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 platinum-

iridium 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

• L X W X H• 1 cm3 = 1 mL = 1 cc

– Density• mass/volume

• DH2O = 1.00 g/mL

• 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

• Significant Figures– Represent measurements– Include digits that are known– One digit is estimated

How Reliable are Measurements?

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 387009 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 NOT significant. Convert to scientific notation.

0.0253 g has 32000 m has 1

Constants and counting numbers have infinite number of significant figures.

6 molecules60 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.

2.5322.53

Digit to immediate right of last significant figure >5, round up the last significant figure

2.5362.54

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 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 down.

2.53502.54

2.52502.52

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