Analyzing Data
Section 2.1 Units and
Measurements
Section 2.2 Scientific Notation and
Dimensional Analysis
Section 2.3 Uncertainty in Data
Section 2.4 Representing Data
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Section 2.1 Units and Measurements
• Define SI base units for time, length, mass, and
temperature.
• Explain how adding a prefix changes a unit.
• Compare the derived units for volume and density.
mass: a measurement that reflects the amount of
matter an object contains
Section 2.1 Units and Measurements
base unit
kelvin
second
derived unit
meter
liter
kilogram
density
Chemists use an internationally
recognized system of units to
communicate their findings.
(cont.)
Units
• Système Internationale d'Unités (SI) is an
internationally agreed upon system of
measurements.
• A base unit is a defined unit in a system of
measurement that is based on an object or
event in the physical world, and is
independent of other units.
Units (cont.)
Units (cont.)
Units (cont.)
• The SI base unit of time is the second (s),
based on the frequency of radiation given
off by a cesium-133 atom.
• The SI base unit for length is the meter (m).
• The SI base unit of mass is the kilogram
(kg), about 2.2 pounds
Units (cont.)
• The SI base unit of temperature
is the kelvin (K).
• Zero kelvin is the point where
there is virtually no particle
motion or kinetic energy, also
known as absolute zero.
• Two other temperature scales
are Celsius and Fahrenheit.
• To convert Celsius to Kelvin
C+273.15=K
Derived Units
• Not all quantities can be measured with SI
base units.
• A unit that is defined by a combination of
base units is called a derived unit.
Derived Units (cont.)
• Volume is measured in cubic meters (m3), but
this is very large. A more convenient measure
is the liter, or one cubic decimeter (dm3).
Derived Units (cont.)
• Density is a derived unit, g/cm3, the
amount of mass per unit volume.
• The density equation is
density = mass/volume.
Section 2.1 Assessment
Which of the following is a derived unit?
A. yard
B. second
C. liter
D
C
A
0%
B
D. kilogram
A. A
B. B
C. C
0%
0%
0%
D. D
Section 2.1 Assessment
What is the relationship between mass
and volume called?
A. density
B. space
D
A
0%
C
D. weight
A. A
B. B
C. C
0%
0%
0%
D. D
B
C. matter
Section 2.2 Scientific Notation and
Dimensional Analysis
• Express numbers in scientific notation.
• Convert between units using dimensional analysis.
quantitative data: numerical information
describing how much, how little, how big, how
tall, how fast, and so on
Section 2.2 Scientific Notation and
Dimensional Analysis (cont.)
scientific notation
dimensional analysis
conversion factor
Scientists often express numbers in
scientific notation and solve problems
using dimensional analysis.
Scientific Notation
• Scientific notation can be used to
express any number as a number between
1 and 10 (the coefficient) multiplied by 10
raised to a power (the exponent).
• Count the number of places the decimal point
must be moved to give a coefficient between
1 and 10.
Scientific Notation (cont.)
• The number of places moved equals the
value of the exponent.
• The exponent is positive when the decimal
moves to the left and negative when the
decimal moves to the right.
800 = 8.0  102
0.0000343 = 3.43  10–5
Scientific Notation (cont.)
• Addition and subtraction
– Exponents must be the same.
– Rewrite values with the same exponent.
– Add or subtract coefficients.
Scientific Notation (cont.)
• Multiplication and division
– To multiply, multiply the coefficients, then
add the exponents.
– To divide, divide the coefficients, then
subtract the exponent of the divisor from the
exponent of the dividend.
Dimensional Analysis
• Dimensional analysis is a systematic
approach to problem solving that uses
conversion factors to move, or convert,
from one unit to another.
• A conversion factor is a ratio of equivalent
values having different units.
Dimensional Analysis
(cont.)
• Writing conversion factors
– Conversion factors are derived from equality
relationships, such as 1 dozen eggs = 12
eggs.
– Percentages can also be used as
conversion factors. They relate the number
of parts of one component to 100 total parts.
Dimensional Analysis
(cont.)
• Using conversion factors
– A conversion factor must cancel one unit
and introduce a new one.
Section 2.2 Assessment
What is a systematic approach to problem
solving that converts from one unit to
another?
A. conversion ratio
A
0%
D
D. dimensional analysis
C
C. scientific notation
A. A
B. B
C. C
0%
0%
0%
D. D
B
B. conversion factor
Section 2.2 Assessment
Which of the following expresses
9,640,000 in the correct scientific
notation?
