Each measurement has some degree of uncertainty associated with it, and it can be vital to know how much uncertainty there is in a given measurement.
For example, when a scientist is measuring the amount of a toxic substance in a river that provides drinking water, she needs to be sure of the uncertainty in measurement especially when that value is very close to the maximum allowed contaminant level.
In this CHEMTOUR you will learn how significant figures allow us to report and propagate uncertainty in measurement.
When you have completed it, you will be able to:
- distinguish between exact and uncertain values
- express values with the appropriate number of significant figures
- record a measurement with correct significant figures
- determine the number of significant figures in any measurement
- use significant figures to express the result of calculations
No scientific measurement can be obtained with perfect precision. The degree of uncertainty in each measurement depends on the instrument used to obtain the measurement.
The more uncertainty there is in a measurement, the less precise the measurement will be. The vessels shown are used to measure volume with different precision.
Choose the vessel which gives the LEAST precise measurement of volume and click on it to pour some liquid into it.
We can see that the liquid level is somewhere between 40 and 50 mL.
Significant figures are comprised of all the certain digits in a measurement and the first estimated digit.
We would report this measurement as 43 mL because we are certain of the 4 in the tens place, and we can estimate the units place as 3. This reported measurement has two significant figures. The uncertainty is indicated by the estimated digit and can usually be estimated as +/− 1. We would not be surprised if we were told that the volume of water in the beaker was actually 42 mL or 44 mL.
Now pour the liquid from the current vessel to the empty vessel that is least precise by clicking on the empty vessel.
For the liquid in this new vessel, type in the certain digits and the first estimated digit of the measurement into the box.
.
The measurement reported with the correct significant figures is now 43.2 mL because the instrument allows us a greater precision. We are certain of 43 and we estimate the 2. Because the uncertainty is given by the estimated digit, we know the possible range of values for the measurement is 43.0 — 43.3 mL.
How many significant figures does this measurement have?
If the liquid in the beaker is poured into the measuring cylinder with markings every 0.1 mL, which of the following would be a possible reading from the measuring cylinder?
If a measuring device has its smallest graduations of 0.1, the correct significant figures will be reported to 0.01.
How many significant figures should be reported for the reading from this thermometer?
A ruler has divisions every 0.01 cm. Which of the following is a possible reading from this ruler that reports the correct significant figures?
Some numbers have no uncertainty associated with them and these are called exact numbers.
For example, you have five fingers on your hand. Because you can count them, you know you have exactly five fingers. When objects are counted, we consider the number exact and think of it as having infinite significant figures.
Another class of exact numbers is defined ratios. For example, there are 100 cm in 1 m. This means a meter is divided into exactly 100 parts and each of these parts is a cm. There is no uncertainty, and hence the numbers also have infinite significant figures.
Identify whether the number in each of the following statements is an exact number.
- There are 12 eggs in the carton.
- The carton weighs 450 g.
- The diameter (d) of a circle is defined as twice the radius (r)
d = 2 x r - There are 1000 g in 1 kg.
- The bottle contains 12 mL of cough syrup.
Note that when dealing with very large numbers of countable objects and estimation is used rather than just counting, uncertainty is introduced and so significant figures become important. For example, the population of the world is 7.3 billion.
Based on the fact that significant figures include all the certain digits in a measurement and the first estimated digit, a number of convenient guidelines can be expressed to count significant figures in a measurement. Note that these guidelines only apply to measurements, not exact numbers. Recall that the significant figures are all the certain and the first estimated digit. Digits that are not significant are shown in red.
- A non-zero number will ALWAYS be significant (either as one of the certain digits or as the first estimated digit).
- Zeros between non-zero digits are always significant, e.g., 3.04 cm
- Leading zeros are never significant, e.g., 0.0232 cm
- Zeros are significant at the end of a number ONLY if the number has a decimal point. For example, 0.230 cm; 400. kg = 3 significant figures BUT 400 kg = 1 significant figure.
Click on the first estimated digit in the following measurements. Recall that the significant figures are all the certain digits and the first estimated digit. Click on the link in the toolbar to review the rules.
