SIGNIFICANT FIGURES &
UNCERTAINTIES IN
MEASUREMENTS
Uncertainties in Measurements
• consider the measurement of the volume of a liquid using a burette (shown in
Fig. 1.7 with the scale greatly magnified).
• Notice that the meniscus of the liquid occurs at about 20.15 mL.
• This means that about 20.15 mL of liquid has been delivered from the burette (if
the initial position of the liquid meniscus was 0.00 mL).
• Note that we must estimate the last number of the volume reading by
interpolating between the 0.1-mL marks.
• Since the last number is estimated, its value may be different if another person
makes the same measurement.
• If five different students read the same volume, the results might be as follows:
Figure 1.7 |
• Measurement of volume
using a burette.
• The volume is read at the
bottom of the liquid curve
called the lower meniscus.
• 20.15 ± 0.01 mL (20.14 – 20.16)mL
• Readings from 5 students
• Rre
• whether or not an instrument is properly calibrated or has sufficient resolution, there are
unavoidable differences in how different people see and interpret measurements
• These results show that the first three numbers (20.1) remain the same regardless of
who makes the measurement; these are called certain digits.
• However, the digit to the right of the 1 must be estimated and therefore varies; it is
called an uncertain digit.
• We customarily report a measurement by recording all the certain digits plus the first
uncertain digit.
Measuring a memory card
• Figure 1.11
Figure 1.11
The width we report for
the memory card depends
on which
ruler we use to measure it.
Significant Figures
• An uncertain number must be reported in such a way as to indicate the uncertainty in its value.
This is done using significant figures.
• Significant figures are the meaningful digits in a reported number.
• Consider the measurement of the memory card in Figure 1.11 using the ruler A above it.
• The card’s width is between 2 and 3 cm, but because there are no graduations between 2 and 3
cm on this ruler, we are estimating the second digit, hence reading is taken as 2.5 cm
• Although we are certain about the 2 in 2.5, we are not certain about the 5.
• The last digit in a measured number is known as uncertain digit; and the uncertainty
associated with a measured number is generally considered to be 1 in the place of the last
digit.
• Thus, when we report the width of the memory card to be 2.5 cm, we are implying that its
width is 2.5 ± 0.1 cm. (2.4 – 2.6) cm.
• Each of the digits in a measured number, including the uncertain digit, is a significant figure.
The reported width of the memory card, 2.5 cm, contains two significant figures.
• Ruler B, with millimeter gradations would enable us to be certain about the
second digit in this measurement and to estimate a third digit. In this case, we
may record the width as 2.45 cm.
• Note that again, we estimate one digit beyond those we can read. The reported
width of 2.45 cm contains three significant figures.
• Reporting the width as 2.45 cm implies that the width is 2.45 ± 0.01 cm.
• It is important not to imply greater certainty in a measured number than is
realistic or necessary..
• For example, it would be inappropriate to report the width of the memory card as
2.4500 cm, because this would imply an uncertainty of 0.0001.
• Therefore, the number of significant figures in any number can be determined
using the following guidelines:
Rules followed in Significant Figures
• 1. Any digit that is not zero is significant (112.1 has four significant figures).
• 2. Zeros located between nonzero digits are significant (305 has three significant figures,
and 50.08 has four significant figures).
• 3. Zeros to the left of the first nonzero digit are not significant (0.0023 has two significant
figures, and 0.000001 has one significant figure).
• 4. Zeros to the right of the last nonzero digit are significant if the number contains a decimal
point (1.200 has four significant figures).
• 5. Zeros to the right of the last nonzero digit in a number that does not contain a decimal
point may or may not be significant (100 may have one, two, or three significant figures— it
is impossible to tell without additional information).
• To avoid ambiguity in such cases, it is best to express such numbers using scientific notation.
If the intended number of significant figures is one, the number is written as 1 x 102;
• if the intended number of significant figures is two, the number is written as 1.0 x 102;
• and if the intended number of significant figures is three, the number is written as 1.00 x 102.
• 200 (2 sf) = 2 x 102 ; 2000 (4 sf) = 2.000 x 103
Sample Problems
1. Determine the number of significant figures in the following measurements:
(a) 443 cm,
= 3 sf
(b)15.03 g,
= 4 sf
(c) 0.0356 kg
= 3 sf
(d) 3.000 10–7 L
= 4 sf
(e) 50 mL = 5.0 x 101 = 1 or 2 sf
(f) 0.9550
(g) 1250
= 4 sf
Sample Problems
1. Determine the number of significant figures in the following measurements:
(a) 443 cm, (b)15.03 g, (c) 0.0356 kg, (d) 3.000 10–7 L,
(e) 50 mL, (f) 0.9550
• Strategy All nonzero digits are significant, so the goal will be to determine which of the
zeros is significant.
• Setup Zeros are significant if they appear between nonzero digits or
- if they appear after a nonzero digit in a number that contains a decimal point.
- Zeros may or may not be significant if they appear to the right of the last nonzero
digit in a number that does not contain a decimal point.
