This material is based upon work supported by the
National Science Foundation
under Grant No. DUE-0336493

 


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Making Good Measurements
 

 

 


Revised: 1/24/2011

 

 

Text Box: “A measurement is a comparison to a standard” - William Shockley (Father of the Transistor)

 

 

 

Contents:
 

 


·         Introduction

·         Standard Units and Measurement Systems

·         Making the (Almost) Perfect Measurement

·         Significant Figures

·         Powers of Ten

·         Rounding

·         Precision V. Accuracy

·         Types of Experimental Error

·         Reporting Results of Measurements

·         Summary

Introduction
 

 

 

 

 

 

 

 


The United States officially uses the English system of measurements. This is also referred to as the U. S. Customary system as well as the British Engineering System. The unit of length is FOOT whereas in the metric system, the unit of length is METER. The United States is the only industrialized country in the world that has NOT officially adopted the metric units as a standard. The roots of the metric system can be traced as far back as 1670 France. Take this link to a Chronology of the metric system.

 

 

A measurement is defined as the ratio of the magnitude (how much) of any quantity to a standard value. The standard value is called a unit. A measurement of any kind requires both a magnitude and a unit.

 

In order to begin the study of a subject, first, we must establish a common language to be used by instructors and students when discussing that subject. In science, an extremely important part of this language is the measurement of various quantities, but this is just not possible unless a good system of units has been established and is understood by everyone. As you begin your studies, you will soon learn the definitions of many new terms that have specific meanings when applied to your course of study. We must develop the basic definitions necessary to take measurements and understand those measurements.

 

Measurements always include error. Part of the error in making measurements is due to the skill of the person making the measurement, but even the most skillful among us cannot make the perfect measurement. Regardless how small we make the divisions on our ruler we can never be sure that the length of the object we are measuring lines up perfectly with one of the marks. No matter how fine the measurement, there is always another decimal places that must be estimated. Therefore, the judgment of the person doing the measurement plays a significant role in the accuracy and precision of the measurement.

 

Making accurate and precise measurements helps us to improve our understanding of the world and enables us to predict the outcome of future measurements. The basics of measurement must be mastered before we can perform meaningful laboratory experiments to test the theory leaned in the classroom. However, when using our senses to make these measurements, we must be careful because this information can sometimes be deceiving. For example, touching 70o water outside on a very cold day may feel warm to the touch. However, the same 70o water on a very hot day may feel cool to the touch.

 

Simply recording a numerical value for the measured quantity is not sufficient to express a physical quantity. A unit must also be indicated. For example, a measurement is taken for the length of the laboratory table and recorded as 183. This number has no real meaning unless expressed with a unit. A correct recording would be 183 centimeters.

 

Standard Units and Measurement Systems
 

 

 

 


There are three fundamental quantities in which all measurements are taken:

·         Length

·         Mass

·         Time

 

Text Box: Standard units are internationally-accepted references used to define the size of the units used when discussing fundamental quantities.

 

 

 

It is very important to differentiate between standard units and fundamental quantities. Scientists, worldwide, use the International System of Units (the SI System) when making measurements and calculations. The SI system is an updated version of the metric system. One obvious difference between the two measurement systems involves units for volume. The metric system uses liter while the SI system uses  for volume. The following table shows the fundamental quantities upon which the SI system is based.

 

FUNDAMENTAL QUANTITIES

SI UNITS (abbreviation)

Length

meter (m)

Mass

kilogram (kg)

Time

second (s)

Electric current

ampere (A)

Temperature

kelvin (K)

Amount of substance

mole (mol)

Luminous intensity

candela (cd)

 

Almost all we know about the universe comes from measuring these quantities. The majority of our knowledge comes from measurements of mass, length, and time alone.

 

The standard units for length, mass, and time in both the metric system and the SI system can be best remembered by using the acronym MKS system. The three letters, MKS, stand for meter, kilogram, and second. Chemists prefer another system, cgs, because it uses smaller units. Here cgs stands for centimeter, gram, and second. One of the greatest advantages of the metric system is that it is a decimal (base-10) system. Therefore, the metric system allows for easy conversion to larger and smaller units by simply shifting the decimal.

 

Making the (Almost) Perfect Measurement
 

 

 

 

 


Text Box: A wise man once said, “With any measurement, comes error.”

 

 

 

 

Of course, in order to make a measurement we first need to specify the quantity that we want to measure. It must, not only, be measurable, but its value must also be re-measurable. Without this "repeatability" of the measurement our results would go against one of the most important requirements of “The Scientific Method”: "empirical replication".