- 458.7s
- 5200.0g
- 100lb
- 6087ft
- 0.004040m
Often measurements are combined or used in equations in science. The result of such a calculation can be no more precise than the measurements that it came from. When we type numbers into our calculator, we get a number that doesn’t always reflect the uncertainty in our value.
When we are performing calculations, we must make sure that we do not increase or decrease the uncertainty. Before we understand how to carry over uncertainty in measurements into calculations, a useful skill is to be able to express any number in terms of a certain number of significant figures by rounding off. To do this, we drop all the insignificant digits and then round (either up or down) the first estimated digit.
E.g. After calculations, a calculator display shows the following number:
2.564528
The final answer should have three significant figures. Click on the first estimated digit in the final answer.
If the number to the right of the first estimated digit is 5 or greater, we round the first estimated digit up; if it is less than 5, we round down. For our example, when rounding to three significant figures, is it appropriate to round up or down?
What about if we had to round 2.564528 to six significant figures? Choose the correct rounded number:
When performing calculations, the final answer will have an uncertainty limited by the uncertainty of the measurements used in the calculation. To do this, we apply the "weak link principle": a chain can be no stronger than its weakest link. We can only know a calculated value as well as we know the least well-known value.
The nature of the calculation determines the weakest link.
When multiplying or dividing, the weakest link is the measurement with the fewest significant figures.
When adding or subtracting, the weakest link is the measurement with the least precise decimal place.
For each of the following problems, click on the number that is the “weakest link” and limits the precision of the final answer.
×
When multiplying or dividing, the weakest link is the measurement with the fewest significant figures. 62 has two significant figures, and 531 has three significant figures. Therefore, 62 is the weakest link.
−
When adding or subtracting, the weakest link is the measurement with the least precise decimal place. 5321 has its first estimated digit in the ones' place and so is correct to +/− 1.
20.2 has its
first estimated digit in the 1/10ths place so is correct to +/− 0.1. Thus, 5321 has the least precise decimal place and is the weakest link. Even though it has more significant figures, because we are subtracting, we need to consider decimal
places. A useful way to visualize this is to write out the subtraction or addition, as shown, with the place values lined up.
For any kind of calculation, first perform the calculation and write down all the digits that your calculator provides. Then consider the appropriate precision of the answer, and round as necessary.
531 × 62 = 32,922
When multiplying or dividing, the final answer must have the same number of significant figures as the weakest link, that is, the measurement with the fewest significant figures.
Which of the following is the answer, with correct significant figures, to the problem?
Since 62 is the weakest link and we are multiplying, the final answer must have two significant figures in it. The significant figures are shown in blue — 32922. To round, look at the digit to the right of the first estimated digit. Because the 922 is being dropped, 33,000 is the correctly rounded answer.
Consider the subtraction problem:
Since the digit to the right of the last significant figure is 5 or greater, we round up.
When adding or subtracting, the final answer must be correct to the same decimal place as the weakest link, the measurement with the least precise decimal place. In this case, 5321 has the least precise decimal place because it is only correct to the ones' place. What is the final answer, correct to the ones' place?
When a calculation involves more than one mathematical operation, order of operations is used to determine the final significant figures in the answer. We don’t round off before the end of the calculation, otherwise we introduce uncertainty.
How many significant figures should be reported in the answer to this problem?
5.61 + 4.54.185 = = 2.41577061 = 2.42
The first operation to do is addition: 5.61 + 4.5.
Remembering the basic rules of significant figures in calculations, what is the weakest link in this step?
4.5 is correct to the 1/10ths place while 5.61 is correct to the 1/100ths place. So, 4.5 is the weakest link, and the final answer must be correct to the 1/10ths place also.
10.11 = 10.1 correct to the 1/10ths place.
If this was the final answer to a calculation, we would report 10.1 as the answer.
However, since it is only one step in the calculation, we do not round, but instead keep track of significant figures as we carry the result into the next calculation. Here, we do this with color coding. All significant figures are blue, others are red, but you can do it on your own by underlining the last significant digit to help you keep track.
