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Data Analysis – measurements and units

Data Analysis – measurements and units. Dr. Chin Chu. Measurements. Temperature demonstration What have been learned here? Human senses are not reliable indicator of physical properties. We need instruments to give us unbiased determination of physical properties.

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Data Analysis – measurements and units

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  1. Data Analysis – measurements and units Dr. Chin Chu

  2. Measurements • Temperature demonstration • What have been learned here? • Human senses are not reliable indicator of physical properties. • We need instruments to give us unbiased determination of physical properties. • A system must be established to properly quantify the measurements. Scales and units.

  3. Measurements • Definition – comparison between measured quantity and accepted, defined standards (SI). • Quantity: • Property that can be measured and described by a pure number and a unitthat refers to the standard.

  4. Measurement Requirements • Know what to measure. • Have a definite agreed upon standard. • Know how to compare the standard to the measured quantity. Tools such as ruler, graduated cylinder, thermometer, balances and etc…

  5. Measurements – Units • SI Units – (the metric system) • Universally accepted • Scaling with 10 • Base Units: • Time (second, s) • Length (meter, m) • Mass (kilogram, kg) • Temperature (Kelvin, K) • Amount of a substance (mole, mol) • Electric current (ampere, A) • Luminous intensity (candela, cd)

  6. Measurements - Temperature • Temperature Scales: • Celsius (°C, centigrade) • Water freezing: 0 °C • Water boiling: 100 °C • Kelvin (K, SI base unit of temperature) • Same spacing as in Celsius scale. • Conversion: Celsius + 273 = Kelvin • Fahrenheit (°F) • Not the same spacing as the other two. • Conversion: Fahrenheit = (5/9)(Celsius -32)

  7. Measurements – Units • Derived Units: • Volume Units: (length)3, such as cm3,m3, dm3 (liter) • Density • Defined as mass per unit volume the substance occupies.

  8. Problem Solving Process ANALYZE THE PROBLEM Read the problem again. Identify what you are given and list the known data. Identify and list the unknowns. Gather information you need from graphs, tables or figures. Plan the steps you will follow to find the answer. THE PROBLEM Read the problem carefully. Be sure to understand what it is asking you. SOLVE THE UNKNOWN Determine whether you need a sketch to solve the problem. If the solution is mathematical, write the equation and isolate the unknowns. Substitute the known quantities into the equation. Solve the equation. Continue the solution process until you solve the problem.. EVALUATE THE ANSWER Re-read the problem. Is the answer reasonable. Check your math. Are the units and the significant figures correct? (Sect. 2.2 and 2.3)

  9. Problem Solving Process – Example THE PROBLEM: A metal cube is 2 cm on each edge and has a mass of 20 g. Is the cube made of pure aluminum? Density of pure aluminum is 2.7 g/cm3. THE APPROACH: Determining the nature of the metal, density is the parameter to compare. WHAT’S KNOWN: Density of pure Al WHAT’S UNKNOWN: Density of the metal cube WHAT ARE NEEDED: Mass and Volume THE MATH: Density = Mass/Volume HOW TO CALCULATE THE VOLUME OF A CUBE? WHAT’S KNOWN: Mass WHAT’S UNKNOWN: Volume THE MATH: Volume=(Edge)3 The edge of the cube is known.

  10. Problem Solving Process – Example THE SOLUTION: Construct the logic flow chart. Write backwards from the flow chart, the last step first then the 2nd last and so on. Actual solution of the problem: Volume of the cube = (edge of the cube)3 = (2 cm) 3 = 8 cm3 Density of the cube = (mass of the cube)/(volume of the cube) = 20 g/8 cm3 = 2.5 g/cm3 Comparison between the densities: density of the metal cube (2.5 g/cm3) is less than the density of the pure Al (2.7 g/cm3) Conclusion: the metal cube is not made of pure Al.

  11. Problem Solving Process – Challenge • What determines whether the object will float or sink in water? • Density of the object relative to water (1 g/cm3). • Sink if density of the object is higher than water. • Float if density of the object is smaller than water. • For a solid piece of aluminum (Al),the density is 2.7 g/cm3. Given a piece of Al that weights 27.0 grams. • Will it float or sink in water? Why? • If your answer is sink, what would you do to make it float?

  12. Data Analysis – significant figures Dr. Chin Chu

  13. 1 2 3 4 5 Significant Figures • Measurements are always done against a standard. • When we measure something, we can (and do) always estimate between the smallest marks. 4.5 mm mm

  14. Significant Figures • The better marks the better we can estimate. • Scientist always understand that the last number measured is actually an estimate, where the level of uncertainty is defined. 4.55 mm object 1 2 3 4 5 mm

  15. Significant Digits and Measurement • Measurement • Done with tools. • The value depends on the smallest subdivision on the measuring tool. • Significant Digits (Figures): • consist of all the definitely known digits plus one final digit that is estimated in between the divisions.

