SIGNIFICANT FIGURES 

N UMERICAL DATA that are used to record 

observations or solve problems are seldom exact. 

The numbers are generally rounded off and, consequently, are estimates of some true value, and the 

mathematical operations or assumptions involved in 

the calculations commonly are approximations. In 

numerical computations, no more than the necessary 

number of digits should be used; to report results 

with too many or too few digits may be misleading. 

To avoid surplus digits, numbers should be rounded 

off at the point where the figures cease to have real 

meaning. Conversely, the number of significant figures may be unnecessarily reduced by choosing the 

less meaningful of several possible methods of calculation. Careful consideration, therefore, should be given 

to the significant digits and arithmetic involved in 

each measurement. 

The number of significant figures resulting from 

any calculation involving simple arithmetic operations 

on measured quantities should not exceed the number 

of significant figures of the least precise number 

entering into the calculation. In the calculation itself, 

one more significant figure may be retained in the 

more precise numbers than exist in the least precise 

number. 

The digits 1 through 9 are always significant, 

regardless of their position in a number. The digit 

0 is significant when it is between other significant 

digits but not when it is at the right or left of the 

number that locates the decimal point, because that 

location may be changed by changing dimensions-for 

example, grams to milligrams. At the right of a number, 0 is significant if it indicates actual precision, but 

not if it is used only to complete a rounded number. 

For example, the number 0.0046 has only two significant figures, but 4,103 has four significant figures. In 

a number such as 53,200 we do not know the number 

of significant figures unless we know whether the 

zeros at the end were actually determined experimentally. To remove this ambiguity the number may be 

written as 5.3200~ lo4 if the zeros are significant, 

and 5.32 x lo4 if they are not. Use of five significant 

figures indicates that the author knows that the two 

zeros have real meaning. Nonsignificant zeros should 

never be used at the right of the decimal part of the 

number. In tabulating data, an alternative is to list 

only the significant figures and absorb the superfluous 

zeros in the general heading, as follows: 

Specimen Tem rature (oE 103) 

A consistent procedure should be followed in rounding off numbers to n significant figures. All digits to 

the right of the nth digit should be discarded, as illustrated in the following six examples of rounded numbers, each of which has only three significant figures: 

Example Original number 

- 

Rounded number - 1 --. 0.32891 0.329 

2 - ----- -- 47,543 

3 -- ----- ----- - ----. .-- 47,500 

11.65 11.6 

4 ---- ---- - ----- ---. 22.75 22.8 

5 ------- -- 18.05 18.0 

6 ---------- - ---- 18.051 18.1 

If the first of the discarded digits is greater than 5, 

add 1 to the nth digit (example 1). If the first of the 

discarded digits is less than 5, leave the nth digit 

unchanged (example 2). If the first of the discarded 

digits is 5 and all the following digits are zero, round 

off to the nearest even number (examples 3-5). If the 

5 is followed by any of the digits 1 through 9, add 1 

to the nth digit (example 6). 

If the difference between successive numbers is 

more important than the total or average, it may be 

desirable to round consistently in one direction all 

numbers in which the first dropped digit is followed 

by zeros only, instead of rounding to the nearest even 

number. 

In presenting numerical data, give only those digits 

that convey actual information. The last digit should 

represent the uncertainty in the data. Unless stated 

otherwise, it is generally assumed that the last significant figure is uncertain by one unit. To illustrate, if 

the length of a drill core is given as 3.12 cm, true 

length is implied to be 3.12k0.01 and is thus somewhere between 3.11 and 3.13 cm. If the uncertainty 

in the last figure is appreciably different from one 

unit, attention can be called to the uncertainty by 

expressing the measurement at 3.12k 0.03 cm. 

Significant figures 
Special problems arise in converting English-to-metric 

or metric-to-English units. These problems can be 

avoided if (1) the precision of the original measurement 

is stated and (2) the author adheres strictly to the concept of significant figures. Most readers will assume 

that the first-listed number represents the system used 

for the actual measurement; hence they should not be 

confused by reconverting the second-listed number. 

Thus, the measurement "500 ft (152 m)" implies a 

precision of "500 f 1 ft," not the "500 * 3 ft" that 

would result from converting the 152 m back to feet. 

(The Survey no longer encourages dual measurements 

in its formal reports.) 

ABSOLUTE AND RELATIVE ERRORS 

The absolute error of a number or measurement generally is defined as the numerical difference between 

the true value and the approximate value as given by 

the number or measurement. The relative error can 

be defined as the absolute error divided by the true 

value of the quantity. The true index of a measurement is expressed by the relative error, which in turn 

is indicated by the number of significant figures required to express the measurement. For this reason, 

the number of significant figures is vitally important 

in reporting measured or computed quantities. 

The following example1 illustrates the difference 

between absolute and relative errors: Assume that the 

length of a carefully prepared core of rock 2 inches 

long has been measured to the nearest thousandth of 

an inch and that a mile of railroad track has been 

measured to the nearest foot. The absolute errors are 

0.0005 inches for the core and 0.5 ft for the track, 

whereas the relative errors are, respectively, 

0.0005 1 --- - 

0.5 1 and -- - 

2 4,000 5,280 - 10,560 

The track measurement is relatively better. 

