# The Significance of Significant Figures

- Page ID
- 175483

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Significant Figures

*See also: Errors in Measurement*

**Table of Contents:**

One of the more troublesome aspects of calculating numerical results based on experimentally measured quantities is the decision of how many significant figures should be retained in the final reported result. The general principle governing this decision is that the final result should convey to the reader the maximum amount of information derivable from the experimental measurements and no misinformation. To illustrate this principle, consider the following example: Suppose we do an experiment to determine how much heat is evolved (per gram) in burning a certain sample. To do this we measure the temperature rise produced in a known amount of water by the heat liberated in burning a weighed quantity of sample. We find that 100 grams of water are heated from 15.5 ◦C to 25.5 ◦C when 0.190 grams of sample are burned. Knowing that it takes 1 calorie of heat to raise the temperature of 1 gram of water 1◦C and in our experiment the temperature of 100 grams of water was raised 10.0◦C, we calculate the quantity of heat evolved in our experiment as:

\[\SI{1}{\calorie\per\celsius} \times \SI{100}{\gram} \times \SI{10}{\celsius} = \underline{\SI{1000}{\calorie}}\]

The number of calories of heat per gram of sample is therefore:

\[\frac{\SI{1000}{\calorie}}{\SI{0.190}{\gram}} = 5263.1579\ldots\SI{}{\calorie\per\gram}\]

or is it? Does the calculated result give the maximum amount of information available from the experimental data and no misinformation?

The reported result, 5263.1579 cal/g, suggests that if we burn 1 gram of sample our experimental apparatus is accurate enough to measure the heat liberated reliably out to the 4th place to the right of the decimal point, or to an accuracy of 0.0001 calorie. Now 0.0001 calorie would raise the temperature of 100 grams of water 0.000 01 ◦C, and to measure this we would need to use a thermometer capable of detecting temperature changes in the sixth place to the right of the decimal point! The fact that the actual temperature measurements are 15.5 and 25.5◦C indicates that our thermometer is only accurate to the first digit to the right of the decimal place, and that under the best of conditions we are able to detect a temperature change no smaller than 0.1◦C. Thus, the reported result conveys to the reader the false impression that our temperature measuring capability is very much greater than is actually the case. In other words the reported result conveys misinformation and is therefore incorrect. Such misinformation will lose points on lab reports!

If our thermometer is only accurate enough to allow us to measure temperature changes no smaller than 0.1◦C under the best of conditions, then we can measure quantities of heat no smaller than 100 g ×1 cal ◦C/g×0.1◦C = __10 cal __if we use 100 grams of water as our heat sink.

Practically this means that our experimental measurements are only accurate enough to measure heat amounts to the second digit to the left of the decimal point and our reported result should have been 5260 cal/g. Another way of saying this is that we have 3 significant digits in our reported quantity. A better way of expressing the result to emphasize that there are only 3 significant digits to use scientific notation: is 5.26×103cal/g.

So far, we have stressed only the limitations of our temperature measuring capability. All of this assumes that the mass of water was accurately measured to at least the first digit to the left of the decimal point; in other words we know the mass of water is between 101 and 99 grams. In our experiment all of the measured quantities entering the calculation of the final result were measured to three significant figures; i.e.

Mass of Water | 100±1 g or 1.00×10^2 g |

Mass of Sample |
0.190±0.001 g or 1.90×10^−1 g |

Temperature Change of Water |
10.0±0.1◦C or 1.00×10^1 ◦C |

Note that in each of the measured quantities the last significant digit is somewhat uncertain. This is generally the case when measured quantities are reported; the last significant digit is a carefully considered estimate and represents the limit of the experimenter’s ability to measure, given the measuring instrument used and the conditions under which the measurement is made. This uncertainty carries over into the result calculated from the measurements; i.e. in the result, 5.26×103cal/g, the last significant digit, 6, is somewhat uncertain. *Thus the number of significant figures in a quantity is the number of trustworthy figures in it, the last trustworthy figure being somewhat in doubt, because it is based upon an estimation.*

Note that when several measured quantities are used to calculate a final result, the least accurately measured quantity limits the accuracy of the final result. For example suppose that in our experiment we had measured the volume of water very crudely so that the best we could do in stating the mass was to say that it was some value between 90 and 110 grams, i.e., 100 ± 10 g or 1.0×102 g. Then it does not matter that we measured the temperature rise of the water and the mass of the sample very accurately, because the accuracy of the final result would be limited by the crudely determined water mass. In this case the final result would be correctly reported as 5.3×103cal/g, the two significant figures of the reported value reflecting the two significant figures in the water mass.

There is a time-saving lesson in all of this for the alert experimentalist. In this example it would have been a waste of time to stand in line waiting to use the analytical balance to weigh the 100 grams of water accurately to the third or fourth place to the right of the decimal, when a simple measurement in a graduated cylinder accurate to the nearest mL (or gram) was adequate to match the accuracy of the temperature measurements. By the same token it was necessary to accurately weigh the sample on an electronic top-loader balance to the nearest milligram to acquire three significant figures (0.190 g). Therefore, an understanding of significant figures is not only necessary to compute correct results from measured quantities, but it is also useful in deciding how much time and care are needed to measure the desired quantities that enter into the final result.