  16. Significant Figures • Only measurements have significant figures. • Counted numbers and defined constants are exact and have infinite number of significant figures. • A dozen is exactly 12 • 1000 mL = 1 L • Being able to locate, and count significant figures is an important skill.

  17. Significant Figures - Examples

  18. Significant Figures: Examples • What is the smallest mark on the ruler that measures 142.15 cm? • ____________________ • 142 cm? • ____________________ • 140 cm? • ____________________ • Does the zero count? • We need rules!!!

  19. Rules of Significant Figures • If there is a decimal point present start counting from the left to right until encountering the first nonzero digit. All digits thereafter are significant. • If the decimal point is absent start counting from the right to left until encountering the first nonzero digit. All digits are significant.

  20. Rules of Significant Figures - Examples Atlantic Ocean Pacific Ocean Example 1 78638 0.00078638 decimal point No decimal point Example 2 78638 78638000

  21. Significant Figures - Exercise In the following measurements , identify the number of significant figures, uncertainty level and estimated digit: • 120 cm • 0.00347 kg • 0.23400 L • 11.24 s • 1100. km • 4.560 x 10-3 m • 0.09720 g/mL

  22. Significant Figures - Answers In the following measurements , identify the number of significant figures, uncertainty level and estimated digit: • 120 cm [2 sig. fig.; ±10cm; 2] • 0.00347 kg [3 sig. fig.; ±0.00001kg; 7] • 0.23400 L [5 sig. fig.; ±0.00001L; the last “0”] • 11.24 s [4 sig. fig.; ±0.01s; 4] • 1100. km [4 sig. fig.; ±1km; the last “0”] • 4.560 x 10-3 m [4 sig. fig.; ±0.001x10-3m; 0] • 0.09720 g/mL [4 sig. fig.; ±0.00001g/mL; 0]

  23. Rounding Rules • Rounding is always from right to left. • Look at the number next to the one you’re rounding. 0 - 4 : leave it 5 - 9 : round up With one exception: when the number next to the one you’re rounding is 5 and not followed by nonzero digits (a.k.a. followed by all zeros) – round up if the number (rounding to) is odd; don’t do anything if it is even.

  24. Rounding Rules • Further explanation for the special rule regarding the last digit to be exactly 5:

  25. Rounding Rules - Examples 2.532 Example 1 2.532 2.53 < 5 > 5 > 5 leave it round up round up round up Last significant digit Last significant digit Last significant digit Last significant digit Example 2 2.536 2.536 2.54 the exception Example 3 2.5351 2.5351 2.54 Example 4 2.5350 2.5350 2.54 odd

  26. Rounding – Exercise Round the following measurements to the specified number of significant figures: • 4.256 cm to 2 sig. fig. • 123500 g to 3 sig. fig. • 0.00374 L to 2 sig. fig. • 2.3451 s to 3 sig. fig. • 5.675 miles to 3 sig. fig. • 0.34625 mm to 4 sig. fig.

  27. Rounding – Answers Round the following measurements to the specified number of significant figures: • 4.256 cm to 2 sig. fig. [4.3 cm] • 123500 g to 3 sig. fig. [124000 g] • 0.00374 L to 2 sig. fig. [0.0037 L] • 2.3451 s to 3 sig. fig. [2.35 s] • 5.675 miles to 3 sig. fig. [5.68 miles] • 0.34625 mm to 4 sig. fig. [0.3462 mm]

  28. Mathematical Operations Involving Significant Figures Addition and Subtraction The answer must have the same number of digits to the right of the decimal point as the value with the fewest digits to the right of the decimal point. Why? The result from the addition or subtraction would have the same precision as the least precise measurement.

  29. Mathematical Operations Involving Significant Figures Addition and Subtraction 28.0 cm 23.538 cm 25.68 cm Example: 28.0 cm Arrange the values so that decimal points line up. 23.538 cm 25.68 cm Do the sum or subtraction. Identify the value with fewest places after decimal point. 77.218 cm Round the answer to the same number of places. 77.2 cm

  30. Mathematical Operations Involving Significant Figures Multiplication and Division The answer must have the same number of significant figures as the measurement with the fewest significant figures.