 

Using a measuring instrument to compare the unknown quantity to a standard and then expressing the comparison as a number first determines the magnitude of the quantity. A number representing the magnitude is not sufficient to express the measurement, however; a unit must also be assigned to the number. For example, to record a person's height simply as 58 is meaningless. We must record it as 58 inches. All measurements, therefore, must be recorded by listing a number and a unit.

 

Many instruments such as rulers, balances, clocks, speedometers, thermometers, voltmeters, etc., are used to make measurements. The information obtained from these instruments must be recorded and evaluated in order to obtain a truer value of the properties of the physical quantity. Such terms as least count, significant figures, precision, accuracy, percent difference, and percent error must be understood and used.

 

Laboratory experiments require the taking and recording of data using a measuring instrument of one kind or another. The fineness of the measuring scale on the instrument is known as the least count of the instrument.

 

 

Text Box: The least count is the value of the smallest scale division of the measurement instrument.

 

 

Example:

 

A ruler with marks 1 millimeter apart has a least count of 1 millimeter. The smaller the least count the more precise the instrument is said to be. For example a metric ruler that has millimeter marks has a least count of 1 mm or 0.1 cm and when properly used it can make more precise measurements than one with only centimeter marks.

 

A measurement of 1.2374 meters was made with a stick having millimeter marks. Significant digits (or significant figures) are digits read from the measuring instrument plus one doubtful digit estimated by the person making the measurement. This doubtful estimate will be a fractional part of the least count of the instrument.

 

Examples:

 

Ø      203.4 cm: least count of ruler was 1 cm, tenth of centimeter was estimated. Here, the 4 is doubtful.

 

Ø      4.07 cm: least count was 0.1 cm. hundredth of centimeter was estimated. Here, the 7 is doubtful.

 

Below are two examples of 'proper' measurements, showing how the last digit is always ‘estimated’ or ‘doubtful’. The doubtful digit is a 'best estimate' of the tenth fraction of the smallest scale division on the instrument. For example, consider this measurement:

 

 

 

 

 

 

 

 

 

 

 

 

 

 


The length of the blue line is "five point something"; more than five but less than six. With certainty we can say it is in the range of 5 to 6. Exactly where in that range is an estimate. We might estimate that it is 20% of the way between 5 and 6 or 5.2.

If we had more marks on the scale we could make a better estimate.

 

 

 

 

 

 

 

 

 

 

 

 

 


It is clear that our estimate was close, but the measurement still does not align with a mark. Also, we may have trouble reading the marks now because they are so fine. Now we can estimate with a higher precision that the measurement is 5.28, although you might say that it is 5.27 or even 5.29.

 

This method of “estimating the last digit” applies to the measurement of many different quantities. Often, we measure a quantity indirectly. For example, we measure the height and diameter of a cylinder, and then calculate the volume using the measured values. Also, we may measure time using a stopwatch. See the figure below. Here, the time is clearly between 1:00 and 2:00. The minute hand indicates the time is between 12 and 13 minutes past the hour. As with measuring distance, we would estimate the last digit, possibly, to be 12.4 minutes.

 

 

 

 

 

 

 

 

 

 

 

 

 


Text Box: When using a digital measuring instrument, you will not estimate the last digit. The instrument has already done that before displaying the answer.

 

 

 

Note the distinction between 5.2, 5.20, and 5.200. The three numbers indicate that the measurements were made with three different instruments. If you do not understand why they are different, carefully read the section on significant figures below.

 

 

Significant Figures 

 


When you make a measurement, it is simply a comparison of an unknown physical quantity with a standard unit. No measurement or answer to a problem is complete until both the number and the units have been specified. You must always give complete answers to problems and numerical questions in such a way that both the number and the units are clearly stated. Also, any measurement of a real physical quantity should also be accompanied by some estimation of the error involved in the measurement.

 

Errors are always present to some degree in any measurement, and the actual accuracy of a measurement must be expressed by the formatting of the number itself. Values read from the measuring instrument are expressed with numbers known as significant figures. The number of significant figures used in the number portion of any measurement indicates how precise the data is.

 

Text Box: A significant figure (abbreviated SF) is one that has been measured with certainty or has been 'properly' estimated. The significant figures in a measured value include all digits as read from the instrument plus one doubtful digit.

