1. SIGNIFICANT FIGURES

1.1 MEASUREMENTS

Physics is called a quantitative science because it describes nature using quantities which are measurable, and seeks to discover mathematical laws relating these quantities. You may be already familiar with some of these quantities: length, time, mass, force, etc. To measure such quantities, we use measuring instruments in a prescribed way in order to assign a number (value) to that quantity. For example, to measure a length we use a measuring instrument (a marked meter stick) in a specially prescribed manner (laying the stick off on the object to be measured) to determine the numerical value we call the "length." Other measurements may require more complicated instruments, or more complex techniques, but the principle is the same.

The meter stick example also illustrates the uncertainties common to all measurements. The smallest division marks on the stick represent millimeters. These are quite small, but with a little practice, most persons can estimate fractions of millimeter. Estimates smaller than l/4 millimeter are less reliable. This limits the accuracy of measurements made with a meter stick.

The importance of this chapter. The rules for significant figures are often presented in textbooks as a way to do error analysis, to determine the appropriate precision for expressing answers. This is a relic of the days when computations were done by hand, and students were not expected to do a more sophisticated analysis of errors.

Today students use calculators and computers, so why should they know this stuff? Calculators spew out answers to many digits, most of them insignificant. But now, as in the past, experimental data and results must be communicated to others, in the form of written reports and published papers. The significant figure rules govern how both data and results should be expressed.

1.2 WAYS TO EXPRESS PRECISION

Suppose a small dust particle lies between two millimeter markings of the stick. We would like to specify the position of the particle. [This scale has its centimeter markings labeled 1, 2, 3, ... etc. Each centimeter is divided into 10 millimeters, which have marks, but are not labeled. Figure one shows a magnified view of a one centimeter portion of the stick.] Figure 1 illustrates the particle's position as seen by the observer. We might specify the position by saying. "The particle lies between 6 and 7 mm on the scale." This statement specifies a range in which the particle is known to lie.

The position might also be specified by stating it "to the nearest millimeter." Since the particle is closer to 7 mm than it is to 6 mm, we would say, "The position is 7 millimeters, to the nearest millimeter." This statement also specifies a range in which the particle is found, but in this case the range is from 6.5 mm to 7.5 mm. This method of stating scale readings is very common in physical science. Usually, we simply state the reading as "0.7 cm" without explicitly specifying that the position was read to the nearest millimeter. Writing the value in this way implies the precision of the measurement, for if we had attempted to read the position to the nearest tenth of a millimeter we might have expressed the value as 0.78 cm. [Bear in mind that on the real meter stick (not this magnified one) the digit "8" might not be completely certain.]

This convention may be formulated as a rule:

Rule 1 (a): Write values of physical measurements so that the last measured digit falls somewhere to the right of the decimal point. This may be done in either of two ways:

1. Use "scientific notation" (powers of 10).

2. Choose larger units of measurement.

Rule 1 (b): The digit representing the smallest measured scale division must be explicitly written, even if it is a zero.

Rule 1 (c): Rounding. Parts (a) and (b) of this rule tell you how to discard insignificant digits, a process called truncating. Some people get fussy about a special case that occurs, when you must truncate (discard) just one digit that happens to be a "5". Should the final result be rounded "up" or "down"? Some advocate rounding (altering) the last digit retained so that it is even. Thus "3.785" would round to "3.78" while "2.755" would round to "2.76". When many numbers are combined, this rule serves to minimize the bias introduced by consistently rounding "5" up (or rounding it down consistently). This is only important if you use many rounded numbers in calculations, which you are not likely to ever do. However, textbooks and published papers use this rule for expressing numbers, so it's good to know about it.

Example: A beam balance read to the nearest 1/10 gram reads exactly 30 grams. That implies that the reading is known to lie between 29.95 and 30.05 gm. The proper way to state this value is "30.0 gm." It is incorrect to simply write "30 gm" for that would imply a value in the larger range from 29.5 to 30.5 gm.

Example: The distance between two towns is measured to the nearest 10 meters and found to be 387,220 meters. To express this correctly we may write it in one of the following ways:

387.22 kilometers (rule l (a), 2)

3.8722 x 105 meters (rule l (a), 1)

3.8722 x 107 centimeters

etc.

1.3 SCALE-LIMITED DATA

When repeated measurements are made of the same quantity, reading to the finest scale division, it may happen that the measurements all yield the same value. This tells us that the precision of the measurement is primarily limited by the measuring scale; there are no other erratic influences causing the measured values to vary. Such data are called scalelimited, and will be treated by rule 1.

