SIGNIFICANT FIGURES
With any calibrated instrument, there is a limit to the number of figures that can be reliably kept in the answer.
For digital instruments, the limitation is the number of figures appearing in the display. For analog (non-digital)
instruments, the general rule for keeping figures is to read the instrument to one figure beyond the smallest
graduation marking. For example, a balance with a beam marked to the nearest 0.1 g should be read to the
nearest 0.01 g, the last figure being estimated.
In Figure 1, a line is shown next to two rulers
with different calibrations. How long is the line?
Measured with the lower ruler, its length is 0.8
unit. Measured with the upper ruler, the line is
0.77 unit. The length of the line has obviously
not changed; however, our ability to measure its
length has changed with our use of a different
measuring instrument. In both cases, the last
Figure 0
figure retained is an estimated one.
Estimating the last figure does introduce
some uncertainty, but gaining the extra digit (with the resultant increase in precision) more than offsets any
error. Standard practice in virtually all scientific measurements is to make the measurements in this fashion, so
that your answer has a group of figures whose numerical value you know exactly, plus one figure at the end
whose value has an implicit uncertainty of ±l or ±2 from the stated value. The number of figures kept in the
measurement using this guideline is called the number of significant figures. Since all measurements are
presumed to have uncertainty in the last figure, writing a number in which all the figures are exactly known will
result in the under-estimation of how precisely the measurement was performed.
When counting the significant figures in data, which you did not personally read, the following rules are
used:
1. All nonzero digits are assumed to be significant figures.
2. Zeros between two non-zero digits (sometimes called captive or internal zeros) are significant, and
zeros to the left of all the nonzero digits (called initial zeros) are not counted as significant.
3. Zeros to the right of all the nonzero digits (the trailing zeros) are significant if the number contains a
decimal point.
4. Trailing zeros are generally not significant if the number has no decimal point (although some people
might or might not count them, depending on the situation).
It is often advantageous to write numbers in scientific notation to eliminate ambiguity. A number in
scientific notation is written as a multiplier (the digit term) times a power of 10 (the exponential term). The digit
term should have only one figure to the left of the decimal point. For example, the number 0.00543 can be
written as 5.43 x 10-3. The number 657,000 (three significant figures) is written as 6.57 x 105, or 6.570 x 105 to
indicate four significant figures.
This last example illustrates another advantage to writing numbers in scientific notation. It is possible to show
explicitly the number of significant figures in a number with trailing zeros. For example, 657,000 would
normally be thought to have three significant figures. If it were actually measured with an instrument capable of
measuring to the nearest 100 units, you would be justified in keeping four significant figures. Writing it as 6.570
x 105 shows this explicitly with no room for misinterpretation, since the number now contains a decimal point.
Some examples of the counting of significant figures:
14.02
0.005490
1.0060
12500
62.4
1.2500 x 104
(4SF)
(4SF)
(5SF)
(3 SF)
(3 SF)
(5 SF)
1
SIGNIFICANT FIGURES IN CALCULATIONS: ROUNDING OFF
The aspect of dealing with significant figures which causes the most trouble for beginning students is in
the handling of significant figures in a calculation. The principle is that the results of a calculation cannot imply
greater accuracy or precision than is in the numbers used in the calculation. How to carry out this principle
depends on the type of mathematical operations involved in the calculation. Suppose only multiplication or
division are involved. The number of significant figures in the result is equal to the smallest number of
significant figures in any of the numbers used in the calculation. For example, in the calculation
42.50 × 0.00914 × 400.98
= 84.496651
36.809 × 0.050080
(shown to the number of figures on a calculator display), the factor with the fewest significant figures is
0.00914, with three, so the answer can be expressed to only three significant figures, as 84.5. In other words, it
must be rounded off. Similarly,
(5.09 ×10 )× (9.104 ×10 ) = 8.6 ×10
53.907 × 0.0030 × (3.33 × 10 )
−7
4
−4
2
(0.0030 has only 2 SF)
8.1605 × 509.5 × 100.0
= 1.83 × 106
3.751× 10 −4 × 607
(
)
(607 has 3 SF)
When rounding numbers off, if the amount to be dropped is less than half of the last figure to be kept, delete
it. If the amount to be dropped is greater than half of the last figure to be kept, increase the last figure by 1. If the
amount to be dropped is exactly half of the last figure to be kept, then do nothing if it is even and increase it by 1
if it is odd. The effect is that the value of the overall result will be lowered in half the cases and raised in the
other half, meaning that any errors introduced by roundoff will cancel out in the long run. For example, 45.550
would be rounded to 45.6, while 192.5 would be rounded to 192. Round numbers off in one step, not in a series
of steps in which you cut back one number at a time.
We now return to the subject of keeping track of significant figures in calculations. If the calculation involves
only additions and subtractions, the number of significant figures in each term in the calculation is less
important than the position of the last significant figure relative to the decimal point in the number. The position
(relative to the decimal point) of the last significant figure in the answer is the last position in which all the
terms in the calculation have a significant figure. Study the following examples:
47.9 + 5.79 - 33.8 = 19.9
105.583 - 105.571 = 0.012
50.33 + 125.80 - 170.311 = 5.82
9.13 + 1.5 = 10.6
The same rules for rounding off apply. Note that subtracting two numbers fairly close in value can lead to the
2
loss of significant figures. In extreme cases, almost all of the significant figures in your data can be lost in
because of this.
When raising a number to a power or taking a root, the same rules apply as for multiplication and division.
When taking a logarithm or antilogarithm, the number of significant figures in the number should equal the
number of figures in the mantissa (that is, the figures after the decimal point) of its logarithm.
If the calculation involves several mathematical operations, keep track of the significant figures as you
perform each step. For example, consider the calculation
[125.9 × (25.8 + 95.7 )] − (1.37 ×104 )
(459.1 − 125) × 0.09802
Working from the inner parentheses outward, we carry out the following series of steps, preserving the proper
number of significant figures at each step:
(
)
(
)
(125.9 × 121.5) − 1.37 × 10 4
334 × 0.09802
=
(1.530 × 10 4 ) − 1.37 × 10 4
32.7
=
1.6 × 103
= 49
32.7
A few words concerning significant figures and calculators (or spreadsheets) are in order. Calculators
(spreadsheets) are not programmed to keep track of significant figures. You are totally responsible for that.
Occasionally (especially when they are zeros), the calculator (spreadsheets) will drop off the final digits, giving
an answer that has too few significant figures. More commonly, though, the number of figures shown on the
calculator (spreadsheets) is many more than is justified by the data used. The calculator (spreadsheets) will do
exactly what it is told-no more, no less. If you give it wrong information, it will do its job of processing the
information. This will indeed give an incorrect result, but this is not the fault of the calculator (spreadsheets). It
is the fault of the person using the calculator (spreadsheets).
ACCURACY AND PRECISION
We have used the terms accuracy and precision without defining
them. These terms are used almost interchangeably by some people,
but mean very different things. The accuracy of a measurement or
experimental result shows how close that result is to some "correct,"
"true," or "accepted" value for that quantity. The precision of a series
of measurements shows how closely that series of measurements
agrees internally—that is, how close one measurement is to another
in the set. The precision of a set of measurements is almost totally a
function of how well that set of measurements was performed by the
experimenter; in other words, it is a measure of the experimenter's
technique. The accuracy of the measurement is partially a function of
technique, but it is also related to the quality of the measuring
instrument and the procedure that the experimenter uses.
To illustrate these concepts, Figure 2 shows four targets. The
precision of the attempt to hit the bullseye is measured by how
Figure 2
3
closely the holes are grouped. The accuracy of the attempt is measured by how closely the "average" shot has
come to the center of the target. Good agreement of the results with one another means that your technique was
reproducible, and that, possibly, the method (or, in this case, your rifle) needs adjustment or correction.
TYPES AND SOURCES OF EXPERIMENTAL ERROR
Three general categories of error are instrumental, operator, and method errors. Instrumental errors are due to
some inaccuracy in the measuring instrument: a poorly calibrated ruler or volumetric instrument, an electrical
malfunction in a meter, a worn pivot in a balance. The user can sometimes control the error (for example, a ruler
that is I cm short can still give reliable results if the experimenter always remembers to add I cm to all the
measurements), but usually the experimenter is limited by the measuring instrument.
Operator errors originate in the technique of the experimenter or operator of the instrument. Misreading the
scale on a buret, failing to set the zero point on a spectrometer, and failure to clean a balance before using are
examples of operator errors. These are the types of errors over which the experimenter has the most control.
Method errors derive from a flaw in the method of analysis or measurement. In other words, the method used
cannot produce accurate and/or precise results. A method that assumes that a material is 100% insoluble when it
is really partly soluble will give poor results no matter what measuring instrument is used or who is using it. As
with instrumental errors, the operator has limited control over this type of error, other than choosing an alternate
method that gives better, more reliable results.
Errors can also be classified as constant or random. A constant error is always the same size and direction. A
buret whose first mark after the zero is 2.00 mL instead of 1.00 mL will always give a volume reading that is
1.00 mL too large (assuming that you zero the buret each time), or a voltmeter may give a reading that is too
small by a constant value due to some electrical circumstance in the system being studied. A random error may
(conceivably) be of any size and of either direction. The fluctuation of a meter reading due to variations in the
electrical current is a random error, as are the fluctuations in a balance caused by drafts deflecting the balance
pan. Random errors can be treated by a statistical analysis of the results.
STATISTICAL MEASURES OF PRECISION
Since there are so many ways in which errors can creep into any experimental measurement, it is highly unlikely
for any set of repeated measurements to have exactly the same result in each trial. There can also be variability
in the sample from one trial to the next. When a quantity has been measured repeatedly and several results
obtained, we need to express the variability of the results in a way that can be easily understood by anyone
interested in the overall reliability and quality of the measurements. Statistics provides several ways to do this.
We will be dealing with a set of experimental results in which N trials were performed, all of which were
supposed to give the same result. The individual results for trials 1, 2, . . . will be denoted by x1, x2,. . . . The first
calculation most commonly done is to find the arithmetic mean, or average, x of the results. This is calculated by
taking the sum of the results and dividing by the number of trials N:
N
x=
∑x
i =1
i
N
(The Greek capital sigma (r) denotes the sum of whatever follows it.) The mean usually represents the most
likely value of the quantity being measured. For example, the following set of data was obtained for a
standardization of 0.02 M KMnO4 solution: 0.01995 M, 0.01997 M, 0.02004 M, 0.02019 M, 0.02001 M. The
arithmetic mean of these five trials is
N
x=
∑ xi
i =1
N
5
=
∑x
i =1
5
i
1
= ( x1 + x2 + x3 + x4 + x5 )
5
4
= 1/5_(0.01995 + 0.01997 + 0.02004
+ 0.02019 + 0.02001) = 0.02003 M
A simple measure of the precision of the set of results is the range w for the set of trials. The range is simply
the difference between the highest and lowest values among the results. In our example, the range is 0.00024 M.
This value does not really communicate any significant information about the precision. It is only when the
range (or the other measures of precision to be discussed) are compared with the value of the quantity being
measured that real meaning can be attributed to these quantities. (For example, knowing that a mass can have an
error of ±1 g is not that significant by itself. If the object weighs 1000 kg, the error is a very small; if the object
weighs 2 g, the error is very significant.) The relative range of a series of measurements is equal to the range
divided by the mean. In this case, the relative range is 0.00024 M/0.02003 M = 0.012.
Relative measures of precision are often expressed in terms of percentages, parts per thousand, and so on, to
obtain numbers that are easily handled. For example, multiplying the relative range by 100 gives it in percent;
multiplying the relative range by 1000 gives it in parts per thousand (ppt); multiplying the relative range by 106
gives it in parts per million (ppm). In our example, the relative range could be expressed as 1.2%, 12 ppt, or
12,000 ppm.
A more meaningful measurement of precision is the deviation of each of the individual values. The deviation
for trial i (di) is found by subtracting the mean of the set from the measured value for that particular trial:
d i = xi − x
Note that if xi < x , the deviation for that trial is a negative number; values greater than the mean have positive
deviations from the mean. If we average the absolute values of the deviations (if we average the values with
their signs, we always get zero), we obtain the average deviation (d) for that set of results. Dividing the average
deviation by the mean gives the relative average deviation (rad) for that set of results. As with the relative
range, the relative average deviation can be expressed in percent, parts per thousand, and so on. For the set of
results given earlier, the deviations, average deviation, and relative average deviation are1
d1 = -0.00008 M
d2 = -0.00006 M
d3 = 0.00001 M
d4 = 0.00016 M
d5 = -0.00002 M
d = 0.000066 M
rad = 0.0033 = 0.33% or 3.3 ppt or 3300 ppm
Note that the range or the average deviation has the same units as the quantity being measured, but the relative
range or relative average deviation has no units.
As a final measure of precision, the standard deviation is often calculated. This number has its greatest
meaning for large samples (with N ≥ 20), but it can be used for smaller samples (down to N = 3) with some
restrictions on its interpretation. The standard deviation s is calculated by taking the sum of the squares (not the
square of the sum!) of the deviations for the individual trials, dividing the result by N - 1, and taking the square
1
Two significant figures have been kept in the calculation of the average deviation, even though only one would be
justified based on the numbers. This practice of adding an "extra' significant figure to the result of the calculation of the
average deviation (and the standard deviation) is generally accepted, even though, technically, it is not justified based on
the rules for significant figures. Any time, in any calculation, that you drop to just one significant figure, a great deal of
precision is lost.
5
root of the result:2
s=
∑ (x − x)
i
N −1
The standard deviation for any set of results is always larger than the average deviation. As with the range, the
relative standard deviation can be calculated by dividing the standard deviation by the mean, and multiplying by
100, 1000, and so on, if desired to obtain the result in percent, parts per thousand, and so forth. For our example,
the standard deviation and the relative standard deviation are:
s=
(−0.00008) 2 + (−0.00006) 2 + (0.00001) 2 + (0.000016) 2 + (−0.00002) 2
5 −1
= 0.000095
relative standard deviation = 0.0047 = 0.47% = 4.7 ppt
One of the most important uses of the standard deviation for a set of trials is that the statistical distribution of
the results can be related to the magnitude of the standard deviation. For a large sample (N ≥ 20), the distribution
of results will be such that 68% of the results will be within one standard deviation of the mean, 95% will be
within two standard deviations, and 99% will be within 2.5 standard deviations. For smaller samples, the
statistics of this "normal distribution" will break down and these figures will not apply exactly.
REJECTION OF A TRIAL
In a set of results, one or more often seem suspicious in that they are quite far away from the others. Are there
any circumstances that let us legitimately throw out these suspicious values so that they don't affect the average?
If there is a known experimental problem with that trial, such as a solution being spilled, the wrong reagent
added, an end-point overshot, you would be justified in eliminating that result from consideration. But you
would not be justified in acting as though that trial did not even exist. You should report all the data and
calculations for that trial, and then exclude the results of that trial from the calculation of the average, average
deviation, and so on. Totally ignoring the result and not reporting it is dishonest reporting of what you did in that
experiment.
Suppose that no valid experimental reason exists for rejecting a result? Can you reject that trial just because it
"looks wrong" or it "just can't be right"? Based simply on your "gut feelings," the answer is no! You must have a
valid basis on which to justify rejecting a trial. It is possible, however, that random errors in the measurements
(in the instrument, method, or operator) have compounded themselves in that trial to produce a result that is so
far away from the mean that it is unlikely that the result is valid. In that case, you may be able to statistically
reject a particular trial.
Two statistical tests are commonly used to reject a trial. The first is called the 4d test. To apply this test,
calculate the mean and average deviation of the set of results without the suspect result. If the deviation of the
suspect result from this newly calculated mean is greater than four times the newly calculated average deviation,
then the result can be rejected on statistical grounds.
For our data, the most suspect value is 0.02019 M (trial 4). After eliminating that value, the new average and
new average deviation are 0.01999 M and 0.000032 M, respectively. The suspect value is 0.00020 M from the
new mean, which is greater than four times the new average deviation. Thus, that result could be rejected.
2
Technically. this formula gives the estimated standard deviation. The true standard deviation σ (Greek lower case sigma)
can be calculated only for samples of infinite size.
6
The second test is the Q test, where Q is the ratio of the difference between the suspect result and its nearest
"neighbor" and the overall range of the results:
Q=
suspect value − nearest value
l arg est value − smallest value
This calculated Q is then compared with a table of critical Q values. If the calculated Q value is greater than the
appropriate value of Q from the table, the suspect result can be rejected with the confidence associated with that
tabulated Q value. A table of critical Q values follows:
N=3
N=4
N=5
N=6
N=7
Q80%
0.781
0.560
0.451
0.386
0.344
Q90%
0.886
0.679
0.557
0.482
0.434
Q95%
0.941
0.765
0.642
0.560
0.507
Q99%
0.988
0.889
0.780
0.698
0.637
In this case, our calculated Q value is 0.00015/0.00024 = 0.625. For N = 5, this value of Q is greater than the
Q80% and Q90% values, but less than the Q95% and Q99% values. Hence, we can reject the value of 0.02019 with
90% confidence that we are doing the right thing, but we could not reject this value with 95% confidence. If you
decided to reject that result, you would still have to report that the result of 0.020 19 M was rejected at the 90%
level by using a Q test, or some similar wording.
7