Albert Einstein

Mr. Mocarski

Measurement in some form is the essential act of science because it is fundamentally an observation. Measurement is a particular kind of observation in which one object is compared to another.

Two types of observations can be made. One is called quantitative and the other is called qualitative. An example of a qualitative observation might be someone saying "Man is it ever hot today" or "Geez that band was too loud and the singer's shirt was so yellow that it hurt my eyes." This type of observation does not produce a number and neither is it based on a firm point of comparison. Qualitative observations do not really qualify as measurements. Qualitative observations, although important, are less useful to the scientist than quantitative observations which are comparisons to some standard which produce a number that tells the difference between one thing and another.

Quantitative observations produce numbers that describe an object or an event and those numbers can be used to accurately describe and predict the properties and behavior of that which has been observed. Quantitative observations, or measurements, are the types of observations that scientists must make if they are going to do useful scientific work.

Lord Kelvin, a very well known 19th century scientist, put it something like this. If you can measure something and express it in numbers then you can say that you really know something about it. If you can't measure what you are studying then your knowledge of it is meager and unsatisfactory.

Lord Kelvin was right. If you want to do science that can be called science you are going to have to make measurements.

A standard of measurement is a quantity (an amount) that has an exact definition and is used as a point of comparison in measurements.

Any definition will do but over time two sets of standards have come to dominate our measurements. In everyday use in the United States and the United Kingdom the British system of standards is used. The rest of the world uses what is commonly referred to as the Metric system, more formally the International System of Units, SI for short. Neither system is inherently better than the other but SI is the system of choice for scientists because it is generally easier to use and is used in more countries than the British standards.

The modern metric system has its roots in the French revolution (1789-1799). During this time the French were trying to revolutionize everything including their system of measurement. A rational system based on physical quantities found in nature and using decimal arithmetic was devised. The intent was to make a system that made scientific sense. After all it was the "Age of Reason."

The fundamental unit was defined as the meter (from a Greek word meaning 'to measure') and this was defined to be 1/10,000,000 of the distance from the north pole to the equator of earth. From this standard of length a standard volume can be derived because volume can be expressed as the product of three lengths. The standard of mass was defined as the amount of matter contained in one standard volume of the substance water. The French even tried to decimalize and change the definition of time standards. This didn't catch on but you can still find antique clocks with 10 hours in a day and a hundred seconds in a minute.

In 1875 17 nations including the United States signed a treaty agreeing to the metric standards. The system has been refined several times, notably in 1960, and 1971, to become SI the modern metric system.

Many quantities can be measured. Light, heat, sound, mass, weight, temperature, density, speed, time, and force are just a few examples that we will be using frequently. Most of these measurements luckily break down into combinations of just three fundamental measurements. These are length, mass, and time. For example speed is a measure of distance and time. Density is mass and volume, and volume is really length cubed. By using the fundamental quantities,length, mass, and time we can derive just about any quantity that we need. This idea separates measurements into two kinds, those that require one measurement, fundamental and those that are combinations of fundamental measurements known as derived quantities.

The SI standards for the 3 fundamental quantities are defined in the following table. By the way, the term unit means 'one' so when we speak of a unit of measurement we are talking about a label that describes 1 amount of something.

The kilogram is the only standard that is still defined by a physical object. Before the modern definition of a meter length was also defined by an actual object, a platinum - iridium bar that was stored in France.

Standards are very important because they allow us to tell how much of something that we have. Governments have agencies that are devoted to their maintenance and refinement. The ultimate authority is the International Bureau of Weights and Measures headquartered quite fittingly in France. The United States maintains the National Institute of Standards and Technology in Washington D.C. So if you really want to know how much of something that you have you can send it to these places for a precise comparison to a standard.

Four other fundamental standards have been defined in the SI. They are ...

• Ampere - electric current
• Kelvin - temperature
• Mole - an amount of substance
• Candela - light intensity

It is a good idea to define some terms here.

Many students try to define width, circumference, and perimeter as separate quantities. They are not. They are really variations on the fundamental measurement of length. There is no difference and this becomes important in solving some types of distance problems. A circumference is a length measured around a circle. Width and height are just lengths measured in different directions. Perimeters are lengths measured around objects.

Derived quantities do need to be defined separately. They are multipart or multidimensional measurements. Area for example has two dimensions. Two measurements of length are combined to indicate an amount of surface.

area- an amount of surface.