• Solution (a) 3; (b) 4; (c) 3; (d) 4; (e) 1 or 2, an ambiguous case; (f) 4
TASK1
Q1. Determine the number of significant figures in the following measurements:
(a) 1129 m, (b) 0.0003 kg, (c) 1.094 cm, (d) 3.5 x 1012 atoms,
(e) 150 mL, (f) 9.550 km.
TASK 2
Q2. Determine the number of significant figures in each of the
following numbers:
• (a) 3.050 x 10-4
• (b) 432.00
• (c) 8.001
• (d) 0.000310
• (e) 150
• (f) 5.0071
Calculations with Measured Numbers
• Since we often use one or more measured numbers to calculate a desired result, a
second set of guidelines specifies how to handle significant figures in
calculations.
• 1. In addition and subtraction,
- the answer cannot have more digits to the right of the decimal point than the
original number with the smallest number of digits to the right of the decimal
point. For example: l
• The rounding procedure works as follows.
• Suppose we want to round off 102.13 and 54.86 each to one digit to the right of
the decimal point.
• To begin, we look at the digit(s) that will be dropped.
• If the leftmost digit to be dropped is less than 5, as in 102.13, we round down
(to 102.1), meaning that we simply drop the digit(s).
• If the leftmost digit to be dropped is equal to or greater than 5, as in 54.86,
we round up (to 54.9), meaning that we add 1 to the preceding digit.
Multiplication and Division
• 2. In multiplication and division, the number 2.
of sf in the final product or quotient is determined by the original number that has
the smallest number of sf.
• Check on the example here →
• 3. Exact numbers can be considered to have
an infinite number of sf and do not limit the
• 125/1065 = 0.117370892 = 0.117 (3 sf)
3.
number of sf in a calculated result.
• E.g, a coin created in 2016 has a mass of 2.5 • Note that it is the number of coins (3), not the
mass, that is an exact number.
g. If we have three such coins, the total mass • The answer should not be rounded to one
significant figure because 3 is an exact number.
is
Multiple Step Calculations
4. In calculations with multiple
steps,
- rounding the result of each step
can result in “rounding error.”
Consider the following two-step
calculation:
• Suppose that A = 3.66, B = 8.45,
and D = 2.11.
• The value of E depends on
whether we round off C prior
to using it in the second step
of the calculation.
• ..
Method 1: 3.66 x 8.45 = 30.927 = 30.9 (C)
30.9 x 2.11 = 65.199 = 65.2 (Not
recommended)
Method 2: 3.66 x 8.45 = 30.927 = 30.93 (C)
30.93 x 2.11 = 65.2623 = 65.3
Recommended method
• In general, it is best to retain at least one extra
digit until the end of a multistep calculation, as
shown by method 2,
to minimize rounding error.
• Sample Problems 1.6 and 1.7 show how
significant figures are handled in arithmetic
operations.
SaSample Problem 1.6mple Problem 1.6
• Perform the following arithmetic operations and report the result to the proper
number of significant figures:
• 5.46 x 102 + 49.91 x 102
Check on this !!
•.
• It may look as though the rule of addition has
been violated in part (e) because the final answer
(5.537 x 103 g) has three places past the decimal
point, not two.
• However, the rule was applied to get the answer
55.37 x 102 g, which has four sf.
• Changing the answer to correct scientific notation
doesn’t change the number of sf,
• but in this case it changes the number of places
past the decimal point.
546 + 4991 = 5537 g, write it in scientific
notation = 5.537 x 103 g
Sample Problems
Q1. Perform the following arithmetic operations, and report the result to the proper
number of significant figures:
(a) 105.5 L + 10.65 L, (b) 81.058 m − 0.35 m,
(c) 3.801 x 1021 atoms + 1.228 x 1019 atoms,
(d) 1.255 dm × 25 dm, (e) 139 g ÷ 275.55 mL.
Q2. Perform the following arithmetic operations, and report the result to the proper
number of significant figures:
(a) 1.0267 cm × 2.508 cm × 12.599 cm,
(b)15.0 kg ÷ 0.036 m3,
(c) 1.113 × 1010 kg −1.050 × 109 kg,
(d)25.75 mL + 15.00 mL,
(e) 46 cm3 + 180.5 cm3.
Sample Question 1.7
An empty container with a volume of 9.850 x 102 cm3 is weighed and found to
have a mass of 124.6 g. The container is filled with a gas and reweighed. The mass
of the container and the gas is 126.5 g.
Determine the density of the gas to the appropriate number of significant figures.
• Strategy This problem requires two steps:
(i) subtraction to determine the mass of the gas, and
(ii) division to determine its density.
Apply the corresponding rule regarding significant figures to each step.
• Setup In the subtraction of the container mass from the combined mass of the
container and the gas, the result can have only one place past the decimal point:
126.5 g -124.6 g = 1.9 g.
• Thus, in the division of the mass of the gas by the volume of the container, the
result can have only two significant figures.