1.4 INDETERMINATE ERRORS IN DATA

When repeated measurements of a quantity do not yield the same value, there may be some erratic influences on the measurement, or in the measuring process, larger than the smallest readable unit on the scale. The repeated measurements show a scatter of values. The scatter may have limited extent, so the measurement isn't completely uncertain. But we cannot predict (determine) exactly what the next measured value will be. Therefore these uncertainties (errors) are called indeterminate. Among many possible causes of indeterminate errors are:

(1) Attempting to read an instrument scale to a very high precision. For example, trying to read a scale to the nearest 1/10 of its smallest division requires difficult estimation which may be highly unreliable.

(2) Mechanical irregularities in the measuring instrument. For example, readings taken from a beam balance may be affected by friction and wear of its mechanical parts. Such effects may be reduced but never completely eliminated.

(3) Uncontrolled (or unnoticed) outside influences on the apparatus.

(4) Careless technique or observation by the experimenter.

Whatever the cause, indeterminate errors reveal themselves when repeated measurements give different values. A typical set of values for a measurement might look like this:

	3.69   3.68   3.67   3.69   3.68   3.69   3.66   3.67

We assume that the measured quantity has just one precise value, independent of the measuring process, and that the variability of the recorded values is caused by imperfections of the instruments or procedure. We want to represent our knowledge, obtained from these measurements, as one "best" value.

We might represent this measurement by the mean (average) of the measured values. The mean is 3.67875 exactly. Recognizing that all of these digits are not meaningful, we round this off to 3.68, retaining the certain digits 3.6 and the first uncertain one, the 8.

The value 3.68 is still somewhat uncertain. Nothing to the right of the 8 was certain enough to keep, but the 8 itself is a borderline case; it is not completely meaningless. We could be more precise and say that our value 3.68 is not likely to be "wrong" by more than ±0.02. The value ±0.02 is an estimate of the uncertainty in the value 3.68. Such results are written in standard form:

3.68 ± 0.02

Estimates or measures of uncertainty are called errors. In this case we have quoted a "maximum error" in the value. We will introduce other, better, kinds of error measures later.

Unfortunately data errors propagate through calculations, usually producing even worse error in the results. In the following discussion we review the "rules for significant figures", a crude method for ensuring that calculated results are stated to a precision consistent with the precision of the data. The student may already be familiar with this method, but this organized review of the rules may be instructive.

1.5 TERMINOLOGY

To illustrate the meanings of the technical terms, consider the number

3586.297

It has seven digits, four to the left of the decimal point and three to the right of it.

Each digit occupies a particular decimal place. The digit 7 occupies the third decimal place to the right of the decimal point. The digit 8 occupies the second decimal place to the left of the decimal point.

This number is said to be "expressed to three decimal places" since three digits are explicitly shown to the right of the decimal point. The numbers which appear in your calculator display may have an many more digits, but we know that many of those are of no significance, and are unreliable.

The fifth decimal place to the left is zero, as are all other places still further left. They are not shown.

When we say a number is expressed "to three decimal places" we mean that three digits are shown to the right of the decimal point. When we say a number is expressed to "seven digit accuracy" we mean that a total of seven digits are certain enough to be explicitly shown, as in the number we are considering here.

Now suppose that the number 3586.297 cm represents an experimental measurement, and we experimentally determined that its uncertainty was 0.2 cm. The size of the uncertainty tells us that the digits 9 and 7 are superfluous, and carry no significant information. They express an amount smaller than the uncertainty. Such digits are called insignificant. Insignificant digits can arise from mathematical calculations. Calculation devices such as electronic calculators display insignificant digits. They may also arise from reading an instrument scale beyond the inherent precision of the instrument.

In example (1) the digits 3, 5, 8, and 6 are called "certain" digits, because the uncertainty is too small to affect their value. The 9 and the 7 are completely uncertain.

What about the digit 2? It is not certain, because the size of the uncertainty ( 0.2) tells us that it might have a value ranging anywhere from 0 to 4. Yet it is not completely uncertain, as are the 9 and the 7. The digit 2 is uncertain, but still significant.

This terminology may be summarized with a diagram.

These names always relate to the first uncertain digit, not to where the decimal point happens to be.

1.6 RULES FOR SIGNIFICANT DIGITS

We customarily drop insignificant digits when recording data and stating results, according to the following rules.