A = L X L (rectangle)
example: 1 cm X 2 cm = 2 cm²

A= ½ bh (triangle)
example: ½ 5 cm X 8 cm = 20 cm²

Notice that the label of the derived quantity area is squared in these examples. It shows that two dimensions of length were measured. Volume is similar but uses 3 dimensions to determine an amount of space.

volume- an amount of space.

V = L X L X L (cube)
example: 2 cm X 2 cm X 2 cm = 8 cm³

Again notice that the label shows the three dimensions that were measured.

density- how tightly packed matter is.

A convenient feature of the metric system is its use of decimal arithmetic. The British system is very awkward to use because of the weird numbers involved in conversions. To convert 15 inches to miles you would have to do the following calculation. 15÷12÷5280. A similar problem in the metric system would be to change 15 centimeters to kilometers. The calculation looks something like this. 15÷100÷1000. Most people, if they know their arithmetic, can do the metric conversion without a pencil and paper by mentally sliding the decimal over 5 places to the left to get 0.00015 km.

In the British system larger multiples and smaller divisions of a basic unit are given new names which denote the factor by which the basic unit is changed. One inch is 1/12 of a foot. One yard is 3 times a foot. One mile is 5,280 times a foot. This gets to be very cumbersome especially since different units of measurement require completely different names to show the changes. For example pounds can become ounces and tons while gallons change into pints, quarts, and cups.

In SI everything is much simpler. Prefixes are attached to the beginnings of labels to show larger and smaller multiples and the meanings of the prefixes are always the same. For example whether you see a millimeter or a milliwatt you know that both measurements are 1/1000 of the basic standard because the prefix milli always means divide by a thousand. The prefixes are very easy to use because they are always multiples of 10. Remember? The metric system was designed as a decimal system so that it would make sense and be easy to use.

The following table lists most of the SI prefixes and their values as powers of 10. 10⁰ means 1 or the basic label. In the table the abbreviations for the prefix is in bold type and the abbreviation for the basic label is in ordinary text.

Sometimes students get confused because the SI prefixes listed above are separated from each other by a factor of 10³ (1000 times) and older metric prefixes that are still in use are separated by a factor of 10¹ (10). You may have seen these prefixes.

Only two of these older prefixes are still in common use. They are deci and centi. Of these two you will most often see centi.

It might help you to visualize the relationship between these prefixes with the help of a backward number line. The largest multiples are to the left and the smallest to the right with the base unit in the center.

It is often necessary to change one label into another so that you can add or subtract or manipulate quantities in some other way. It is easy to do this using metric prefixes. You simply determine the decimal difference between the two labels and move the decimal point that many times to the left or right. That's where the backward number line becomes a very useful mnemonic device. Start out with the prefix that you want to change and look for the prefix into which you want to change it. If it's to the right you move the decimal to the right that many places. If it's to the left the decimal slides to the left. This makes sense if you think about it because when you want to change a big label into a smaller label, for example, you've got to multiply. That means the decimal place moves right. That's exactly what the backward number line shows you.

Here are two examples.

Since centi is 10^-2 and Mega is 10⁶ you'll have to move the decimal 8 places to the left. You move it to the left because Mega is way bigger than centi. The line helps you to see this because Mega is to the left of centi. The answer is 2.6 X 10^-8Mm or .000000026 Mm.

As easy as this might seem students often make mistakes in converting from one label to another. The two most common errors are forgetting about the weird lower values of deci and centi and moving the decimal the wrong way. In other words multiplying when they should divide or vice versa.

Try the following examples.

1. 2.7 mV = _______ V
2. 0.09 pm = _______ mm
3. 1021 m = _______ km
4. 0.97 GW = _______ W
5. 81,852 ml = _______ l

The answers are ...

1. 2,700 V or 2.7 X 10³ V
2. 0.00000000009 mm or 9.0 X 10^-11mm
3. 1.021 km or 1.021 X 10⁰ km
4. 970,000,000 W or 9.7 X 10⁸ W
5. 81.852 L(l) or 8.1852 X 10¹ L

Because of the way volume and mass were derived from the meter there are a couple of complications that commonly arise in science problems that involve volume measurements and mass and volume measurements where the substance measured is water. Remember these two facts when using metric units.

