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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.