So, 5.61 + 4.5 = 10.11
To complete the calculation, we do the last division step.
Which is the weakest link in this final step?
When multiplying or dividing, the weakest link is determined by the number with fewest significant figures. Although 10.11and 4.185 both appear to have four significant figures, 10.11came from a prior calculation and only has three significant figures. Thus, 10.1 1 is the weakest link, and the final answer should only have three significant figures.
The final answer to this calculation has three significant figures. Note that it would have been difficult to determine how many significant figures should be in the final answer without actually performing the calculation. This is often true when addition and subtraction are involved.
How many significant figures should be used for the result of the following calculation?
The mass of one molecule of hydrofluoric, HF, is obtained by adding the mass of one atom of hydrogen (1.00782 atomic mass units) to the mass of one atom of fluorine (18.9984) atomic mass units).
Which of the following is the mass of one molecule of HF to the correct number of significant figures?
What is the answer to the problem reported to the correct number of significant figures? !
g/ml
A thermometer was used to measure a temperature of 54.56°F. What was the smallest marking on the thermometer to report the given degree of precision?
How many significant figures are there in the following: 10 bags of leaves?
By Riverhugger (Own work) [CC BY-SA 4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons
Congratulations! You have completed the significant figures ChemTour.
You have learned:
- how significant figures are used to report the value and the uncertainty in a measurement
- how to read a measurement and report the correct significant figures
- which numbers significant figures apply to, and which are considered exact with infinite significant figures
- how to determine the number of significant figures in any measurement
- how to use significant figures in calculations using the weakest link principle
The reading using correct significant figures is 207.5. 207 is certain and the first estimated digit is 5.
This number has four significant figures.
With divisions every 0.01 cm, we can be certain up to and including the 1/100ths place. As a result, the 1/1000ths place will be the first estimated digit and will be included as a significant figure. Hence, 2.22 2cm is the correct reading.
This calculation contains two multiplications and a division. When multiplying or dividing, the weakest link is the number with fewest significant figures, and the answer must have the same number of significant figures.
2.531 has four significant figures (all non-zero digits are significant).
350 has two significant figures (it has a zero at the end of the number, but no decimal place, so the zero is NOT significant — rule 4).
0.00254 has three significant figures (leading zeros are NOT significant (rule 3).
So, 350 is the weakest link with two significant figures. Following the rule for multiplication, the final answer should also have two significant figures in it.
This problem is solved by addition:
When the mathematical operation is addition or subtraction, the weakest link is the number with the least precise decimal place. 18.9984 is correct to the 1/10,000ths place, while 1.00782 is corrected to the 1/100,000ths place. Hence, 18.9984 is the weakest link, and the answer should be correct to the same decimal place, the 1/10,000ths place.
18.9984 + 1.00782 = 20.00622 = 20.0062 atomic mass units.
25.1 g / (103.49 mL − 99.4 mL)
This problem contains subtraction AND division. We follow order of operations when we solve it. First, do the subtraction.
When subtracting, the weakest link is the measurement with the least precise decimal place. 99.4 mL is only correct to the 1/10ths place while 103.49 is correct to the 1/100ths place. 99.4 mL is the weakest link. The answer must be correct to the same decimal place as the weakest link, so after the subtraction step the answer must be correct to the 1/10ths place. Because this is not our final answer, we use color to keep track of significant figures and do not round off at this point.
The final step is division: 25.1 g / (103.49 mL − 99.4 mL) = 25.1g/4.09 mL. Since we are dividing, our weakest link is the measurement with the fewest significant figures, 4.0 9mL (two), and the final answer must have the same number of significant figures as the weakest link.
25.1 g / (103.49 mL − 99.4 mL) = 25.1g/4.09 mL = 6.136919 g/mL = 6.1 g/mL
The significant figures are all the certain digits and the first estimated digit. In 54.56°F, the first estimated digit is 6, and so the thermometer must have had divisions every 0.1°F for the 1/10ths place to be certain.
Since 10 bags can be counted without any uncertainty, this number is considered exact, with infinite significant figures.