A. 9.64  104
A
0%
D
D. 9.64  610
C
C. 9.64 × 106
A. A
B. B
C. C
0%
0%
0%
D. D
B
B. 9.64  105
Section 2.3 Uncertainty in Data
• Define and compare accuracy and precision.
• Describe the accuracy of experimental data using
error and percent error.
• Apply rules for significant figures to express
uncertainty in measured and calculated values.
experiment: a set of controlled observations that
test a hypothesis
Section 2.3 Uncertainty in Data (cont.)
accuracy
percent error
precision
significant figures
error
Measurements contain uncertainties
that affect how a result is presented.
Accuracy and Precision
• Accuracy refers to how close a measured
value is to an accepted value.
• Precision refers to how close a series of
measurements are to one another.
Accuracy and Precision (cont.)
• Error is defined as the difference between
and experimental value and an accepted
value.
Accuracy and Precision (cont.)
• The error equation is
error = experimental value – accepted value.
• Percent error expresses error as a
percentage of the accepted value.
Significant Figures
• Often, precision is limited by the tools
available.
• Significant figures include all known digits
plus one estimated digit.
Significant Figures (cont.)
• Rules for significant figures
– Rule 1: Nonzero numbers are always significant.
– Rule 2: Zeros between nonzero numbers are
always significant.
– Rule 3: All final zeros to the right of the decimal
are significant.
– Rule 4: Placeholder zeros are not significant. To
remove placeholder zeros, rewrite the number in
scientific notation.
– Rule 5: Counting numbers and defined constants
have an infinite number of significant figures.
Rounding Numbers
• Calculators are not aware of significant
figures.
• Answers should not have more significant
figures than the original data with the fewest
figures, and should be rounded.
Rounding Numbers (cont.)
• Rules for rounding
– Rule 1: If the digit to the right of the last significant
figure is less than 5, do not change the last
significant figure.
– Rule 2: If the digit to the right of the last significant
figure is greater than 5, round up to the last
significant figure.
– Rule 3: If the digits to the right of the last significant
figure are a 5 followed by a nonzero digit, round up
to the last significant figure.
Rounding Numbers (cont.)
• Rules for rounding (cont.)
– Rule 4: If the digits to the right of the last significant
figure are a 5 followed by a 0 or no other number at
all, look at the last significant figure. If it is odd,
round it up; if it is even, do not round up.
Rounding Numbers (cont.)
• Addition and subtraction
– Round numbers so all numbers have the same
number of digits to the right of the decimal.
• Multiplication and division
– Round the answer to the same number of significant
figures as the original measurement with the fewest
significant figures.
Section 2.3 Assessment
Determine the number of significant
figures in the following:
8,200, 723.0, and 0.01.
A. 4, 4, and 3
A
0%
D
D. 2, 4, and 1
C
C. 2, 3, and 1
A. A
B. B
C. C
0%
0%
0%
D. D
B
B. 4, 3, and 3
Section 2.3 Assessment
A substance has an accepted density of
2.00 g/L. You measured the density as
1.80 g/L. What is the percent error?
A. 20%
A
0%
D
D. 90%
C
C. 10%
A. A
B. B
C. C
0%
0%
0%
D. D
B
B. –20%
Section 2.4 Representing Data
• Create graphics to
reveal patterns in data.
• Interpret graphs.
independent variable:
the variable that is
changed during an
experiment
graph
Graphs visually depict data, making it
easier to see patterns and trends.
Graphing
• A graph is a visual display of data that
makes trends easier to see than in a table.
Graphing (cont.)
• A circle graph, or pie chart, has wedges
that visually represent percentages of a
fixed whole.
Graphing (cont.)
• Bar graphs are often used to show how a
quantity varies across categories.
Graphing (cont.)
• On line graphs, independent variables are
plotted on the x-axis and dependent
variables are plotted on the y-axis.
Graphing (cont.)
• If a line through the points is straight, the
relationship is linear and can be analyzed
further by examining the slope.
Interpreting Graphs
• Interpolation is reading and estimating
values falling between points on the graph.
• Extrapolation is estimating values outside the
points by extending the line.
Interpreting Graphs (cont.)
• This graph shows important ozone
measurements and helps the viewer visualize
a trend from two different time periods.