Sample Question 1.7
An empty container with a volume of 9.850 x 102 cm3 is weighed and found to
have a mass of 124.6 g. The container is filled with a gas and reweighed. The mass
of the container and the gas is 126.5 g.
Determine the density of the gas to the appropriate number of significant figures.
• Strategy This problem requires two steps:
(i) subtraction to determine the mass of the gas, and
(ii) division to determine its density.
Apply the corresponding rule regarding significant figures to each step.
• Setup In the subtraction of the container mass from the combined mass of the
container and the gas, the result can have only one place past the decimal point:
126.5 g -124.6 g = 1.9 g.
• Thus, in the division of the mass of the gas by the volume of the container, the
result can have only two significant figures.
• Solution
• Check on this !!
• In this case, although each
of the three numbers we
started with has four
significant figures,
• the solution has only two
significant figures.
• ..
TASK
Q1. An empty container with a volume of 150.0 cm3 is weighed and found to have
a mass of 72.5 g. The container is filled with a liquid and reweighed. The
mass of the container and the liquid is 194.3 g.
Determine the density of the liquid to the appropriate number of sf.
Q2. Another empty container with an unknown volume is weighed and found to
have a mass of 81.2 g. The container is then filled with a liquid with a density
of 1.015 g/cm3 and reweighed. The mass of the container and the liquid is
177.9 g.
Determine the volume of the container to the appropriate number of sf.
Accuracy and Precision
• Figure 1.12
• shows - The distribution of
papers shows the difference
between accuracy and
precision.
• (a) Good accuracy and good
precision.
• (b) Poor accuracy but good
precision.
• (c) Poor accuracy and poor
precision.
• (a)
• (c)
(b)
Accuracy and Precision
• Accuracy and precision are two ways to gauge the quality of a set of measured
numbers. Although the difference between the two terms may be subtle, it is
important.
• Accuracy tells us how close a measurement is to the true value.
• Precision tells us how closely multiple measurements of the same thing are to
one another (Figure 1.12).
• Suppose that three students are asked to determine the mass of an aspirin tablet.
Each student weighs the aspirin tablet three times. The results (in grams) are
Accuracy and Precision
• The true mass of the tablet is 0.370 g.
• Student A’s results are more precise than those of student B, but neither set of results is very
accurate.
• Student C’s results are both precise (very small deviation of individual masses from the
average mass) and accurate (average value very close to the true value).
• Figure 1.13 shows all three students’ results in relation to the true mass of the tablet.
• Highly accurate measurements are usually precise, as well, although highly precise
measurements do not necessarily guarantee accurate results.
• For example, an improperly calibrated meter stick or a faulty balance may give precise readings
that are significantly different from the correct value.
Graphing
• Figure 1.13 Graphing the students’ data illustrates the difference between
precision and accuracy.
• Student A’s results are precise (values are close to one another) but not accurate
because the average value is far from the true value.
• Student B’s results are neither precise nor accurate.
• Student C’s results are both precise and accurate
•.
Graphing Data
More Questions
Uncertainty in Measurement
Q1. What volume of water does the graduated cylinder contain (to the proper
number of significant figures)?
a) 32.2 mL b) 30.25 mL c) 32.5 mL
d) 32.50 mL e) 32.500 mL
Q2. Which of the following is the sum of the following numbers to the correct
number of significant figures?
3.115 + 0.2281 + 712.5 + 45
• a) 760.8431
b) 760.843 c) 760.84 d) 760.8
e) 761
Q3. The true dependence of y on x is represented by the line. Three students measured y as a
function of x and plotted their data on the graph. Which set of data has the best accuracy and
which has the best precision, respectively?
• a) red, green
•
•
•
•
b) green, green
c) green, purple
d) purple, purple
e) purple, green
Q4.What is the result of the following calculation to the correct number of
significant figures? (6.266 − 6.261) ÷ 522.0
a) 9.5785 x 10–6 b) 9.579 x 10–6 c) 9.58 x 10–6
d) 9.6 x 10–6
e) 1 x 10–6
Dimensional Analysis—Tracking Units
• Dimensional analysis or the factor label method is the use of conversion factors
in problem solving. For example, changing 12.0 inches to m.
• It takes two steps;
(i) one to convert inches to centimeters,
(ii) to convert centimeters to meters.
• The additional conversion factor required is derived from the equality
• 1 m = 100 cm and is expressed as either
• _We must choose the conversion factor that will introduce the unit meter and
cancel the unit centimeter (i.e., the one on the right).
• Since 1 in = 2.54 cm
Q. An average adult has 5.2 L of blood. What is the volume of blood in
cubic meters?
1 L = 1000 cm3
(1 m = 100 cm)3
1 m3 = 1000 000 cm3
• Setup 1 L 1000 cm3 and 1 cm = 1 x10–2 m. When a unit is raised to a power, the
corresponding conversion factor must also be raised to that power in order for the
units to cancel appropriately.
• Solution: