally, when considering variations in the climate,

you want to have a good idea of the time scale. Are the changes occurring over months, years,

decades, or centuries? Again, our response depends on this. Another way of saying all this is that

units provide the ability to determine what is large, small, fast or slow because ultimately these

concepts are only meaningful relative to some standard. The units tell us what standard we are

comparing to.

In addition to units telling us the scale of the measurement, units tell the kind of quantity being

measured. For the time being, we will start with three basic units (length, mass and time) that can

be used to construct all other physical units. The units for these quantities are called fundamental

units (discussed previously in Chapter 2). For example, the units of distance (a fundamental

quantity) can be meters, inches, feet, kilometers, and so on. Here we see the two uses of units

coming together. If scale (the size of things) did not matter, we would have just one unit for

length, to distinguish it from the other quantities, like time and mass. Instead, we have many

different units because they help immediately identify the scale that is involved. Hence, inches

versus feet immediately provide information of the size of interest. (There is also the historicalissue of different systems of units metric and English!) The units directly relate to the

operational definition of the quantity of interest and form the building blocks for all

measurements. The combination of numbers to indicate multiples (or fractions) of the referenced

unit, coupled with an indication of the errors, forms the complete quantitative representation of

data in science.

How do we determine the basic building blocks for our units? These come directly from our

operational definitions. Historically, physicists used to maintain standard building blocks in

Paris, France. The reference standards for length and mass were made out of a tough alloy

whose shape and dimensions changed only slightly with temperature; they were kept in a

carefully controlled dry environment. Today we use more invariant and more precise standards.

These may include the mass of an electron, the oscillation of a molecule (for time), and the

wavelength of a specific laser light. Standards are continually improving.

Here are some widely used fundamental units:

Quantity Units

Distance Meter, inch, footTime Second, minute, hour, day, year


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Mass Kilogram, gram, slug

Note that most of the units are frequently expressed with abbreviation: foot = ft, meter = m,

second = sec or s, minute = min, kilogram = kg, etc. The choice of a set of units is generally

referred to as a system of units. There are many different systems of units. The two common

systems that we will most often use in the textbook is the metric system (also referred to as the

MKS system or as the SI system) or the English system. (SI means Systeme

Internationale, the international system of metric units.) These systems are people-sized. We

are familiar with things whose size is expressed in these units, and as a result we usually feel

comfortable with these units. This sense of comfort frequently is helpful when developing an

intuitive understanding of a physical system.

System Distance Time Mass

Metric (SI) Meter Second Kilogram

English Foot Second Slug (or pounds for

weight, a topic for

later)

For objects that are slightly smaller, it is common to use a variation on each of these:

System Distance Time Mass

Metric (cgs) centimeter Second gram

English inches Second Ounces (for weight)

Addition and subtraction involve operations only on the numbers (the units are unchanged).

Thus physical quantities can be added (or subtracted) only if they have the same units. These

guidelines need to be kept in mind whenever mathematical operations are performed withphysical quantities. Once again, we have a signal to look for in the use of numbers in scientific

reports. By definition, if a report attempts to add quantities of different units, it just cannot be

done, and you should be suspicious! We also see that it makes no sense to measure area in inches:

square inches is the appropriate unit. Likewise, volume is measured in cubic inches or cubic

meters, etc. If you can remember the basic units for quantities, you can use this as a check on the

science that you are evaluating.

In contrast, numbers with units can be multiplied and divided to make more complex units. The

reader is probably already familiar with the concepts of area and volume an example of

multiplying units, and with speed an example of dividing units However, these are such a

common use of numbers with units that it is worth briefly reviewing the two concepts.

The area of rectangle is one of the first things students learn. For example, the length ( L ) of a

rectangle might be 4 inches, and the height ( H) might be 2 inches. Then the area ( A ) is1:

2(4 in)(2 in) 8 in (which reads "8 square inches")A L H= = =

1 Notice how we use a letter in parenthesis to indicate the variable that will represent a certain quantity. This isreviewed in Appendix A.


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Note we have multiplied both the numbers and the units. We can demonstrate area and volume

graphically with stacked squares and blocks.

Figure illustrating the concept of area and volume by breaking a surface or volume

into smaller pieces. One question this helps answer is w many square inches are

there in a square foot? Answer: 1 ft2 = (12 in)2 = 144 in2. This often causes

confusion, but the key is to remember that whatever mathematical manipulation

you perform on the number 12 must also be performed on the units: (12 in)2 = 122

in2. Thus a cube that is one foot long on each side has a volume of 1 cubic foot or

(123 =) 1728 cubic inches.

Surprisingly, our example of dividing units, speed, is something that many people seem to forget

when in a science classroom and there is a test in front of them. But, driving on the freeway, they

can generally tell you how long it will take to get to a particular exit if they know their speed and

the distance to the exit (assuming no traffic jams occur!) They use the fact that average speed is

defined to be distance traveled divided by time elapsed:


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distancespeed = ,

time

sometimes written asspeed /x t=

,

where x is the distance traveled and t is the time it takes (more on symbols and equations in

Chapter 4).If distance is measured in meters (or miles) and time in seconds (or hours), then theunit of speed is m/s (or mi/hr), which is read meters per second (or miles per hour). If this

seemed confusing, just do the exercise mentioned above if the exit you want on the freeway is

20 miles away and you are going 60 miles per hour, how long will it take to get there?

Example: my car traveled 400 miles in 8 hours. What is the average speed?

average speed = distance/time = 400 mi / 8 hr = 50 mi/hr. This result is read Fifty

miles per hour.

Notice that in ordinary speech, people are often careless with compound units and it may not

matter for the application at hand. This is especially true for the compound units we discussed.

For example, a store might sell you 3 yards of material. This is really 3 yards long and, say, 1

yard wide, so the area is really 3 square yards. Similarly, some people might say that the speed

limit is 55, but that overlooks the units entirely. They assume that you know the units are miles

per hour, but the appropriate units might be mi/hr, or kilometers/hr, or even meters/sec. It

cannot be overlying emphasized: when doing science, it is important to specify units correctly

when describing measurable quantities. If you have ever done any baking, you know how true

this is! Just confuse tablespoons and teaspoons, or dry measure and liquid measure and see how

your cake turns out. (This could be a fun experiment try baking something with the wrong

units. Worst case scenario, you produce something inedible. Best case scenario, you discover a

new awesome dish!)

We will take a brief detour to examine in more detail the character of derived units. We noted

earlier that length, mass, and time are the fundamental quantities that are most commonly used,

and each of them can be associated with an operational definition. This makes the quantities

precise and unambiguous. They can be used to do work that is relatively objective. Velocity is a

derived unit. You might not measure velocity by a direct operation; you might make a distance

measurement the value of which is divided by the value of a time measurement. The business of

division is an operation that can be precisely and unambiguously described. In other words, the

operational character of the quantity (velocity) is still highly reliable; its just a bit more

complicated.

Similarly, we find that some fundamental quantities cant be measured directly. For example, we

cannot lay out a measuring tape to the stars to find out how far away they are. But we can

perform some operations that will provide such information. Specifically, we can use the

triangulation method to measure a nearby stars distance from us. In the triangulation process,

we measure the angle to the star at one time and again six months later. These two angles are

different, and trigonometric operations (along with the radius of the earths orbit) will allow us to

calculate the stars distance. In other words, if direct measurement is impractical, the quantity of

interest might be obtained by indirect means. Measurement of other quantities can be used (like


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the angles toward stars), provided that operations exist that can relate these other measurements

to the desired quantities. We just need to have clear and precise operational rules to connect

measurements with theory. As long as we have these operational rules, science proceeds reliably.

Modern Aside: Length, mass, and time are not entirely independent. The theory of special

relativity shows that space and time are related to each other; in fact it is common to talk of the

spacetime continuum. The general theory then proceeds to show that there is a link between

mass and spacetime.

3B-2 People-Sized: The Scale of Things

We use the phrase people-sized to describe quantities that people are familiar with in everydayexperiences. We dont need microscopes or telescopes to examine such things. Note, as a

reference point, that a meter is slightly larger than 1 yard (which is 3 feet long). I am 1.75 (or one

and three-fourths) meters tall. My mass is 77 kilograms. It takes me about 33 seconds to walk

from one end of a football field to the other. My best sprinters speed is almost 9 meters per

second. These routine quantities (typical of daily life) are expressed in comfortable numbers

when the MKS (or SI) system is used. In different units they are less appealing. Consider the

following: I am also 17,500,000,000 Angstroms tall; my mass is 77 106 milligrams; and it takes

me 0.00038 days (or 1.05 106 years) to walk from one end of a football field to the other. My

best sprinters speed is an extremely small fraction (about 3 108) of the speed of light. The

speed of light is definitely not part of our experience base. (P.S. 106 =10 x 10 x 10 x 10 x 10 x 10 ...,

i.e., 10 multiplied 6 times, AND 106

=1

/10

x1

/10

x1

/10

x1

/10

x1

/10

x1

/10

..., i.e.,1

/10

multiplied 6times)2.

It is possible to convert from one system of units to another. For example, there are 0.3049 meters

in a foot. Whenever conversions are needed, we perform an operation that is nothing more than

multiplying by 1, but a special form of 1. For example: 0.3048 meter = 1 foot. Therefore, 1 =

0.3048 m / 1 ft. = 0.3048 m/ft. Also 1 = 1 ft / 0.3048 m. All we did here was invert the fraction,

because 1 divided by 1 is still 1. Either form of 1 is useful as a multiplier for conversions when

going from one system of units to another.

Example: I am 5 ft 9 in. tall = 5912 ft = 5.75 ft x (0.3048 meters/ft) = 1.75 meter.

Note that we discard ft/ft, which equals 1.

Exercise: What is your height in meters?

2 Not comfortable with scientific notation? See Appendix A.


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Example: Peggy Sue is 1.52 meters tall = 1.52 meters (3.28 ft / meter) = 5.00 ft.

Here we discard meters/meter

Exercise: A famous NBA basketball player is 2.14 meters tall.

What is his height in feet and inches?

Although we will not make a big issue of unit conversions, they do come in handy when making

certain estimates, as we will see in a later example. Also, scientists and engineers must be adroit

at these manipulations. Earlier, we also gave some masses and times using different units. The

reader should be aware that all kinds of quantities (velocity, force, energy, etc.) can be expressed

in a wide variety of units, and that conversion from one set of units to another is straightforward.

Thought Exercise: How big a number can you imagine or visualize? When do numbers becomeso big or so small that they lose their meaning to you? Here is one way to think of a million

people-sized things. Consider a pattern of balls one meter in diameter (people-sized). Space

the balls along the sideline of a football field, and then extend the pattern to three-dimensional

space with a pattern of one hundred balls in each direction. This pattern has 100 100 100 (or

one million) balls. So it is not too hard to imagine one million somethings. Now imagine this

cube becoming the single item that is used to repeat the 3-D pattern (100 100 100) for a trillion

one-meter balls. This is harder to grasp.

It is unfortunate that 25 years after Congress mandated a switch to the metric system, few groupsin the U.S. use it or are familiar with it. This presents problems in trade, since (for example) it

may be hard for a French repairman to find a metric wrench that is suitable for a 12-inch bolt.

This could be one reason we do not sell many different kinds of products in France. The only

other country that has not adapted metric units is Liberia. The emergence of the global village

might force the U.S. to accept the metric system.

3B-3: Advanced Estimates

It is one thing to estimate the size of common objects. The more challenging skill is to estimate

things that require simple mathematical operations. For example, the typical acceleration of a car

can be estimated from television commercials3

. In the U.S., you regularly hear about cars goingzero to sixty in a few seconds. Lets take if to be 0 to 60 mph in 4 s. Using conversion of units (go

back and review the unit section if need be!):

3 For this example, it helps to remember that acceleration is the change in velocity during a given interval of time. If you

do not remember this, it is ok for now, as we will review these ideas in later chapters. The point of this example is to show

the value of remembering simple formulas and sizes of key things to make basic estimates.


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2miles 1 hr 5280 ft 160 22 ft/shr 3600 s 1 mile 4 s

converts mph to ft/s divide by time to estimate acceleration

=

When we compare this to the acceleration due to gravity, which is232 ft/s , we can get a sense for

the acceleration of a car. We would call this acceleration 0.7 gs, because it is 0.7 times theacceleration due to gravity. For comparison, roller coasters can generate 2 4 gs! So, we can

immediately see why many people can get sick or do not like roller coasters, but fewer people

complain about accelerating their care!

Now, at this point you might be ready to give up acceleration, gravity, units, gs - do I really

need to know all this? The real answer is yes and no. We will review these topics here and in later

chapters, because it is very powerful to understand and use these basic concepts to make

estimates. You will find that with just a few ideas like these you can estimate a wide-range of

important things:

1)

What do you expect the cost of energy in your house to be?2) How much do you expect a typical school to cost? Given this, are your taxes being

wasted or well-spent?

3) What might the impact be of a few degrees of average heating? Should I really be

worried?

Estimates provide a quick check on claims in the media. They are not the final word, and can be

wrong in important ways, but they are a first step. In order to get better at them, there are two

useful ideas that we will review here. In preparation for these sections, if you are not completely

comfortable with significant figures and scientific notation, please see Appendix A for a review.

Example: Suppose you wanted an order of magnitude estimate of what the daily costs to runyour local 24-hour fast-food restaurant is? At first glance, this seems almost impossible as there

are some many aspects to the business: rent, salary, energy costs, food and supplies! But, if we

made it a bit easier and asked what is the minimum costs? There are some obvious estimates you

can make:

Salary for workers: average 5 workers at $10/hour: $1200

One manager at $100,000 for the year: $300

overhead (rent/energy/food/supplies): $500

___________________________________________________________________

Total $2000

There are a number of important features of this estimate that are important to notice. First, I did

not bother to figure out exactly how many workers or what they got paid. I used my experience

that there generally 3 8 workers in a fast food place at any time, and I picked a nice round

number for a minimum salary. Some workers are certainly making more, especially when benefits

are included, but this will give me a good minimum. I know the place has at least one manager,

probably more, but again, to get a minimum estimate, I went with one. Once again, no idea of

their salary, but it is unlikely to be off by factors of 10. Also, when I divided by the number of


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days in a year, I rounded the answer. Finally, I really have no idea about the other costs, but a

good rule of thumb (some estimates require either these rules of thumb or outright guesses) is

that overhead costs are at least 50% of the other costs you know. If you did not want to use my

50% rule, you could make some educated guesses based on what you pay for rent and utilities,

and assume they serve at least 100 hamburgers in a day and pay a few dollars per pound for their

meat. This will give you a good order of magnitude for the minimum costs. Notice, the actual daily

costs could be as much as $6000 or as little as $1000, and I am still within my factor of 10 for my

estimate!

An important hint when making an estimate for things is to focus on using integers that are a

good approximation to the quantity of interest. In fact, the language you most often hear is good

to an order of magnitude or good to a factor of two. An order of magnitude is another way

to say factor of ten, and it is one reason for the popularity of scientific notation. So, factor of 2

estimates are generally more accurate than order of magnitude ones, but order of magnitude is

much easier to achieve! Consider again the example of the acceleration of a car. There are two

parts to it: (1) the change in speed from 0 to 60 mph; and (2) the time this takes. The time is thepart we are approximating. We know it is greater than 1 s and less than 10 s, so to make a

reasonable estimate that is probably good to a factor of 2, we picked 4 s. It is still unlikely that the

time was less than 2 s or more than 8 s, numbers that are half and twice the time we picked.

Hence, we are able to claim that this estimate is good to a factor of two.

Challenge: One of the most common issues in public debates is the funding for school systems.

Using ideas in this section, estimate the minimum cost to run a school system. The challenge is to

NOT look up the actual spending in your area now, but to honestly estimate what you expect the

minimum cost to be. Then, compare that to what your area spends. If your estimate is higher than

your area spends, ask yourself how that might be? Could this explain problems with education inyour area? If your estimate is lower, now ask the really hard questions. Since you went for a

minimum, are there clear benefits from the additional spending? Can you determine if you missed

key areas of spending, or perhaps you actually identified waste in the system? Just a fair

warning: if you are not a teacher, most people under estimate what it takes to pay for qualified

teachers!

3-B. Errors: quantifying reliability


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Figure showing two different ways to measure a foot:

an actual foot and a ruler! Which method is more

accurate, and which is more precise?

We cannot repeat too much that science, especially physics, involves measurement. This

measurement process is an operation that is well defined; it is the operational property of

measurements that gives physical science its reliability. Reality in the philosophical sense is notthe issue! What does matter is the fact that no measurement is perfectly accurate. Even

theoretical equations are never perfect because they always contain some physical constants

whose values were determined by experiments. It is important to remember that physics is

ultimately based on human sensory observations and measurements performed according to

precisely prescribe operations (sometimes aided by instruments) of natural phenomena. Again

we note the fundamental fact that physical science is an experimental endeavor.

3-B.1. Precision versus Accuracy

There are two main classes of error in scientific measurements. We refer to these as: precision and

accuracy. They are different things and should not be confused with each other. As you considerthis, you might wonder, what does a detail like this have to do with being a good science

spectator? Lets briefly return to the sports analogy. For example, in baseball there are at least two

ways to get a person out: force out or tag out. Both are an out, and it might not seem to matter as

a spectator to know the different types of outs. But, it certainly increases your ability to enjoy the

sport and provide intelligent commentary and critiques of the sport if you know this difference.

Ironically, it is less important to know the specific names, then the fact that there is more than one

way to get an out! Precision and accuracy are somewhat analogous to this situation they both

describe errors in measurements, but sufficiently different types of errors that it is important to

know the difference. And, having the correct name for the two main types of errors just makes

you that much better at evaluating science when it is necessary to do so!

Precision is the easier of the two to define carefully. It helps if you are familiar with significant

figures. If not, it is useful to review this topic in Appendix A. The precision of a measurement is

your ability to tell the difference between two numbers. For example, if you are sure that an

object is 0.10 cm long and not 0.09 cm or 0.11 cm, your precision is 0.01 cm. In generally, the last

significant figure in a measurement generally determines the precision. Thus 9.613 kilograms has

a precision (or uncertainty) of 0.001 kilograms (because the measurement was not 9.612 or 9.614

kilograms.) Another way to say this is that precision is an estimate of the smallest increment you

have access to when you make a measurement (usually due to the limitations of the measuring


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instrument). Hence 9.613 kilograms and 0.026 kilograms have exactly the same precision (0.001

kilograms) because it is the last significant figure that matters, not the size of the measurement.

Precision is also referred to at times as the resolution of the measurement.

There are times that the precision may be less than the last significant figure but greater than the

second to last significant figure. For example, you might be using a ruler that is marked precisely

in increments of 1 cm. In this case, you can certainly estimate the result to some fraction of a cm.

Depending on your confidence in your ability to estimate the values less than a cm, you might

state your results as a certain number of centimeters plus or minus 0.5 cm. For example, for the

object in the figure, you would the length as 4.5 0.5 cm to indicate that even though you are

quoting the result with a significant figure of 0.1 cms, your estimate of the precision is actually

0.5 cm.

Figure: The ruler in the figure is marked in centimeter increments. The objectbeing measured falls between 4 and 5 cm. A reasonable estimate of the

precision is 0.5 cm. If one was really confident, you might even use a

precision of 0.2 cm!

Often you do want to know the precision with respect to the actual size of the measurement itself.

Relative precision refers to the size of the error or uncertainty relative to the size of the measured

quantity. Hence 0.001 kg out of 9.613 kg is a relative precision of 0.01%, while 0.001 kg out of

0.026 kg is about 4%. Thus, the relative precision associated with 9.613 kg is better than that

associated with 0.026 kg. Note that relative precision is dimensionless (usually given as a

percentage), but precision is expressed with units. Similar arguments apply to theoretical

calculations.

Accuracy expresses the size of the error or departure of a measurement from the exact answer.