  31. Mathematical Operations Involving Significant Figures Multiplication and Division 28.0 cm 23.538 cm 25.68 cm Example: 3 28.0 cm Carry out the operation. 23.538 cm Identify the value with fewest significant figures. 25.68 cm 16924.76352 cm3 Round the answer to the same significant figures. 16900 cm3

  32. Math Operation – Exercise Complete the following math calculations with proper significant figures: • 12.45 m + 34 m = _____ m • 1100 g + 123 g + 823.6 g = _______ g • 23.45 L - 5.572 L = ______ L • 24.1 mm x 2.7 mm = _______ mm2 • 0.965 m x 2.63 m x 0.5472 m = _______ m3 • 45.76 kg ÷ 25.67L = _________ kg/L

  33. Math Operation – Answers Complete the following math calculations with proper significant figures: • 12.45 m + 34 m = [46]m • 1200 g + 123 g + 823.6 g = [2100] g • 23.45 L - 5.572 L = [17.88] L • 24.1 mm x 2.7 mm = [65] mm2 • 0.965 m x 2.63 m x 0.5472 m = [1.39] m3 • 45.76 kg ÷ 25.67L = [1.783] kg/L

  34. Data Analysis – accuracy versus precision Dr. Chin Chu

  35. How Reliable are Measurements? • Multiple measurements are taken to ensure data integrity. • Assessments have to be made regarding how close the data are to the actual value (accuracy) and how close those multiple measurements are relative to each other (precision).

  36. Let’s use a golf analogy

  37. Accurate? No Precise? Yes

  38. Accurate? Yes Precise? Yes

  39. Precise? No Accurate? Maybe?

  40. Accurate? Yes Precise? We cannot say!

  41. Accuracy vs. Precision - Exercise • Three students measure the room to be 10.2 m, 10.3 m and 10.4 m across. • Were they precise? • Were they accurate?

  42. Accuracy vs. Precision - Answers • Three students measure the room to be 10.2 m, 10.3 m and 10.4 m across. • Were they precise? [Yes] • Were they accurate? [Not sure since the actual width of the room was not provided.]

  43. Data Analysis – scientific notations Dr. Chin Chu

  44. Scientific Notations • Mass of a proton is 0.00000000000000000000000000167262 kg • Mass of an electron is 0.000000000000000000000000000000910939 kg • Which one has more mass? • Hard to handle those numbers, right? • A better way has to be somewhere!

  45. Scientific Notations • THE MATH OF 10’s 1 = 1 = 100 10 = 10 = 101 100 = 10x10 = 102 1,000 = 10x10x10 = 103 10,000 = 10x10x10x10x10 = 104 0.1 = 1/10 = 1/101 = 10-1 0.01 = 1/100 = 102 = 10-2 0.001 = 1/1000 = 1/103 = 10-3 0.0001 = 1/10000 = 1/104 = 10-4 Did you see the pattern?

  46. Scientific Notation • The exponent of 10 indicate the number of digits away from the decimal point: • Positive value: to the left of the decimal point. • Negative value : to the right of the decimal point. • It provides us a way to shorten very large (or small) numbers into manageable parts. • Scientific Notation: multiple of two factors • Factor 1: a number between 1 and 10. • Factor 2: 10 raised to a power, or exponent

  47. Scientific Notation Example 1: 12670000 12670000 = 1267000 x 10 = 1267 00 x 10 x 10 = 12670 x 10 x 10 x 10 = 1267 x 10 x 10 x 10 x 10 = 126.7 x 10 x 10 x 10 x 10 x 10 = 12.67 x 10 x 10 x 10 x 10 x 10 x 10 = 1.267 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1.267 x 107 Alternatively, 12670000. 12670000 3 2 1 7 6 5 4 1.267 x 107

  48. Scientific Notation Example 2: 0.0000001267 0.0000001267 = 0.000001267 x (1/10) = 0.00001267 x (1/10) x (1/10) = 0.0001267 x (1/10) x (1/10) x (1/10) = 0.001267 x (1/10) x (1/10) x (1/10) x (1/10) = 0.01267 x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) = 12.67 x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) = 1.267 x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) x (1/10) = 1.267 x 10-7 0.0000001267 Alternatively, 1 2 3 4 5 6 7 1.267 x 10-7

  49. Scientific Notation – Prefixes Used with SI Units

  50. Scientific Notation – Operations • Addition and Subtraction • How does one add 34562 and 76541290? • Add them in columns! 3.4562 x104 34562 Obviously wrong answer! +) 76541290 7.6541290x107 11.1103290x10? 76575852 3.4562 x 104 Notice that the first digit 3 of the 1st number is right on top of the forth digit 4 of the 2nd number. How about move digits to line up? +) 7654.1290 x 104 7657.5852 x 104 The right answer! 76575852 When adding or subtracting numbers written in scientific notation, one must be sure that the exponents are the same before doing the arithmetic!

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