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Chapter 2
Data Analysis
Units of Measurement
• Measurement
– Comparison to a standard
• Standard
– Well defined
– Make consistent measurements
• Useful measurement
– Number
– Unit
• SI Units
– Système Internationale d’Unités—SI
– Standard unit of measure
Units of Measurement
• Base units
– 7 base units (p. 26 Table
2-1)
– Defined unit
– Based on object or event
in physical world
– Independent of other
units
– Time
• Frequency of microwave
radiation given off by
cesium-133 atom
• Second, s
– Length
• Distance light travels
through a vacuum in
1/299792458 of a second.
• Meter, m
– Mass
• Defined by the platinumiridium metal cylinder
• Kilogram, kg
– Volume
• Measure of the amount of
a liquid
• Liter, L
Units of Measurement
• Prefixes
– Table p. 26
•
•
•
•
mega- micro
hecto (h_): 102
deka (da_ or dk_): 10
decimeter
– 1 dm = .1 m
– 10 cm = 1 dm
– 1000 cm3 = 1 dm3
• King Henry Died By
Drinking Chocolate
Milk
• Yotta (Y_): 1024
– 1 septillion
• Yocto (y_): 10-24
– 1 septillionth
Units of Measurement
• Derived Units
– Require a combination of base units
– Volume
• LXWXH
• 1 cm3 = 1 mL = 1 cc
– Density
•
•
•
•
•
mass/volume
DH O = 1.00 g/mL
2
D = m/v
M = DV
V = M/D
• Practice p. 29 #1-3; p. 30 #4-11; p. 50 #51-57
Units of Measurement
• Temperature
– Measure of how hot or cold an object is relative to
other objects
– kelvin, K
• Water
– freezes about 273 K
– boils about 373 K
Scientific Notation and Dimensional
Analysis
• Scientific notation expresses numbers as:
–
–
–
–
M x 10n
M is a number between 1 & 10
Ten raised to a power (exponent)
n is an integer
• Adding & subtracting
– Exponents must be the same
• Multiplying & dividing
– Multiply or divide first factors
– Add exponents for multiplication
– Subtract exponents for division
• Practice Problems p. 32 #12-16; p. 50 #75-78
Scientific Notation and Dimensional
Analysis
• Dimensional analysis
– Solving problems with conversion factors
– Conversion factor
• Ration based on an equality
• Ex. 12 in./1 ft. or 1 ft./12 in.
• Ex. 7 days/1 wk
– Focuses on units used
48 km =? m
(48 km)X (1000 m/1km) = 48,000 m
Scientific Notation and Dimensional
Analysis
What is a speed of 550 m/s in km/min?
• Practice Problems p. 35 #19-28; p. 51 #79-80
How Reliable are Measurements?
Accuracy and Precision
• Accuracy
– The nearness of a measurement to its accepted value
• Precision
– The agreement between numerical values of two or
more measurements that have been made in the
same way.
• You can be precise without being accurate.
• Systematic errors can cause results to be precise
but not accurate
How Reliable are Measurements?
Accuracy and Precision
• Percent error
– Compares the size of an error to the size of the
accepted value
• Calculating Percent Error (Relative Error)
– Percent error =
error
X 100
Value Accepted
– Error = Value Accepted – Value Experimental
– Take the absolute difference
• Ignore if positive or negative integer
How Reliable are Measurements?
• Error in Measurement
– Some error or uncertainty exists in all
measurement
• No measurement is known to an infinite number of
decimal places.
– All measurements should include every digit
known with certainty plus the first digit that is
uncertain
• Practice Problems p. 38 #29-30; p. 51 #81-82
How Reliable are Measurements?
• Significant Figures
– Represent measurements
– Include digits that are known
– One digit is estimated
How Reliable are Measurements?
Significant Figures
Rule
Examples
Non-zero numbers are always
significant
72.3 g has 3
Zeros between non-zero numbers
are always significant
40.7 L has 3
87009 has 5
All final zeros to the right of the
decimal place are significant
6.20 g has 3
Zeros that act as placeholders are
0.0253 g has 3
NOT significant. Convert to scientific 2000 m has 1
notation.
Constants and counting numbers
have infinite number of significant
figures.
6 molecules
60 s = 1 min
How Reliable are Measurements?
• Rounding off numbers
Rule
Example
Digit to immediate right of last significant figure <5, do not
change the last significant figure.
2.5322.53
Digit to immediate right of last significant figure >5, round up 2.5362.54
the last significant figure
Digit to immediate right of last significant figure = 5 AND
followed by a nonzero digit, round up last significant figure.
2.5351--.2.54
Digit to immediate right of last significant figure = 5 AND is
2.53502.54
not followed by a nonzero digit, look at last significant figure.
If it is an odd digit, round it up; if it is an even digit, round it
2.52502.52
down.
How Reliable are Measurements?
• Rounding off numbers
– Addition and subtraction
• Answer must have same number of digits to right of the
decimal place as value with fewest digits to the right of
the decimal point.
• Example:
1.24 mL
12.4 mL
+ 124 mL
137.64 mL = 138 mL
How Reliable are Measurements?
• Rounding off numbers
– Multiplication and division
• Answer must have same number of significant figures
as the measurement with the fewest significant figures
• Practice problems: p. 39 #31-32; p. 41 #33-36;
p. 42 #37-44; p. 51 #83-85
Representing Data
• Graphing
– Circle graphs
• Also called pie chart
• Show parts of a fixed whole, usually percents
– Bar graph
•
•
•
•
•
Show how a quantity varies with factors
Ex. Time, location, temperature
Measured quantity on y-axis (vertical axis)
Independent variable on x-axis (horizontal axis)
Heights show how quantity varies
Representing Data
• Line Graphs
–
–
–
–
Points represent intersection of data for two variables
Independent variable on x-axis
Dependent variable on y-axis
Best fit line
•
•
•
•
•
Equal points above and below line
Straight line—variables directly related
Rises to the right—positive slope
Sinks to the right—negative slope
Slope = y2-y1 = Δy
x2-x1 Δx
Representing Data
• Interpreting Graphs
– Identify independent and dependent variables
– Look at ranges of data
– Consider what measurements were taken
– Decide if relationship is linear or nonlinear
• Practice problems p. 51-52 #86-87