Unlike precision, which is well-defined, we never know the exact answer; thus we can only guess

or estimate the accuracy of any given quantity. Two common examples of the difference between

precision and accuracy are the use of scales and watches. For example, it is not uncommon for a

inexpensive household scale to measure with a certain precision, say to 0.1 pounds, but to have

an unknown offset that is larger. For example, the scale may be offset by 2 pounds and we do not

know it. This is why most scales have a way to zero the scale, which is an attempt to get the

measurement as accurate as possible by making sure the scale measures zero when nothing is on

it. The other common example of accuracy is when people set their watch forward a few minutes

to make sure they are not late for things. The precision of the watch is not affected by this, and

remains seconds (if it has a second hand), but the accuracy is only minutes.

Often, the best that we can do is to infer that the accuracy is no better than some given value

based on our experience. However, as errors are eliminated (or when corrections are made to

account for errors), precision and accuracy tend to converge. The reason for this is that once you

have eliminated the sources of error not related to the resolution of your measuring device, the

only error left is the precision. At this point, the precision becomes a good estimate of accuracy

because it tells you how likely the number is to be different from the real result. But, how do


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we go about eliminating or accounting for error? For this, we need to understand the different

sources of errors and the ways to account for them.

3-B.3 Systematic versus statistical errors

When evaluating sources of error, especially issues with accuracy, there are two types of errorswe need to consider: systematic and statistical. Systematic errors arise when there is a specific and

consistent mistake that is made in a measurement. For example, a meter stick would expand at an

elevated temperature required for a certain experiment. If we did not notice or account for this

expansion, then length measurements using this ruler would produce values that are too small.

Also, a systematic error might happen if an experimenter consistently uses an instrument poorly.

Many students learn to use protractors to measure angles in school. Generally, one has to line the

protractor up correctly, or the angle that is measured is wrong. Systematic errors are often the

most difficult to identify because they are by definition mistakes you make that you are not aware

of! This is why having multiple people review an experiment is incredibly useful for identifying

systematic errors.

A common method for dealing with systematic errors is to design experiments measure thedifferences in the quantities of interest. By computing the difference of two measurements with the

same potential systematic error, the error cancels out. Lets consider the case of our ruler that

expanded so that any given measurement of length is 2 cm longer than the actual length. Suppose

we want to measure the height of two people: Susan and John. We measure Susan to be 150 cm,

and John to be 160 cm. In reality, Susan is 148 cm and John is 158 cm. This represents the

systematic error in each measurement. But, if we ask for the difference in the two heights, the

answer using our ruler is:

height of John - height of Susan 160 cm 150 cm 10 cm= = .

If we do this using the actual heights, we arrive at the same answer:

height of John - height of Susan 158 cm 148 cm 10 cm= = .

The most useful feature of using differences to account for systematic error is that it works even

when you do not know the magnitude of the systematic error! Consider our above example.

Suppose we just knew that the systematic error was some value E, and that John had a real

height that we will call jH and Susan had a real height that we call sH . Then what we know is

the following:

measured height of John - measured height of Susan 160 cm 150 cm 10 cm= =

Rewriting this using our variables and the fact that4:

measured height of John ( )

measured height of Susan ( )

j

s

H E

H E

= +

= +

4 Refer to Appendix A to review the use of variables.


15/38



( ) ( ) ( ) ( ) ( ) 10 cmj s j s j s

H E H E H H E E H H+ + = + = = ,

we see that the common systematic error always cancels out, and we get the actual height

difference. We put actual in quotes because there may be other errors that do not cancel; these

are the statistical errors discussed next.

Statistical errors occur if conditions in a measurement fluctuate randomly. There are two basic

types of fluctuations in measurements. The first type comes from variations in the measurement

technique. For example, consider measuring the diameter of a uniform cylindrical bar with a high

precision micrometer. The measuring tool will not always return to exactly the same spot on the

bar, and the diameter of the cylinder will not be perfectly uniform. Also, it is impossible to use

the micrometer in exactly the same way every time; and this may produce a random fluctuation

in the measured value. (Note that it could also produce a systematic error that is not random.)

The second class of statistical error comes from intrinsic fluctuations. There are many physical

quantities that are inherently fluctuating in time or position. For example, consider the weight of

a person. If you were to measure it throughout the day, it would fluctuate based on whole host

of factors such as, how much the person just ate, how recently the exercised, etc. The interesting

thing about intrinsic fluctuations is that sometimes they need to be treated as a source of error and

sometimes they are the thing you want to measure! Consider our weight example. Usually, you

just want to know the average change in your weight on a daily or weekly basis. In this case, the

fluctuations mentioned here are potential sources of error because they can lead to you

concluding that your average weight changed between two days when what really changed was

the time you weighed yourself! However, if you are interested in how quickly eating and

exercises changes your weight, these are the quantities you would want to measure!

In physics, we have learned that there are even different classes of intrinsic fluctuations. A

particular important breakthrough that we will return to in later chapter is the idea of quantum

fluctuations. These are fluctuations in systems that exist no matter what you do to get rid of them!Another important class of fluctuations that will be discussed in the content chapters is

associated with large numbers of objects acting together, such as molecules in a gas. In this case,

it is easier to treat the fluctuations in a statistical sense then to track the motion of every particle!

Because of this (and other important phenomena), statistics is a critical area of mathematics for

the serious science spectator, and we will spend an entire section reviewing some key concepts in

statistics. But, lets quickly consider two more examples of fluctuations, the types that occur, and

why they are important.

Physical Example: We have already addressed some of the issues with measuring the average

temperature of the Earth when we looked at operational definitions. The temperature is an excellentexample of a physical process with intrinsic fluctuations. Just think about your daily experience.

The temperature fluctuates throughout the day in fairly regular way. It warms up as the sun is

out, and then cools down once the sun goes down. But, on top of that regular variation, it can

fluctuate as clouds come and go, or as you move in an out of the shade, or as the wind moves the

hot and cold air around from different places. If you spent the day walking around with a

thermometer, you would see all of these variations as fluctuations in your measurement. Some of

these fluctuations are what we would consider actual changes in temperature (such as the variation


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from day to night) and others might be viewed as sources of statistical error (such as the wind

moving the air around your thermometer). However, understanding and correctly accounting for

these fluctuations and determining under what conditions they represent a measurement of a

real change or an error is one of the major challenges of doing careful experiments.

Fiscal Example: Suppose you are planning to invest in the stock market. The prices on the stock

market vary in time: they fluctuate. There are two types of fluctuations. There are short term

fluctuations that really are random chance due to the fact that large numbers of people are

buying and selling stock at any given moment, causing small fluctuations in prices. There are

long term fluctuations. These tend to be more deterministic. They are not completely

predictable in detail because are models are still limited. But, one can always count on an

eventual downturn after an excessive increase in prices that are associated with a bubble in the

market. Finally, there is the long term average trend which is consistently increasing.

Depending on your goals and the time you plan to invest for, you are going to worry more or less

about the short term random fluctuations, longer variations, or even longer long-term average

behavior.