Rule 2: First discard all insignificant digits except the leading one. Then round off this uncertain digit.

So the proper way to write the measurement discussed above are rounded to four significant digits: 358.3 cm. This is straightforward ecept in the case where the leading insignificant digit happens to be a 5. Some books advocate a rule to handle this in an unbiased way:

Rule 3: If the first uncertain digit is a 5, round it up or down as necessary to make the result even. [You could round up or down to make the result odd, so long as you adopted the same rule consistently everwhere in the calculation.]

This rule ensures that over many calculations you won't be introducing systematic error in one direction (up or down).

Example of rule 3: A set of data has an average value of 3.645987 cm. The uncertainty is found to be 0.03 cm. Discard the 987. The first uncertain digit is 5. Round it down, to give the result 3.64 if you are using the "even" rule. You may object that the 5987 would suggest rounding up. But remember, we first discarded those digts 987 as insignificant, which means that digit is totally unreliable for any purpose. We must take the word "insignificant" seriously. It would be a waste of time to apply rounding rules tediously to a string of insignifant digits.

Rule 3 is a bit of "overkill", considering that significant figures rules are themselves only an approximate indication of quality of a result. Little is lost by simply discarding all insignificant digits. If you are taking a course which expects you to use this rule, you may find your results sometimes differ from the "book values" by 1. That's no "big deal".

1.7 PROPAGATION OF UNCERTAIN DIGITS

An important feature of experimental data is that the errors combine and propagate through calculations to produce errors in the calculated results. The concept of significant figures is a crude and inadequate tool for dealing with this (we'll introduce better ways later). But it is instructive to consider how insignificant figures are generated in simple calculations. The operations of these examples are not ones you would normally do longhand.

Consider this multiplication example, in which uncertain digits are shown in bold italics.

    3954
x    286
 ———————
   23724
  31632
  7908
————————
 1139844

Multiplying 3954 by the uncertain digit 6 gives a number in which every digit is uncertain. In the other sub multiplications, digits resulting from multiplication by the uncertain digit 4 must be counted as uncertain. Any column addition containing uncertain digits gives an uncertain result. Therefore only the first two digits of the answer are certain. The three is uncertain, and the remaining digits are completely uncertain.

Therefore this result should be rounded to 114. Notice that even though the multiplicand had four significant digits, the result has only three. This illustrates a rule:

Rule 4: Multiplication or division. Results of multiplication or division are rounded to the same number of significant digits as the least accurate data quantity.

The rule for addition is different because the decimal location of the first uncertain digit determines the location of the first uncertain digit in the sum.

Rule 5: Addition or subtraction. Find the data quantity whose last significant digit occupies the leftmost decimal place. This is the position of the last significant digit of the result.

Example.

 0.52865
39.42
15.1
 0.02896
—————————
55.07761, which should be rounded to 55.1

Rule 6: When other operations are encountered, analyze the computational procedure as we have illustrated above.

Example. Find the square root of 9.4263. Analyze the longhand calculation:

The seven and the zero in the answer are certain because their value does not depend on the uncertain digit 3. The 2 is the first uncertain digit.

A practical common-sense method to determine the number of certain figures is to use the calculator directly. The digit 3 in the number 9.4263 in the above example is uncertain, but how uncertain? With such crude information, we can only say, conservatively, that its uncertainty might be as large as ±0.0005. Add this to 9.4263 to get 9.44268. The square root of 9.4268 is 3.0703094. Now subtract 0.0005 from 9.4263 to get 9.4258. The square root of 9.4258 is 3.0701466. Examining these results, we see that we should express the square root as 3.0702, retaining the first uncertain figure. It happens to have the same number of significant figures as the number we began with, but its percent uncertainty appears to be about three times as great.

These examples demonstrate the inadequacy of any significant figure rules for a serious analysis of uncertainties. Later, when we introduce better mathematical tools for dealing with uncertainties, we will see that the process of taking a square root reduces the percent uncertainty by about half.

Electronic calculators are so inexpensive these days that every student has one, and no one would likely do a square root longhand, as we have done here. The square root displayed on the calculator screen is 3.070228, which gives not the slightest clue how many of the digits are significant. Obviously we need to examine this matter of uncertainties in more detail, with the goal of developing a relatively simple and reliable mathematical process for estimating the precision of results of mathematical calculations. This will be done in subsequent chapters.

© 2004 by Donald E. Simanek