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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. 2.5322.53 Digit to immediate right of last significant figure >5, round up 2.5362.54 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 2.53502.54 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 2.52502.52 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