3-B.3. Sources of Error

Figure illustrating some potential sources of error with a meter stick.

Before focusing on characterizing fluctuations, it is worth taking a tour of potential sources of

error. Consider the measurement of the diameter of a penny. It can be done with a meter stick or

a micrometer. Both are referenced to the same standard, but with different degrees of precision.


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Both contain some inherent physical imperfections, again with different degrees of precision.

Each has different limitations. Both can be used with bad technique. The penny also must be

considered. It will not be perfectly round; thus the diameter is not the same taken in different

locations. Any given penny will have its dimensions change with temperature. And every

penny is slightly different from every other penny, so measurements of one penny are not

perfectly representative of all others. This simple example is typical of the type of error

considerations that are encountered in every experiment. There are several sources of uncertainty

in measurements, and we will take a brief look at each of these.

1. Uncertainty in the standard used to define the unit of measure. For example, the kilogram used to

be defined as the mass of a reference object that was kept in Paris. This object could be compared

with other objects by using a pan balance. The error that would arise from such comparisons was

fairly small for most applications. Today, the basic unit of mass is referenced to the hydrogen

atom. When standards are poorly defined, considerable error can occur. An example of this

problem may arise because the standard is not constant. The foot, an English length unit, was

initially defined as the length of the foot of a certain English king. (This may be why English

monarchs are often referred to as rulers.) The kings foot also was not of constant length from

morning to night or from day to day (or even as the temperature changed). The kings foot alsomight change over longer time periods due to growth or shrinkage (or because of gout, etc.).

These days, however, the foot is defined in terms of the metric system (1 foot = 0.3048 meters); the

meter is referenced to the wavelength of light emitted by an atom when it changes from one

energy level to another energy level.

2. Imperfections in the equipment being used. No measuring equipment can operate perfectly. For

example, if I use a meter stick on an extremely cold day, it will shrink slightly, increasing the

errors in its readings. Of course, a correction can be made to account for expansion or

contraction. This implies that we need to know at what temperature the meter stick is accurate,

and that we know how the length of the meter stick changes with temperature.

Figure of different types of equipment for measuring the mass of

objects. Each of these has different sources and types of errors.


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3. Limitations in the equipment being used. When I measure my weight on a bathroom scale, I

might get 157 pounds. But I wouldnt get 157.357 pounds, because the scale is not capable of

measuring to this precision, although some other scale might be. When my bathroom scale says

157 pounds, the precision is probably about one pound, because I can tell it doesnt mean 158

pounds, but I would have a hard time measuring precisely 157.5 pounds. However, it is also

possible (perhaps even likely) that the scale could be in error by as much as two or three pounds.

4. Inaccurate techniques in using equipment. When I stand on the bathroom scale some mornings, I

get 173 pounds. But then I notice that the scale says 11 pounds even when I am off it. This means

that someone has tinkered with the knob that sets the zero reading. If I had relied on the value

173 pounds, I would have gotten my inaccurate result because of improper technique. Making

sure that the equipment is properly calibrated is standard procedure when setting up experiments,

and it prevents the above problem. Other flawed techniques are also possible. Someone

standing near me might get a different reading than I do because they are looking at the scale

from an angle, changing the geometrical relationship between the numbers and the indicator

pointing at them. This is like the difficulty encountered when trying to read a car speedometer from the

passenger seat next to the right front window. The driver has a better (more accurate) viewpoint.

5. Improperassumptions concerning the system being measured. If you measured my height early in

the morning, you would get 69.5 inches. But if you assume that this is always my height, you are

wrong, because you would be assuming that a person always has the same height, which is not

true. In fact, when you measure my height in the evening, you would get a number like 69.2

inches. The reason I and other people usually get shorter during the day is that the vertebrae in

our backbones are separated by pads called discs, which tend to compress during the day. At

night, when we sleep, the discs are not under stress, and so they expand slightly, making us a

little taller when we get up in the morning.

Here is another example of an improper assumption about the system being studied. Suppose

you wanted to study the influence of gravity on falling pieces of paper. So you design anexperiment in which you time the fall of pieces of paper from different heights above the ground.

Your results would be inaccurate because falling paper is influenced not only by gravity but by

air resistance. So your claim that the results showed the influence only of gravity would be

wrong. The proper way to study the effect of gravity of falling pieces of paper is to drop them in

a vacuum chamber, eliminating or greatly reducing the influence of air resistance. This illustrates

a significant challenge in science. Scientist must develop models describing a system which are

simple and yet complete enough to predict the behavior of the system.


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Figure showing your height changing with the time of day. Humans are slightly

taller in the morning than in the evening because of the compression that spinal

disks are exposed to during the day because we are primarily vertical.

6. Variations in the system being measured . If you were to measure how tall people in a class are,

you would find considerable variation. These are actual variations; they are not the result of how

we use our equipment. So, if you were to measure the height of only one student, you would not

be able to draw any reliable conclusions about the range of heights in the class.

Thought Experiment: If I step on a scale in Death Valley, my weight may be

different than when I step on the same scale at the top of Pikes Peak. This could

be due to almost any of the 6 preceding categories of uncertainties. How couldthis be?

A number of important mathematical methods have been developed to study systems

with natural variations or fluctuations. It is this last example that we will turn our

attention to now how to understand and think about fluctuations and uncertainties that

are either intrinsic in the system, due to randomness, or due to some lack of information

on our part.

3-B.4 Error bars and notation

When we want to tell someone what our expected error is for a given measurement, we already

saw in our discussion of precision that we write our answer as:

6.45 0.5 m

This expression has three parts:


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1) 6.45: this is our best estimate of our measurement, or the average value of our

measurement. The last digit gives a good indication of our precision. Though as we

mentioned, sometimes our precision is a little worse than this digit, but not as bad as the

next digit!

2) 0.5: this is our best estimate of the error in our measurement. Notice that this will often

include both the accuracy and the precision. Because in this case, the result is quoted to

the nearest 0.01 and the error is given as 0.5, there is probably a statement being made

about precision and accuracy. Often, if the accuracy is as bad as this error indicates, the

precision is no longer relevant, and you would write the measurement as 6.5 0.5 m .

3) The final piece is the units. In this case, the unit is meters.

There are a few other issues to be aware of with this notation. In some types of measurements,

the number after the is the standard deviation, a quantity we will review in the next section. This

is not always the true expected error in the measurement, but it is often a way to communicate the

expected range of values, which is very closely related to error. From a science spectator point of

view, the main goal is to see that a reasonable effort was made regarding the general precision

and accuracy of the measurement, and to report it. Subtle distinctions involving propagation of

errors, standard deviation versus standard error, etc., are things you can explore more if

interested, but not essential.

It is worth noting that sometimes the error is not symmetrical. Maybe there is a iron-clad reason

the number can not be negative for example. So, you might get a situation where the answer is:

1

0.50.5 m+

.