 

 

 

To keep from implying unwarranted precision in a number than has actually been measured or calculated, it is often necessary to drop digits from the end of a calculated quantity. For example, I tell you that I weigh 170 LB. If you record this as 170.00 LB, you imply unwarranted precision. I told you my weight to the nearest pound. However, you indicated that you know my weight to the nearest hundredth of a pound. Similarly, attempting to measure the width of a human hair using a meter stick with 1 cm as the smallest division is absurd. It is imperative that you know, and can use, the rules for rounding a number to the specific number of digits that are required in any given situation.

 

Below is a good example of why you cannot simply report your answer using all digits displayed on your calculator. The sides of the yellow box are being measured and the area is then calculated.

 

The least count of the ruler is 1 mm (0.1 cm). Doubtful digit is hundredth of 1 cm (0.01 cm).

 

Length is between 1.7 cm and 1.8 cm. We will estimate 1.72 cm, which is 3 SF. The doubtful digit is 2.

 

Width is between 1.5 cm and 1.6 cm. We will estimate 1.58 cm, which is 3 SF. The doubtful digit is 8.

Calculated Area = length (width)

                        = 1.72 cm (1.58 cm)                            = 2.7176 cm2

                        = 2.72 cm2       (to 3 S.F.)

 

The true area lies somewhere in the range of:

A = 1.7 cm (1.5 cm) = 2.6 cm2

and

A = 1.8 cm (1.6 cm) = 2.9 cm2

 

Once the measurements have been made and recorded, it may be necessary to perform arithmetical operations using the significant figures. You cannot increase the precision of measurement by doing arithmetic operations such as addition, subtraction, multiplication, and division. This can be confusing because the calculator will return a large number of decimal places, especially when performing division. Go here, for a review of Significant Figures.

 

Powers-of-10 Notation
 

 

 

 


Another important mathematical tool is the use of powers-of-10 notation when very large or very small measurements are recorded. Many areas of physical science have come to rely heavily on this form of notation, and you will regularly see it in scientific writing. The exponent on the 10 is used simply to place the decimal point; large positive exponents represent very large numbers and large negative exponents indicate very small numbers (less than one). Powers-of-10 notation is also needed to specify the proper number of significant figures in a number when there are leading or trailing zeros needed to properly locate the decimal point. In such cases, powers-of-10 notation assures that no ambiguity in notation is present when the number is properly written.

 

Your textbook also explains the use of prefixes on units to adjust the size of these units so that they are more appropriate in certain measurements. As an example, the prefix kilo can be applied to a unit such as meter to construct a unit for length measurement (kilometer) that is 1000 times larger than the basic unit (meter) and which is more appropriate for discussing the distances between cities or the speed of an automobile than the base unit, meter, would be.

 

 

Rounding
 

 

 


Once you have determined how to report your answer, that is, how many significant figures it should contain, use the following procedures to round the number to the correct number of significant figures.

 

Starting with the left-most digit, if the first non-significant digit is less than 5, drop it.

 

Example:

 

22.361 rounded to 4 significant figures is 22.36 (the 1 is the first non-significant digit).

 

Starting with the left-most digit, if the first non-significant digit is greater than 5, drop it and increase the preceding digit by one.

 

Example:

 

15.47 rounded to 3 significant figures is 15.5 (7 is the first non-significant digit).

 

Starting with the left-most digit, if the first non-significant digit is 5, drop it and round up the preceding digit only to make it even.

 

Example:

 

23.85 rounded to 3 significant figures is 23.8 (the 8 is already even).

 

Example:

 

23.75 rounded to 3 significant figures is 23.8 (the 7 is rounded up to make it even).

 

The rationale for this last rule involves skewing your data when making a large number of measurements. Except for the instances where you drop a 5, you will round up about half the time and round down about half the time. Consequently, if you always round up when dropping the 5, your data will be slightly skewed to the high side. However, using the rule mentioned above when dropping a 5, you will round up approximately half the time and round down approximately half the time.

 

Precision vs. Accuracy
 

 

 


Most dictionary definitions of these two words do not clearly make the distinction as it is used in the science of measurement.

 

Text Box: Accurate: "capable of providing a correct reading or measurement." In physical science it means 'correct'. A measurement is accurate if it correctly reflects the quantity being measured.
Precise: "exact, as in performance, execution, or amount." In physical science it means "repeatable, reliable, getting the same measurement each time."

 

 

 

 

 

 

 

We can never make a perfect measurement. The best we can do is to come as close as possible within the limitations of the measuring instruments and our ability to make the measurement.

A digital clock may display the time as 9:28:36:21 which is extremely precise (to the nearest hundredth of a second) but it may be incorrect by one full hour which is quite inaccurate.

