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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
Section 2.1 Units and Measurements
• Define SI base units for time, length, mass, and temperature.
mass: a measurement that reflects the amount of matter an object contains
• Explain how adding a prefix changes a unit.
• Compare the derived units for volume and density.
Section 2-1
Section 2.1 Units and Measurements (cont.)
base unit
second
meter
kilogram
Chemists use an internationally recognized system of units to communicate their findings.
kelvin
derived unit
liter
density
Section 2-1
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.
Section 2-1
Units (cont.)
Section 2-1
Units (cont.)
Section 2-1
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 distance light travels in a vacuum in 1/299,792,458th of a second.
• The SI base unit of mass is the kilogram (kg), about 2.2 pounds
Section 2-1
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.
Section 2-1
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.
Section 2-1
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).
Section 2-1
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 2-2
Section 2.2 Scientific Notation and Dimensional Analysis
• Express numbers in scientific notation.
quantitative data: numerical information describing how much, how little, how big, how tall, how fast, and so on
• Convert between units using dimensional analysis.
Section 2-2
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.
Section 2-2
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.
Section 2-2
Scientific Notation (cont.)
800 = 8.0 102
0.0000343 = 3.43 10–5
• 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.
Section 2-2
Scientific Notation (cont.)
• Addition and subtraction
– Exponents must be the same.
– Rewrite values with the same exponent.
– Add or subtract coefficients.
Section 2-2
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.
Section 2-2
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.
Section 2-2
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.
Section 2-2
Dimensional Analysis (cont.)
• Using conversion factors
– A conversion factor must cancel one unit and introduce a new one.
Section 2-3
Section 2.3 Uncertainty in Data
• Define and compare accuracy and precision.
experiment: a set of controlled observations that test a hypothesis
• Describe the accuracy of experimental data using error and percent error.
• Apply rules for significant figures to express uncertainty in measured and calculated values.
Section 2-3
Section 2.3 Uncertainty in Data (cont.)
accuracy
precision
error
Measurements contain uncertainties that affect how a result is presented.
percent error
significant figures
Section 2-3
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.
Section 2-3
Accuracy and Precision (cont.)
• Error is defined as the difference between and experimental value and an accepted value.
Section 2-3
Accuracy and Precision (cont.)
• The error equation is error = experimental value – accepted value.
• Percent error expresses error as a percentage of the accepted value.
Section 2-3
Significant Figures
• Often, precision is limited by the tools available.
• Significant figures include all known digits plus one estimated digit.
Section 2-3
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.
Section 2-3
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.
Section 2-3
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.
Section 2-3
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.
Section 2-3
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 2-4
Section 2.4 Representing Data
• Create graphics to reveal patterns in data.
independent variable: the variable that is changed during an experiment
graph
• Interpret graphs.
Graphs visually depict data, making it easier to see patterns and trends.
Section 2-4
Graphing
• A graph is a visual display of data that makes trends easier to see than in a table.
Section 2-4
Graphing (cont.)
• A circle graph, or pie chart, has wedges that visually represent percentages of a fixed whole.
Section 2-4
Graphing (cont.)
• Bar graphs are often used to show how a quantity varies across categories.
Section 2-4
Graphing (cont.)
• On line graphs, independent variables are plotted on the x-axis and dependent variables are plotted on the y-axis.
Section 2-4
Graphing (cont.)
• If a line through the points is straight, the relationship is linear and can be analyzed further by examining the slope.
Section 2-4
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.
Section 2-4
Interpreting Graphs (cont.)
• This graph shows important ozone measurements and helps the viewer visualize a trend from two different time periods.
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