A simple arithmetic operation, such as addition or 

multiplication, may affect the number of significant 

figures in the result. In addition and subtraction, the 

placement of the decimal point is important in the 

retention of significant figures. The general rule can 

be illustrated thus: Suppose you want to add the numbers 120.632, 8.14, 980.3, and 1,401.0023, each number being correct to its last figure. Inasmuch as the 

third number listed is correct only to the first decimal 

place, it is meaningless to retain more than two 

decimal places in the other numbers. Consequently, 

and the result is rounded to 2,510.1, or to five significant figures. Note that only one decimal place is 

retained in the sum and that the number of significant figures in the sum is less than the number of 

significant figures in two of the original numbers. The 

procedure of rounding off applies to measurements 

but not to whole numbers that are correct to the last 

digit. If the whole numbers in the example given 

above applied to individual persons or digits and represented counts that were correct to the last digit, 

they would be shown as: 

and the total would not be rounded off. 

If small numbers are added to (or subtracted from) 

large numbers of limited accuracy, the total should 

retain no more significant figures than are justified 

by the accuracy of the larger numbers. For example, 

in adding 356,000 (good to only three figures) and 

1,420 (good also to three figures), the sum is 357,000, 

not 357,420. The figures that are dropped are within 

the limits of error of the larger number and are 

meaningless in the sum. By the same reasoning, the 

addition of a very large group of numbers of limited 

accuracy cannot produce a total more accurate than 

the respective items. Therefore, if several hundred objects have been weighed individually with an accuracy 

of three figures, the total weight of all the objects 

should be rounded off to three significant figures. 

In subtraction, the number of significant figures in 

the difference may be considerably reduced if the 

numbers are close to each other in numerical value. 

Suppose 0.1189 is subtracted from 0.1204. The difference is 0.0015, which contains only two signxicant 

figures. 

In the multiplication or division of two or more 

approximate numbers of different accuracies, the 

more accurate numbers should be rounded off so as 

to contain one more significant figure than the least 

accurate number. In this procedure, the error of the 

product is due almost entirely to the error of the least 

accurate number. Therefore, the final result should be 

given to as many significant figures as are contained 

120 Arithmetic operations 
in the least accurate number, and no more. As illustrations, two calculations may be given: 

103.24 x 0.0081 = 103 x 0.0081 = 0.83 

and 

In computing with logarithms, no more decimals need 

be retained in the mantissa of the logarithm than the 

number of significant figures in the numerical factors 

that enter the computation. Thus, log 352.3 = 2.5469. 

It is sometimes easier to use logarithms directly from 

the tables without rounding off, but the results of 

computation should never be presented as being more 

accurate than the original data. 

A result cannot be more accurate than the data 

used to obtain it. Thus the number of significant figures of the result cannot be greater than is justified 

by the least accurate number entering into the calculation. Despite this rule, many published data contain 

incorrect significant figures. 

Many estimates of ore reserves are carried to as 

many as six significant figures-for example, 123,415 

tons. Such a number gives a spurious impression of 

accuracy, if not a suspicion that the estimator is 

incompetent. To see the fallacy, just consider how 

reserve tonnages are calculated. The estimated 

volume, which is usually determined from drill-hole 

information, is multiplied by the density of the ore. 

At best, the volume can be determined accurately to 

only three significant figures, and probably to no 

more than two. The density of the ore may be correct 

to two significant figures. Consequently, the calculation of the estimated tonnage can produce no more 

than two sigmficant figures. The figure in the foregoing example should be given as 120,000 tons. 

Again, the depth to a geologic structure, as computed from gravity determinations, might be given as 

13,016 ft. If, as is usual, this figure was calculated on 

the assumption of a density contrast for the ore body 

good to only two significant figures, the figure should 

be reported as "about 13,000 ft." 

Reports on results of chemical analyses provide yet 

another illustration. Typically the results might be 

reported as 1,061.39 for SO,, 880.90 for Na, and 

205.62 for C1, all in milligrams per liter (mgIL). Each 

of these numbers contains five or six significant figures, whereas the analytical procedures used justified 

only two or three. Moreover, concentrations of more 

than 1,000 mgL are customarily reported to only 

three significant figures; for concentrations between 

10 and 1,000 mgL only whole numbers are reported. 

It follows that the above results should be listed as 

1,060, 881, and 206 mgIL. 

Certain other field measurements, some of them 

crude, are improperly reported to a greater number 

of significant figures than would be justified by even 

the most refined laboratory methods. In these, as well 

as in laboratory measurements, care should be taken 

to use only as many significant figures as are 

justified. 

Some published stratigraphic measurements1 indicate unrealistic accuracies. The calculated thickness 

of a sedimentary formation of Tertiary age might be 

given as 14,633 ft, but if the top and bottom are as ill 

defined as most Tertiary units, a more acceptable 

figure would be "about 15,000 ft." Calculations of the 

thickness of such rock units based on measurements 

of strike and dip along a measured traverse inevitably 

contain many uncertainties (exact amount and direction of dip, magnetic declination, nature of exposure, 

and others), which are almost impossible to evaluate 

and which limit the acceptable value to a few significant figures. Mining geologists have been known to 

pace the length of an adit but to use a steel tape to 

measure the last few feet and to record the total 

distance in fractions of a foot. So too have stratigraphers been known to measure the poorly exposed 

parts of a section by hand leveling but to measure 

cliff-forming beds by tape and then to construct a columnar section in which some units, and the total 

thickness, are reported in inches or even in fractions 

of an inch. 

lSTA 7 was in press before the Survey adopted the policy of 

using metric units in all its formal reports. 

Misusing significant figures 121