Suppose you are trying to hit the center of a target with four blue darts. Consider the pictures below.

Neither Precise Nor Accurate

 

This is a random pattern, neither precise nor accurate. The darts are not clustered together and are not near the bull's eye.

Precise, Not Accurate

 

This is a precise pattern, but not accurate. The darts are clustered together but did not hit the intended mark.

Accurate, Not Precise

 

This is an accurate pattern, but not precise. The darts are not clustered, but their 'average' position is the center of the bull's eye.

Precise and Accurate

 

This pattern is both precise and accurate. The darts are tightly clustered and their average position is the center of the bull's eye.

 

Attempting to reduce error in accuracy makes for good experimental technique, but there is little we can do about errors in precision, except using a more precise measuring device.

 

Types of Experimental Errors
 

 

 

 


Experimental errors can be generally classified as being of three types:

 

·         Illegitimate

·         Systematic

·         Random

 

ILLEGITIMATE ERROR

 

These type errors are due to carelessness in reading an instrument, in recording observations, mathematical calculations, or possibly an accident.  Examples of illegitimate errors:

 

·         Error in reading a scale, usually due to incorrect alignment of the line of sight.

·         Recording the wrong measured value.

·         Not observing the significant figures in a calculation.

·         An experiment may call for a ball to be dropped and timed for how long it falls. The ball may strike another object during its fall, negating the validity of that “run”.

 

A measurement should never be included if it is known to be faulty. If the cart is halfway down the track before starting the clock, the measurement must be discarded. However, be careful with this. You must never take a series of measurements and choose the ones you like. All reasonable measurements must be included. This is the purpose of the lab activity – to determine if the observation agrees with the theory.

 

SYSTEMATIC ERROR

 

Systematic errors are associated with particular measurement instruments or techniques, such as an improperly calibrated instrument or bias on the part of the observer.  Examples of systematic errors:

 

·         An improperly "zeroed" instrument or an instrument that is not properly calibrated.

·         Human reaction time when starting or stopping a clock. You may repeatedly stop the clock too soon or too late.

·         Personal bias of an observer, who, for example, always takes a low reading of a scale division.

 

Avoiding systematic errors depends on the skill of the observer to detect and to prevent or correct them. Experimental physics isn't just about making measurements; it's about making meaningful measurements.  Think hard about whether the number you obtain might include significant systematic errors. 

 

RANDOM ERROR

 

Random errors result from unknown and unpredictable variations in experimental situations.  Random error does not have any consistent effects across the entire collection of measurements taken. Instead, it causes measured values to be above and below the actual value. If the number of measurements is sufficiently large, there will be as many values above the actual value as there are below the actual value. Again, if the number of measurements is sufficiently large, these random fluctuations in the measurements would sum to zero. The effect of random errors can be reduced and minimized by improving and refining experimental techniques and repeating the measurement a sufficient number of times so that the erroneous readings become statistically insignificant.

 

Reporting Results of Measurements
 

 

 

 


The process of taking any measurement always involves some uncertainty. Such uncertainty is usually called experimental error. It is expressed as a ratio of the amount of error compared to the size of the thing being measured. For example, an error of 1 meter is not much when we are measuring the distance from the Earth to the Moon but it is a large error when measuring the length of a table that is two meters long.

 

The following two methods may be used to calculate the amount of error.

 

·         Percent Error

·         Percent Difference

 

1.         When an accepted or standard value of the physical quantity is known, the percent error is calculated to compare an experimental measurement with a standard.

 

Example:

The well-known value of acceleration due to gravity, to 3 SF, is . We can employ various ways to measure this quantity. We might measure it to be . Calculating the percent error yields:

 

 

 

Many texts will recommend using the absolute value. However, if we retain the sign, we will know if our error is above or below the accepted, or known, value.

 

2.         Percent difference is calculated in a comparison of two or more experimental measurements. This is used when no standard exists, or when it is desired to measure the precision of an experiment.

 

 

These techniques are suitable for students just beginning to learn about errors in measurement. However, more involved statistical techniques are employed for a more rigorous treatment of the subject.

Summary
 

 

 


Error in measurement is normal. We strive to increase the precision and accuracy of our measurements by using a precise measuring instrument and taking great care in estimating the doubtful digit. Also, we must be mindful of random and systematic errors. Our ability to recognize these errors increases with experience. However, it is fundamentally impossible to make an absolutely exact measurement.

 

 

 

 

 

                                  

 

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the
National Science Foundation.