This reflects the fact that even though your error may be 1 m, the answer can not possibly go

below zero, so 0.5 1 m makes no sense.

Finally, it is worth pointing out that when graphing data, the error is indicated by error bars that

represent the range of values suggested by the error. This is illustrated in the following figure:

Figure illustrating the use of error bars. For each of the measurements of

height in this plot, the error is 0.1 cm .

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

Height(cm)

time (days)


21/38



3-C Fluctuations, Statistics, and Probability

In this section, we will be tackling two of the most important, and perhaps the most challenging

topics in mathematics that is relevant to science literacy: probability and statistics. The challenge

with these topics is that to apply them correctly often involves subtle ideas that can be easily

overlooked by the non-expert. This is coupled with the fact that our intuition of these topics isoften unreliable. We will consider these two topics in the context of an important physical

phenomenon:fluctuations. To help with this discussion, we will focus on two examples for which

people do have some intuition: financial issues (money) and weather.

Given the inherent difficulty of these topics, it is worth explicitly recalling that the goal of our

discussion is to provide essential tools for basic science spectatorship. To return to our sports

analogy, to be a science spectator, it is not necessary to go into all the subtle aspects of using an

off-side trap in a game of soccer. But, you need to know the basic facts about off-sides to

intelligently watch the game. As we deal with the concepts of randomness, probability, statistics,

and fluctuations, the reader needs to understand that this is really the subject for whole textbooks

and there are many details we will gloss over by necessity. We will focus on key conceptual

ideas and a few central calculations that are handy to have for analysis of scientific reports (and

occasionally daily life).

3-C.1 Definitions

Fluctuations in a quantity are changes in time or space. There are two basic types of fluctuations:

random and predictable. The day to day fluctuations in temperature and the price of the

stock market are basically random. We might have some ability to predict their general trends,

but detail prediction is not possible. However, the long term fluctuations are predictable

summer is hotter on average then winter in Northern latitudes! Another example of a predictable

fluctuation is given by the voltage in an alternating current system. The voltage fluctuates from

positive to negative in a smooth and predictable manner. Generally, predictable fluctuationsoccur in systems that are periodic. In fact, once it has been determined that the variations in a

quantity are sufficiently regular and periodic, one generally does not use the term fluctuation,

but the more specific term of periodic. Periodic variations are generally well described by

sinusoidal functions, a topic we will review in Chapter 4. So, for most of this section, we focus on

random fluctuations, and just use the term fluctuation.

Distributions refer to the collection of numbers that are the result of a series of measurements.

These can take many forms. They can be measurements of a single quantity taking over time. For

example, a set temperature measurements that is taken every hour forms a distribution. If one

measures the height of individuals in a group, this will also be a distribution. These are examples

of discrete distributions where you have measured a countable number of objects. A challenge for

these types of distributions is determining how best tograph them. This is best illustrated with asimple example.

Example: Consider measuring the heights of a group of people. (Names have been removed to

protect their identity!) Here are the results in the form of a table:


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Person Height

A 1.2 m

B 0.9 m

C 1.3 m

D 0.7 m

E 1.5 mF 1.2 m

G 1.4 m

H 1.0 m

I 1.1 m

If we want to look at the distribution of the results, we need to count how many times we get the

measurement of the height in a certain range or bin. We will show the results of using three

different choices:

All three of these graphs are of the same data from the table. The first one plots the number of

heights in bins of width 0.3 m. So, there are two people with a height between 0.6 m and 0.9 m,

the first bin in the plot. The second plot uses bins of 0.2 m, and the last one uses bins of 0.1 m.

In the last case, one of the bins goes from 1.1 m to 1.2 m, and this is the bin with three people in it.

Notice how the visual impression of the distribution can vary with the choice of bin size. This is a

key feature of reporting data, and something to be extremely aware of as a science spectator!Make sure that the person presenting the data is not using their bin choice to convey a falseimpression!

Probabilistic event is one where the outcome is a matter of random chance. Consider a box filled

with half red balls and half blue balls. Putting your hand in without looking and grabbing a ball

gives you a 50% of getting a red ball and a 50% chance of getting a blue ball. Similarly, the case

flipping a perfectly balanced coin is a 50-50 situation5. There is a 50% chance of getting heads and

a 50% chance of getting tails. In this case, any model used to describe the behavior predicts the

chance of particular outcomes occurring and not the outcome itself. The field of probability deals

5 There is a subtle issue here that the careful reader will notice. For a single coin flip, if we truly tracked every elementof launching the coin, we could predict whether or not it would come up heads or tails. For the ball example, if wetracked every motion of the balls and the exact location of your hand, we could predict your outcome. So, strictlyspeaking it is deterministic. However, in practice, there are so many variables involved, we can consider it

probabilistic. In contrast, quantum mechanics is a truly intrinsically probabilistic theory in many ways. We will revisitthis in the content chapters.

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Numberofpeople

Height (m)

0.6 0.8 1.0 1.2 1.4 1.60.0

0.5

1.0

1.5

2.0

2.5

3.0

um

erop

eople

Height (m)

0.6 0.8 1.0 1.2 1.40.0

0.5

1.0

1.5

2.0

2.5

3.0

umberop

eople

Height (m)


23/38



with the rules for predicting various combinations of random events. For example, if you wish toknow the probability of getting four heads in a row when flipping a balanced coin, you woulduse probability theory to compute this. This is one area where our intuition is often flawed.

Intuition Test: Write down on a piece of paper what you might expect to get if you flipped a coin50 times in a row. Use H to indicate heads and T to indicate tails. After writing down this series,try flipping an actual coin 50 times. One thing to look for is the longest streak of head or tails inyour fake series (most number of heads or tails in a row) and the longest one in the real series ofheads and tails. Also, compare the number of times you gets streaks of 3, 4, 5, 6, etc., this will letyou know how good your intuition is. Hint: This exercise works best if you can get a group offriends to do it independently from each other and compare notes. Then the differences reallyshow up!

A deterministic event is one in which the outcome itselfis predictable (within the stated precision)

with the use of a mathematical model, and not just the probability of different outcomes. Thiswould be the case of throwing a ball. Given the initial speed and direction of the throw (and a

few other variables, such as wind and air-resistance), a person can reasonably predict where the

ball will land. There is some error associated with the prediction, but this can be characterized as

well.

