As chemists we learn about matter and the changes in matter through observation and experiment. Many times such studies require the measurement of physical quantities in order for them to be meaningful. Therefore, it is necessary for us as chemists, and students of chemistry, to understand the convention used in recording and interpreting scientific measurements.

First, we must acknowledge that in order for a measured quantity to have meaning it must have two components; a numerical value (number) and, a unit. The unit is necessary in order for us to understand the scale on which the value was determined. We cannot properly interpret the dimensions of a reported value if it is unitless.

Imagine that you have asked me to measure the volume of a liquid sample and I inform you that it has a volume of 2.10. How do you interpret this information?Was the volume 2.10 mL; or perhaps it was 2.10 pints? These are two very different quantities of liquid! You would need to inquire about the unit of measurement employed in order to gather any meaning from this reported value.

We will focus more on the importance of units and their manipulation in the Dimensional Analysis tutorial. Now, however, we will direct our attention towards the numerical value recorded in a measurement.

There are two kinds of numbers:
1) Inexact- those that are measured and have limited precision.
2) Exact- those that are defined or counted and have unlimited precision.
(Precision refers to how finely we know a value.)

Let's look at a few example measurements so that we can learn to understand why measured values are said to be inexact. Three thermometers are pictured below. The thermometers were used to measure the temperature of the same sample but, as you will see, the resulting readings are quite different due to how they are calibrated.

Thermometer A        Thermometer B        Thermometer C

Thermometer A is calibrated such that the smallest division shown represents two Celsius degrees. When reading this thermometer we can see that the temperature of the sample was for certain greater than 18^oC and less than 20^oC. We need to guess the temperature between these two values (we are guessing in the ones place). We can correctly record the temperature as 19^oC.

Thermometer B is calibrated such that the smallest division represents one Celsius degree. When reading this thermometer we can see that the temperature of the sample was for certain greater than 21^oC and less than 22^oC. Again, we can give our best estimate of the temperature as lying between these two values. This time we will be able to record the temperature in the tenths place. We can record the temperature as being 21.2^oC.

Thermometer C is more finely calibrated; the smallest division represents one tenth of a Celsius degree. According to this thermometer, the temperature of the sample was for certain greater than 20.6^oC and less than 20.7^oC. We could record the temperature as  being 20.65^oC, giving the temperature of the system to the one hundredth of a degree.

Using three different thermometers we have obtained three different temperature readings for the same system; 19, 21.2, and 20.65^oC. Thermometer C was the most finely calibrated and resulted in our knowing the temperature with the most certainty. It is the most precise of the three thermometers. (Note: The thermometer readings also vary due to the quality of the thermometers. In general, a more finely calibrated thermometer will be of higher quality and will result in a more accurate temperature reading. As you might expect, as the quality of thermometer increases so does it cost. Therefore, in our lab exercises we will be working with thermometers similar to Thermometer B. They meet our experimental needs.)

Perhaps you noticed that in all of the values we recorded above, we included all certain digits plus one uncertain digit in the temperature reading. All scientific measurements are recorded in this way and have some degree of uncertainty, that is why they are inexact. A properly recorded measurement expresses both the magnitude and precision in its value. The reading given for thermometer A correctly implies to another scientist that the temperature of the sample was 19^oC (magnitude) plus or minus (+) 1^oC (precision). Scientists use significant figures to keep track of the precision of measured/calculated values.

Significant Figures:include all certain digits plus one uncertain digit.

This graduate cylinder is calibrated such that the smallest division represents one milliliter (mL) of liquid. Since the cylinder is calibrated to the ones place we can say with certainty that there is 37 mL of liquid and then we get to estimate by what fraction of a mL the liquid volume is greater than that. An appropriate reading would be 37.5 mL.  (Remember, when reading a graduate cylinder we record the liquid volume at the bottom of the meniscus.) This recorded value has three significant figures (3, 7, 5) and indicates that the level of uncertainty of the measurement was + 0.1 mL.
 
 
 
 
 
 

The smallest division on this ruler represents 1 mm or 0.1 cm. The length of the paper clip is 4.72 cm. This value has three significant figures and suggests that we know the length of the paper clip to + 0.01 cm. (Note: We often report length measurements with a ruler such as this to the nearest tenth of a centimeter (+0.1 cm) because it is very difficult to consistently record values to the nearest hundredth of a centimeter (+0.01 cm).)
 

Since all scientists should be following this convention when reporting their measurements, there exists a set of rules to determine the number of significant digits in a recorded value.

Rule 1:  All non-zero digits are always significant.

8.94 cm3 has 3 sig. figs.
144.14m has 5 sig. figs.
18 in has 2 sig. figs.

Rule 2:  Leading zeros (zeros that precede non-zero digits) are not significant.  These zeroes are merely holding the decimal place.

0.00178 has 3 sig. figs.
0.02 has 1 sig. fig.
0.0008236 has 4 sig. figs.

Rule 3:  Confined zeros (zeros between non-zero digits) are significant.

2.0094 has 5 sig. figs.
0.1103 has 4 sig. figs.
0.00605 has 3 sig. figs.
208 has 3 sig. figs.

Rule 4:  Trailing zeros (zeros to the right of the last non-zero digit) are significant if:
1) there is a decimal point in the number, 2) there is an overbar. Otherwise, they are insignificant.
 

Practice Set A:
How many significant figures are in the following numbers?
65,405
40.0
400
11.08
0.027
20,450.

If you are told that your sample contained 25.20 mg Ca²⁺, how would you interpret the level of precision of this this reported value?
(A)  25.2 +0.1 mg Ca²⁺    (B)  25.20 +0.01 mg Ca²⁺

Answers
 

Once we have made measurements of physical quantities, it is often necessary for us to manipulate them mathematically in order to obtain useful information. Mathematical manipulations should never increase or decrease the level of precision implied by the measurements. As a result, we must keep track of precision (significant figures) throughout calculations by applying the following rules.

Rules governing precision in calculations:

In multiplication and division, the number of sig. figs. in the product or quotient is the same as in the value from the calculation that contains the fewest sig. figs.

5.26 x 2.0 = 10.52
3 sig. figs. x 2 sig. figs. = 2 sig. figs. in answer
The second significant figure falls in the ones place, so we have to report our answer to the ones place which requires rounding. (Here the digit shown in bold type is the last significant figure.)
answer = 11

6.174 / 2.01 = 3.07164
4 sig. figs. / 3 sig. figs. = 3 sig. figs. in answer
The third significant figure is in the hundredths place, so the answer can only be reported to the hundredths place.
answer = 3.07

In addition and subtraction there cannot be any significant digits further to the right than the least precise number being added or subtracted.

                                    24.26                            14.211                               400
                                 +   2.1                            -  6.12                               + 206
                                    26.36                              8.091                               606
with rounding the
answers become:          26.4                               8.09                                  600
 

Rule 1:  If the first digit to be dropped is < 5 then it, and all digits following it, are simply dropped. It may be necessary to an add an insignificant zero to hold the decimal place.

65.132 to three sig. figs. = 65.1
124.56 to two sig. figs. = 120

Rule 2:  If the first digit to be dropped is > 5 or 5 followed by nonzero digits, then all unwanted digits are dropped and the last retained digit is increased by one unit (round up).

 2.654 to two sig. figs. = 2.7
 0.0076 to one sig. fig. = 0.008
2,318 to three sig. figs. = 2,320

Rule 3:  If the first digit to be dropped is a 5 not followed by other digits or only followed by zeros, then apply the even-odd rule.

Even-Odd Rule: Drop the 5 and any zeros present and
a) increase the last retained digit by one unit if it is odd or,
b) do not change the last retained digit if it is even.
(Note: The even-odd rule favors even numbers.)

42.1500 to 3 sig. figs. = 42.2
38.025 to 4 sig figs = 38.02

Practice Set B:

Do the following math and report the answers with the appropriate number of significant figures.

a) 0.145 / 1.2 =

b) 4.100 x 3.45 =

c) 91.6 + 42 + 210 =

d) 610. - 14 + 542.8 =

(answers)

When performing multiple step calculations, an overbar should be used to keep track of the significant figures from step to step. Rounding should only be performed when reporting the final answer. (You need to pay particular attention to significant figures when working with a calculator!)
 

Practice Set C:

Do the following math and report the answers with the proper number of significant figures.

(answers)

Answers to practice problems:
Set A: 5, 3, 1, 4, 2, 5, B (return to text)
Set B: a) 0.12, b) 14.1, c) 340, d) 1139 (return to text)
Set C: a) 55, b) 9,700, c) 1.3 (return to text)