Section 2.4 Assessment
____ variables are plotted on the
____-axis in a line graph.
A. independent, x
B. independent, y
D
A
0%
C
D. dependent, z
A. A
B. B
C. C
0%
0%
0%
D. D
B
C. dependent, x
Section 2.4 Assessment
What kind of graph shows how quantities
vary across categories?
A. pie charts
B. line graphs
D
A
0%
C
D. bar graphs
A. A
B. B
C. C
0%
0%
0%
D. D
B
C. Venn diagrams
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Section 2.1 Units and Measurements
Key Concepts
• SI measurement units allow scientists to report data
to other scientists.
• Adding prefixes to SI units extends the range of
possible measurements.
• To convert to Kelvin temperature, add 273 to the
Celsius temperature. K = °C + 273
• Volume and density have derived units. Density, which
is a ratio of mass to volume, can be used to identify an
unknown sample of matter.
Section 2.2 Scientific Notation and
Dimensional Analysis
Key Concepts
• A number expressed in scientific notation is written as a
coefficient between 1 and 10 multiplied by 10 raised to a
power.
• To add or subtract numbers in scientific notation, the
numbers must have the same exponent.
• To multiply or divide numbers in scientific notation,
multiply or divide the coefficients and then add or
subtract the exponents, respectively.
• Dimensional analysis uses conversion factors to solve
problems.
Section 2.3 Uncertainty in Data
Key Concepts
• An accurate measurement is close to the accepted
value. A set of precise measurements shows little
variation.
• The measurement device determines the degree of
precision possible.
• Error is the difference between the measured value and
the accepted value. Percent error gives the percent
deviation from the accepted value.
error = experimental value – accepted value
Section 2.3 Uncertainty in Data (cont.)
Key Concepts
• The number of significant figures reflects the
precision of reported data.
• Calculations should be rounded to the correct number
of significant figures.
Section 2.4 Representing Data
Key Concepts
• Circle graphs show parts of a whole. Bar graphs
show how a factor varies with time, location, or
temperature.
• Independent (x-axis) variables and dependent (y-axis)
variables can be related in a linear or a nonlinear
manner. The slope of a straight line is defined as
rise/run, or ∆y/∆x.
• Because line graph data are considered continuous,
you can interpolate between data points or
extrapolate beyond them.
Which of the following is the SI derived
unit of volume?
A. gallon
B. quart
D
A
0%
C
D. kilogram
A. A
B. B
C. C
0%
0%
0%
D. D
B
C. m3
Which prefix means 1/10th?
A. deciB. hemi-
C. kilo-
D
C
A
0%
B
D. centi-
A. A
B. B
C. C
0%
0%
0%
D. D
Divide 6.0  109 by 1.5  103.
A. 4.0  106
B. 4.5  103
C. 4.0  103
D
C
A. A
B. B
C. C
0%
0%
0%
D. D
B
0%
A
D. 4.5 
106
Round the following to 3 significant
figures 2.3450.
A. 2.35
B. 2.345
D
A
0%
C
D. 2.40
A. A
B. B
C. C
0%
0%
0%
D. D
B
C. 2.34
The rise divided by the run on a line graph
is the ____.
A. x-axis
B. slope
D
A
0%
C
D. y-intercept
A. A
B. B
C. C
0%
0%
0%
D. D
B
C. y-axis
Which is NOT an SI base unit?
A. meter
B. second
C. liter
D
C
A
0%
B
D. kelvin
A. A
B. B
C. C
0%
0%
0%
D. D
Which value is NOT equivalent to the
others?
A. 800 m
B. 0.8 km
D
A
0%
C
D. 8.0 x 105 cm
A. A
B. B
C. C
0%
0%
0%
D. D
B
C. 80 dm
Find the solution with the correct number
of significant figures:
25  0.25
A. 6.25
A
0%
D
D. 6.250
C
C. 6.3
A. A
B. B
C. C
0%
0%
0%
D. D
B
B. 6.2
How many significant figures are there in
0.0000245010 meters?
A. 4
B. 5
D
A
0%
C
D. 11
A. A
B. B
C. C
0%
0%
0%
D. D
B
C. 6
Which is NOT a quantitative measurement
of a liquid?
A. color
B. volume
D
A
0%
C
D. density
A. A
B. B
C. C
0%
0%
0%
D. D
B
C. mass
Click on an image to enlarge.
Table 2.2
SI Prefixes
Figure 2.10 Accuracy and Precision
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