Significant Figures and Measurement Accuracy
You are probably familiar with the equation for density, D=M/V (density equals mass over volume). For instance, water has a density of one gram per milliliter, meaning one ml of water weighs 1 g. Suppose you are weighing a block of an unknown metal in a chemistry lab. You place the block on your brand-new digital balance, which is good to the ten-thousandths place, and get a mass of 1056.748 grams. Now, you must find the volume. Whipping out your trusty centimeter ruler, you find the block measures 10.2 cm by 4.4 cm by 3.9 cm. Some quick math yields a density of 1056.748 divided by 175.032 cubic centimeters, or 6.037 g/cm3. You turn in the lab, confident in another "A."
You may already see the problem with this experiment. While the electronic balance is accurate to the ten-thousandths place, you ruler is not. Your measurement on 10 cm of one side of the metal block may not have been accurate to the tenths place (it could have been 9.9 or 10.2 cm, for instance), let alone the ten-thousandths. When dividing the very accurate result from the digital balance by your inaccurate ruler measurements, it follows that the end result would also be inaccurate, because the volume measurement could have been off by quite a margin.
In fact, the density estimation would only be as accurate as its most inaccurate component. Let's assume the ruler measurement was only good to the ones place; the tenths place may have been wrong (it was a cheap $.10 ruler). The table below shows the effect the inaccurate digit has on the volume measurement using two numbers given above, with the uncertain figures are shown in red.
As you can see, even a few uncertain figures in your data can cause large errors in the results. The figures that are certain in a measurement, as the 4 in the tens place above, are called the significant figures, and are very important in determining accuracy in calculations.
To determine the number of significant figures you should keep when performing a mathematical operation, first determine whether you will be adding/subtracting or multiplying/dividing. To add or subtract, simply keep decimal places that are significant (accurate) in both numbers. In the example below, only the hundreds and tens places are significant in both quantities. Therefore, the reported result should be 120.
To multiply or divide, first count up the number of significant figures in each quantity. Next, multiply the two quantities as normal. Then, take the lower number of "sig figs" of the two original quantities and apply this number of significant figures to the result. In the example discussed above, one number has two significant figures and the other has only one. Therefore, only one significant figure (since one is a smaller number than two!) should be kept in the result.
As a final note on significant figures, some facts are known to infinite significant places. For instance, the atomic number of lithium is 3.000000 (with zeroes going on to infinity) because there are exactly three protons in its nucleus by definition. Likewise, there are always 1000 milliliters in a liter, 1000 milligrams in a gram, etc. Many other commonly-used figures, like pi and how many kilometers are in a mile, are known to more significant figures than will ever be used in an experiment; some have been computed to hundreds of decimal places!
Example Problem 1
Identify the result, compensated for significant figures, in the problems below. Assume each decimal figure of the given quantities is significant.
A. 875.62 * 25.4
B. 78.2 - 23.13
C. 25.1536516816849 + 2.00
D. 25 / 5.0
A. Since there are five significant figures in the first number and three in the second, we should keep only three in the final result. The product of the multiplication is 850.22, or 850 when compensated for accuracy.
B. In addition and subtraction, simply keep decimal places that are certain in both numbers. The smallest significant decimal place is the tenths, so we end up with 55.1 (rounded up).
C. Using the above rule, we can determine that the accurate result is 27.15.
D. Since both figures have two significant digits, keep two figures after dividing, for a result of 5.0.
Many instruments will be certified to a certain number of significant figures, such as electronic balances or spectrometers. In particular, computerized instruments should have a predetermined level of accuracy. However, other instruments must be read by the chemist's eye, such as rulers, conventional balances with sliding weights, graduated cylinders, burets, etc. In this case, you should read the instrument to one place beyond the demarcations on the instrument. For instance, if a graduated cylinder has marks for every deciliter of liquid, try to record the level to the millimeters place. Always be sure, when measuring liquids, to record the level at the bottom of the small dip, called the meniscus. The Flash movie below provides a graphical example of how to read a graduated cylinder.
Example Problem 2
Identify the unit to which you should read when using the instruments below.
A. A balance marked to the grams place.
B. A voltmeter with marks every tenth of a volt.
C. A computer-operated pH meter accurate to the hundredths place.
D. A graduate marked to every two milliliters.
E. An electronic balance certified to the tenths of milligrams.
Answers: (read one beyond the indicated level except on digital devices)
A. Tenths of a gram.
B. Hundredths of a volt.
C. Hundredths place (read only to the indicated number, since it is digital).
D. Tenths of a milliliter (one decimal place beyond milliliters).
E. Tenths of milligrams.