1. 1 cm³ = 1 ml
2. 1 ml of water = 1 g of water

When the metric system was devised in the late 18th century the fundamental starting point was the meter. The meter was used to define the standards of volume and mass. The liter was defined as a volume (amount of space) equal to a cube measuring 10 cm on each side. If you think about it 10 cm X 10 cm X 10 cm equals 1000 cm³. Since a liter was defined as the basic label of volume (10⁰) a liter contains 1000 ml according to the prefix scheme. That means that 1000 cm³ must equal 1000 ml so that 1cm³ is equal to 1 ml. A ml and a cm³ are exactly the same volume.

Similarly a gram of mass was defined as being the amount of matter in 1 cm³ of water and therefore in one ml of water. So a gram of water has a volume of either 1 ml or 1 cm³.

You can test your understanding of these special relationships with the following examples.

1. 1 L = _______ cm³
2. 1 dL = _______ cm³
3. 1 L of water = _______ g of water
4. 2.7 g of water = _______ L of water
5. 17.5 L of water = _______ kg of water

The trick is to recognize that these are special relationships that don't follow the regular prefix scheme. In number 1 you see that since cm³ are also ml the problem really is to change L into ml and express the answer as cm³. the same thing is true of example 2. In 3, 4, and 5 you need to see that a ml of water is a gram of water so that for instance in number 3 you are really changing liters to ml and expressing it as grams. Number 5 is sort of special because it illustrates that in 1 liter of water there are 1000 ml which are 1000 g which is equal to 1 kg. Whew!

The answers are ...

1. 1000 cm³ or 1 X 10³ cm³
2. 100 cm³ or 1 X 10³ cm³
3. 1000 g or 1 X 10³g
4. 0.0027 L or 2.7 X 10^-3 L
5. 17.5 kg or 1.75 X 10¹ kg

It is impossible to make a perfect measurement. The only exact quantities are ones that we define. The only kilogram that is exactly one kilogram, for example, is the standard kept in France and it's only exact because we say it is. It is impossible to make a perfect measurement. The way that matter and energy are put together actually preclude even the possibility of a perfectly accurate and precise measurement. You'll learn more about this later.

There are two ways that you can make errors in measurements one is bad and the other we can live with.

The bad kind is called a systematic error. This kind of error is the result of making the measurement incorrectly. It is an error in the design of the experiment or the instrument making the measurement. It is an error in the system that you are using to make the measurement. A couple of examples might help you understand.

If you are riding in the front passenger seat of a car and look over at the speedometer you might think that the driver is going faster than he is. The reason for this is that you are looking at the speedometer from an angle and you see the needle and the numbers in a different relationship to one another than the driver. By the way, this is called parallax error. The important point is that no matter how many times you make the measurement you'll always read the speed as being higher than it is. You can never average out your error by taking many measurements. The error is built in and always occurs. Very bad.

Another example can be illustrated with a pendulum. The rate at which a pendulum swings back and forth is mostly determined by the length of the pendulum. The length is the distance between the pivot point and the center of mass swinging at the end. Suppose that you unknowingly measure the length from the pivot to the end of the mass. Your length measurements will always be longer than the measurement for which you are looking. Always. No matter how many times you make the measurement you'll have a pendulum that is too long. Very bad. There is no way to average out your error.

Systematic errors have to be carefully guarded against. The trouble is sometimes we can't recognize that we are doing anything wrong. Evil things these systematic errors.

The kind of errors that we can live with are called random errors. These are errors that are unpredictable and unavoidable facts of the universe. If we are careful in making our measurements we will sometimes get one that's too low and sometimes get one that's too high. We can never be perfectly accurate and precise but at least if we make a lot of observations we can average them so that the high ones cancel out the low ones and we wind up somewhere in the middle. That's why we can live with these types of errors. They exist because we can't avoid the fact of imprecision, it seems to be built into the universe, but we can average it out.

Measurement is an integral component of scientific observation. Measurements are performed to explore the characteristics of unknown phenomena and also to corroborate theoretical conjecture. To perform measurements, scientists employ "instruments." Measuring devices are called instruments because, much like musical instruments, they must be calibrated against known standards to ensure acceptable performance. This is analogous to using a pitch fork to ascertain that the notes of a piano are at the correct tone.