### Examples of Significant Figures

The mass indicated by 367 g is known to be definitely less than 368 g and more than 366 g, but the figure 7 is somewhat in doubt. Thus in the number 367 there are three significant figures. Often a more precise meaning could be given by writing 367 ± 1, meaning that the true value is known to lie between 366 and 368 g. To illustrate with other examples:

- 4.9 has 2 significant figures
- 6.75 has 3 significant figures
- 60.75 has 4 significant figures
- 49275 has 5 significant figures

Zeros may or may not be significant depending on the circumstances. When a zero occurs between two digits other than zero or to the right of another digit beyond the decimal point it is always taken to be significant.

- 2051 has 4 significant figures
- 2005 has 4 significant figures
- 25.00 has 4 significant figures

Zeros with no digits other than zero to their left are never significant (they fix the decimal point).

- 0.251 has 3 significant figures
- 0.025 has 2 significant figures
- 0.205 has 3 significant figures

Zeros with no digit other than zero to their right in whole numbers must be present to fix the decimal point. They are usually considered as not significant unless other information is given. For example: 2500 has 2 significant figures as in a 2500 mL beaker whose volume may vary by ± 100 mL (2500 ± 100 mL). If this volume were a 2500 mL volumetric flask which would be accurately calibrated to ± 1 mL, then 2500 has 4 significant figures and is usually written as 2500 ± 1.

Note: The problem of this type of zero is readily solved by exponential notation (which is always used by scientists when they wish to avoid ambiguity), or by changing units; then only significant digits are recorded. The 2500 mL beaker would contain 2.5 x 103 mL or 2.5 liters while the volumetric flask would contain 2.500 x 103 mL or 2.500 liters. Note: It is easy to discern which figures are significant when the number is written in scientific notation.

Other examples:

- 3 sig. fig. \(2.51 \times 10^3\)
- 2510 ± 1 4 sig. fig. \(2.510 \times 10^3\)
- 25,100.000 8 sig. fig. \(2.5100000 \times 10^4\)
- 0.000251 3 sig. fig. \(2.51 \times 10^-4\)

## Mathematical Operations Involving Significant Figures

Normally, in mathematical operations involving significant figures, the answer is reported in such a way that it reflects the reliability of the least accurate operation.

**Addition:**

247 | .3 | g | |

42389 | g | ||

+ | 6 | .973 | g |

42652 | .273 | g |

/( \begin{align} & 247.3 & g &\\\\& 42398 & g &\\\\+ & 6.973 & g &\\\\\cline{1-3}& 42652.273 & g & = 42652 g\end{align} /)

The answer is recorded to the nearest gram, since the number 42398 was determined only to the nearest gram.

**Subtraction**:

**(a)** 17.524 g

__− 17.006 g__

0.518 g = 0.518 g

**(b) **20.5 g

__− 8.396 g__

12.104 g = 12.1 g

In addition and subtraction the number of significant figures in the answer is not necessarily the same as in the values added or subtracted. Five significant figures were required in the weighings shown in example **(a)** to give an answer which is reliable only to three significant figures.

**Multiplication:**

In multiplication, the same number of significant figures are generally carried in the product as are contained in the factor with the least number of significant figures. (See exception below.)

*200.61×3.65 = 732*

3.65 has only three significant figures so the answer is recorded to three figures.

**Division:**

In division the same number of significant figures are carried in the quotient as are contained in the factor with the least number of significant figures. (Also see exception below.)

121/83.15 = 1.46

**Exception:**

In certain cases where the counting of significant figures would give an answer less trustworthy than the factors involved, the retention of one additional figure is justified. In this case the answer is recorded to three significant figures. In 95 there are only two digits but the accuracy of 95 is nearly 1 part in 100 (really 1 part in 95). If the answer were recorded as 16 (two significant figures) this would indicate an accuracy of only 1 part in 16. This is not as true an indication of the accuracy of the measurement as the answer 16.2, which implies an uncertainty of 1 part in 162. Similarly in multiplication 9.8 x 1.368 = 13.4, not 13.

1535

95 = 16.2

9.8×1.368 = 13.4

### Rounding off

Inexperienced students often round off measured numbers to give too few significant digits and then use these excessively rounded off numbers to calculate final results. This often leads to serious error in the result. When several multiplication and division operations involving measured quantities precede the calculation of the final result, it is recommended that at least one extra digit in excess of the number considered to be significant be carried through the calculation until the final result and then rounded off to the appropriate number of significant figures. For example: suppose a sample of nitrogen gas (N2) weighing 0.1079 g is found to occupy a volume of 0.987 L. The density of the gas is calculated to be

d = 0.0179 g/ 0.987L = 0.109 321 g/L

If one were to report the density of the gas to the correct number of significant figures it would be 0.1093g L−1. If, however, we wished to use this datum to calculate the pressure of the gas from the ideal gas equation of state knowing that the temperature of the gas is 298.26 K we could proceed as follows:

P = dRT/M, where M = 28.013 g/mol

P =0.109 32 g/L ×0.082 056 L/atmK mol ×298.26K/

28.013 g mol= 0.0955 atm

Note that in this example we carried along an extra digit in each quantity until the final result was obtained before doing the round off to the appropriate number of significant figures. If the student had rounded off the density to 0.1 g/L and used this value to calculate the pressure s/he would have calculated P = 8.74×10−2 atm.

## Conclusion

The correct application of significant figures involves nothing more than the use of common sense in mathematical computation. The main purpose of stressing the subject is to eliminate the tendency of students to carry out their calculations to numerous decimal places (reported by hand calculators, for example) ignorant of the fact that a large portion of the answer is meaningless. Conversely, some students throw away their experimental precision by improper rounding midstream in a calculation and thus obtain

an inaccurate answer. Be sure to always retain your full experimental precision in your calculated results.