Statistical events refer to the analysis of the properties of many events. One challenge in looking at

the outcome of different systems and behaviors is that when considering a large number ofevents, the results can be random even when the individual events are deterministic. In this

case, it is often best to use a statistical description of the system. For example, if we throw a ball

many times, each individual throw is deterministic. But, we do not have the ability to throw the

ball with the same initial angle and velocity every time. So, if we measure the landing spots, we

will get a distribution of locations. We will treat this problem in a probabilistic fashion, where therandomness is due to unknown information (initial angle and speed of the throw) in an otherwisedeterministic process (throwing the ball). In this case, we can analyze the statistical properties of

the landing locations, and learn useful information about our throwing process. On the otherhand, if we flip our coin many different times, we will get a distribution of heads and tails. In thiscase, the distribution is from an intrinsically random process. We will use the same statisticalmethods in these two cases, but often we will be asking different questions. Therefore, justbecause you see the same words like mean or standard deviation, it is important to understand thecontext I which they are being used to fully understand what they mean!

Generic fluctuations would be ones that are not easy to classify as strictly speaking probabilistic ordeterministic. One example is cases where the quantity of interest is being studied for different

objects. For example, we already mentioned looking at the height of different people in a room.Depending on your perspective, you might think of this as probabilistic (the heights are randombecause you do not know anything about the people) or deterministic with a degree of error (youknow something about the genetics and history of the people in the room). Another way to thinkof these categories is that the distinction betweenprobabilistic and deterministic is often an issue ofinformation regarding the system. As with anything, there are clear-cut cases where theinformation required is so overwhelming, we declare the situationprobabilistic, as we did for our

ball and coin examples. However, consider the case of the amount of money I have in my pockets

from day to day. This is not strictly speaking random it occurs because I take money in and out


24/38



for specific reasons. To the degree that I can track and predict these reasons, the process is

deterministic. But, there always seem to be surprise events that occur that prevent the

development of a fully predictive model for the behavior (yet!). Therefore, these fluctuations are

often best described in aprobabilistic model.

Errors in measurement techniques often produce a distribution of results that are well described

by statistical methods. In fact, sources of error can create fluctuations even when the quantity

appears to be constant. For example, the length of a rod might be unchanging; but the measured

value of the length may fluctuate due to variations in the process of measurement. This could

include the case where different measurement tools are used.

3-C.2 Mathematical Formulas

Given the importance of statistical analysis, it is the one area of mathematics where it is worth

reviewing the relevant formulas and doing a few computational examples. To make this easier, it

is important to introduce some notation. To help with the abstract notation, we will use the

specific example discussed with regard to distributions of the students heights in a class. The

first step is to define a variable that represents each measurement. It is common to use x, but wecould just as easily use any other letter. Often, we pick the first letter of the quantity that we are

measuring. The next step is to number the different measurements and use that as a subscript on

are variable to keep track of each element. We can do that in our table of heights:

Person Height

1x 1.2 m

2x 0.9 m

3x 1.3 m

4x 0.7 m

5x 1.5 m

6x 1.2 m

7x 1.4 m

8x 1.0 m

9x 1.1 m

We generally refer to the number of the measurements as N or n, which for our example is

9n = . A final notation point: if we want to refer to ageneric measurement and a specific one, we

will use the notation ix where the variable i as a subscript tells us that we want one of the 9measurements that we made, but at this point, we do not care which one!

Here is where for many readers their math fear kicks in just take a deep breath and relax!

Remember, we are learning the essential features of a new language. This is a chance to start

practicing converting between symbols and English.


25/38



There are two main terms that we want to be able to computes: average (or mean) and standarddeviation.

We define the average (or mean) xave of our measurements to be

1

1 n

ave i

i

x x x x

n =

= = = .

Hereavex , x and x are just different symbols that can be used for the average value of x, and

is a symbol that indicates an operation that adds all the values of x together, where i indicates

each separate value and takes on all values from 1 to n. You can see how this takes significantly

more space in English to write then it does in math symbols. Remember, if you are not happy

with the symbols, go with the English: The average is the sum of all the measured values divided

by the number of measurements that were made. So for our example, it looks like this:

9

1 1

1 1

9

1.2 m 0.9 m 1.3 m 0.7 m 1.5 m 1.2 m 1.4 m 1.0 m 1.1 m1.14 m

9

n

ave i i

i i

x x xn = =

= =

+ + + + + + + += =

One feature to notice about the calculation is that we followed our rules for units. We only added

numbers with the same units, meters! Also, because the number of measurement does not have

any units, the final answer has the same units as the measurements, as we would expect for the

average height.

We often want to have a measure of the spread or the width of the distribution of the measured

values. You could just compute the difference between the highest and lowest values, but this

does not capture the fact that you often have many measurements grouped around a central

value with just a few values that are very large or very small. For this case, we define the

standarddeviation and use it to estimate the width of the distribution. This is illustrated in the

figure that represents the measurement of a large number of heights.


26/38



Figure illustrating a distribution of heights. This distribution is plotted as aprobability

distribution where the number of results for each height has been divided by the total

number of heights. The red arrows represents the width as measured by thestandard deviation: the width is given as twice the standard deviation and represents

the mean plus and minus the standard deviation. Notice, this captures most of the

measurements, though certainly not the full range. It is common to use this as the

estimate of the range of likely values. In this case, the range given by the red arrow

represents approximately 65% of the data. If we went to plus and minus twice the

standard deviation, we would be at 95% of the data.

For our purposes, we will define the standard deviation x (called the standard deviation of x)

by the following formula6:

2

1

1( )

n

x i

i

x xn

=

=

Again, the formula may appear scary, but the English is more complicated in some ways:

Take each measurement that you made and subtract the mean from it. Take each of these

numbers and square them. Now, take all these new numbers and add them up (that is the symbol again). Now divide this number by the number of measurements and take the square

root of this number.

Here is an example showing how to calculate the standard deviation of the height of fourstudents (we use a small number to focus on the arithmetic involved):

6 Strictly speaking this is an approximation. Because our focus is on the concept that numbers have uncertainties, and the

different physical meanings associated with the standard deviation, we will use this approximation for all problem

examples that we discuss in this text. This does ignore some subtle issues in a rigorous study of statistics, but these do not

impact the main conclusions we will be making. If you are interested in a more detailed discussion, you can refer to any

basic book on statistics.

0.5 1.0 1.5 2.0 2.5 3.00.00

0.02

0.04

0.06

0.08

0.10

mean minus one

standard deviationmean plus one

standard deviation

Mean =1.5 m

Probabilityof

height

Height (m)


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i xii

x x 2( )

ix x

1 1.50 0.10 m 0.0100 m2

2 1.55 0.05 m 0.0025 m2

3 1.60 0.00 m 0.0000 m2

4 1.75 +0.15 m 0.0225 m2

2

1

( )n

i

i

x x=

= (0.0100 + 0.0025 + 0.0000 + 0.0225) m2

= 0.0350 m2 ,

and so the standard deviation is

2 2

1

1 1( ) (0.0350m ) 0.09 m

4

n

x i

i

x xn

=

= = =

As with our earlier example, we can also compute the average height of these four students. You

should check that it is 1.60 m. In this case, if we ask the question what is the average height of a

student in the class, we might present the answer as:

Student height = 1.60 0.09 m.