It is next important to distinguish between accuracy and precision in making measurements. Contrary to common usage, these words are not interchangeable. Nor are they mutually exclusive. Accuracy is the term used, in scientific context, regarding the agreement of a particular measurement with the correct value. An accurate value is a true value. Precision is the term used in science to describe the reproducibility of a measurement. The more precise a measurement, the closer each value is on repeated reading.

Measurements are much more likely to be completely precise than completely accurate. Given the limitations of most measuring devices, particularly those we use in introductory chemistry, measurements are typically only certain to a few significant figures. Within the limitations of these devices, repeated measurement will very often be reproducible. On the other hand, for these same reasons no measuremnt is totally accurate. We simply cannot "see" all possible decimal places using any measurement tool. When scientist study new phenomena, there are no benchmarks by which to tell if measurements are accurate. In this instance, a scientist has no way of telling if a value is accurate, other than there is precision upon repeated measure by him or others.

So there is always a degree of error in the accuracy of a measurement. This error is called a random error, because the measurement is equally likely to be high or low. Sometimes this is called an indeterminate error, because it is solely the fault of the limitations of the measurement tool. The last significant figure of any measured quantity contains this random error, and may be equally high or low.

A measurement may also have a systematic error. This is the result of either a faulty measurement or an inappropriate measuring device. A systematic error is simply wrong. This is sometimes called a determinate error, because it can be corrected with a properly functioning measuring device, or the correct choice of apparatus.

The dartboards to the left provide classic examples of both the differences in precision and accuracy, and the differences in random and systematic errors. The top board shows large systematic and large random errors. The values are simply all over the place. The middle board displays small random errors, but large systematic errors. The measurement is reproducible but not true. The bottom dartboard represents the case of both small random and small systematic errors.

• A measurement is PRECISE if it is reproducible after multiple attempts at reading it.
• A measurement is ACCURATE if it is close to the true value.
• A RANDOM ERROR may be equally high or low. This error can be minimized but is unavoidable.
• A SYSTEMATIC ERROR is consistently wrong because of limitations of experimental design or apparatus.

These two terms are often misused. They have definite meanings to scientists. If you are in the secret science club, that is if you know all the little cool details, you will never misuse these words.

Accuracy is how close you actually come to a standard in your measurement. For example I can accurately guess how much money is in your pocket right now. It is somewhere between 0 and 5000 dollars. Right? This is an accurate statement. I came close, within the margin of error that I defined, to the actual value. The trouble with my guess is that it is not very precise. Another similar example is if you wanted an accurate balance, you could purchase one pretty cheaply if it measures to the nearest gram. On the other hand if you want one that measures accurately to the nearest ten thousandth of a gram you'll have to pay dearly for it.

Precision is how tightly many measurements cluster together around the true value. Remember that you can't ever be perfect but precision in your measurements means that your knowledge of a measurement is close to the true value. You can usually see that someone is saying how precise they are by noting how many decimal places they include in their measurement. Precision does not guarantee accuracy. You can measure a mass to the nearest .0001g but if your expensive balance is not accurately set (calibrated) against some standard your measurement will be a bad one.

Good measurements are both accurate and precise. Scientists usually include an estimate of how good their measurements are. You'll see this as a range indicating how much they think that they might be off. This is usually expressed as a ± value or a percent.

It is meaningless to report a number with a precision of 17 decimal places when your cheapo economy balance only measures to the nearest gram. The only numbers that are a result of the measurement, that really matter, in this case would be the whole numbers to the left of the decimal.

In scientific measurements you have to be careful to tell everyone what numbers you actually got from your observations. For example if you measure the density of an object with a balance that measures to the nearest gram and a ruler to the nearest centimeter you might get the following data. Mass = 9 g. Volume = 7 cm³. When you divide that through, your calculator will tell you that you have a density of 1.28571428571 and some more g/cm³. If you report your findings as such other scientists would assume that you had very precise instruments. They'd surely be jealous and start to complain to their superiors about the abysmal equipment that they are being forced to use.

To avoid such envy and strife you should only report the numbers that you actually get from your instruments. These are called significant figures. The basic idea is that you can never be more precise than your least precise measurement. In our density example the number would have to be rounded to 1 g/cm³ because your instruments are not precise enough for you to be sure about any of those decimals.

There are some simple rules for finding and using significant numbers in your calculations and measurements that are best left to your chemistry teachers. You will learn them eventually but we will be content here with rounding to the decimal place indicated by our least precise measurement.