This is a statement about two facts that we have determined: (1) The mean or average height is

1.60 m and (2) the standard deviation is 0.09 m. In this case, the standard deviation is telling us

about the spread or range of heights in the class. One might think of it as an error in the sense of

how accurately the mean reflects a true measurement of height in the following sense. For a

class where the students are all essentially the same height, the standard deviation will be reallysmall and the average height will provide an accurate picture of the height of a student in the

class. For a class with a wide range of heights, the standard deviation will be large, and the

mean will not necessarily provide an accurate picture of a typical height because such a quantity

does not exist!

ASIDE:

Another important issue is one of judgment. Sometimes data sets contain values that simply do not seem

credible. What is to be done in such cases? Consider the heights of four students that were presented previously,

and suppose that a fifth height had been recorded of 16.1 meters. This means that one of the students was about

as tall as a five-story building! Surely this is ridiculous, and the value should be discarded. But on second thought,it looks like a decimal was misplaced. Do we simply move the decimal? Do we repeat all five measurements?

If we have names associated with the heights, then it would only be necessary to re-measure one student.

Suppose that the measurementcould not be repeated, what would be done then? The point of this paragraph issimple. Sometimes it is necessary to reject data, and this is a matter of judgment. There is no general rule ofthumb that applies other than vigilance by the experimenter.


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In contrast to measuring four different people, we can consider the following interesting twist.

Suppose the four heights in the table were four measurements of the height of the same person

taken four different times. In this case, we would compute the same mean and standard

deviation. We would still say that the standard deviation represents the accuracy of the

measurement. But, notice there is now a different sense to this accuracy it is the accuracy with

which were able to make the particular measurement! It might represent the variation in the

persons height with the time of day; it might be our ability to use a ruler; or it might be some

other error. Though very similar to the interpretation of standard deviation for the heights of the

different people in the class, it is useful to note the subtle differences. For people in the class,

there is clearly a real distribution of heights even when our measurement technique is perfect! The

standard deviation is a way to measure how wide this distribution is. For a single person, it is

conceivable that the measurements should represent a single value and the only reason for the

variation in measurements is some source of error in the measurement.

We can see from this example that a blind use of calculational formulas should be avoided.

When we interpret the standard deviation in the above equations as a measure of the accuracy of

the mean, we assume that the x values are reliable. But we know that an individuals height

changes during the day. Students may also be wearing shoes of differing thicknesses, or mighthave differing posture. All of these conditions affect the final result. Could a standard system be

adapted for obtaining this quantity of average height? Would it be sensible (or worth the trouble

and expense) to do so?

The mathematics of statistical processes is a highly developed and sophisticated branch of

mathematics, and perhaps of the most poorly understood and misused! Given this, our example

provides a good idea of what you should take away from our discussion of averages and

standard deviation. At its most basic level, the fact that an average and standard deviation is

being reported tells you that there is a range of possible values for the measurement. The

standard deviation provides a good estimate for how wide the range of values is. If you want to

be even more specific, the results of mathematical studies of systems that fluctuate in this wayimply that two thirds of the samples will fall within one standard deviation of the average when

the resulting distribution of results agrees with a normal curve of error. This is also called a

Gaussian distribution or a bell-shaped distribution. Surprisingly, many things in the physical

world are described by such a curve, and so this is a good rule of thumb to use for the

quantitative meaning of standard deviation. The next figure displays these properties.

3-C.3 Statistics of Basic Probability

There is another characteristic of a set of fluctuating or random data that plays an important role

in scientific literacy the most likely or most probable value. Often, this is also the average of the

set of numbers, but this is not always true. This introduces the idea of probability. Probabilities

are represented as fractions, because a probability of one is considered an event that isguaranteed to happen. In general, the public is presented with probabilities all the time,

especially in the context of risk. As with the more general issue of statistics, probabilities are often

poorly understood and even completely misused at times. For simple cases, the probability is

straightforward to compute (or estimate) by taking the number of ways you can get your result

and dividing by the total number of outcomes. Unfortunately, one quickly finds that it can be

difficult to correctly count things like the number of possible outcomes, even when it looks easy!

Also, this argument relies on all the outcomes being equally likely.


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Warning Examples: Consider the challenge of computing the probability that an asteroid will hit

the Earth. An incorrect and nave argument would say: There are two possibilities, it hits or it

does not, so the chance of being hit by an asteroid is 50% (1 out of 2). The errors in this argumentshould be reasonably obvious, as the probability of the asteroid hitting the Earth involves a

number of factors, starting with the question of the time frame you want to consider! Another way

in which counting can cause problems is highlighted by the following example. Suppose you

determined the probability of a US citizen dying from a shark attack in a year by taking the

number of US citizens killed by sharks last year and dividing by the total number of US citizens.

What is one obvious problem with using this probability to assess your risk? I doubt anyone was

killed by sharks further than 1 mile from the water. So, if you are never within a mile of the

water, this calculation makes no sense.

As with statistics, probability is full of subtle issues. We can get a sense of some of the issues by

considering four standard cases, but as with statistics, it is most useful to know what you do not

understand, and then to seek expert help when it really matters!

CASE 1: Flipping a coin: In this case, there are two possible outcomes (heads or tails), and each

outcome can be achieved in a single way. So, the probability of each outcome is or 50%. Also,

in this case, the two outcomes are equally likely. So, this is the classic case where counting works

because each outcome is equally likely.

CASE 2: Rolling two normal six-sided dice: In this case, the outcomes are a one to six on one

dice and a one to six on the other dice. In this case, there are 36 different outcomes (6 x 6). The

argument is as follows. A one on the first dice can goes with any of the six values on the seconddice, and the same with a two, and so on, until you reach 36 possibilities. Now, compare the

probability of rolling a two, a three or a six. There is only one way to roll a two, so this has a

probability of 1/36. There are two ways to roll a three a two on one dice and a one on the other,

or vice versa. So this has a probability of 2/36, or 1/18. So, a three is twice a likely as rolling a two.

Finally, for a six, you can have the pairs (5,1), (1,5), (4, 2), (2, 4) and (3, 3). So there are five

possible outcomes, for a probability of 5/36. If you do this for all the values between two and

twelve, you will find that the outcome with the highest probability is seven (6/36 = 1/6). Also, this

is an example of a symmetric case. For example, the probability of getting six is the same as

getting eight. It turns out, for a case like this, if you rolled the dice enough times and took an

average of your results, it would also equal seven! This is an example of the average and most

likely result being the same.

CASE 3: Consider a room with 10 people in it with the following ages: 15, 15, 20, 25, 25, 25, 30, 50,

74. The average age is 31. This is not even one of the choices! But, if you were to pick a person at

random, the most likely age you would get is 25, something very different from the average. You

should notice that this case is very asymmetric. There are a few very large ages (50 and 74), that

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