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DATA ANALYSIS

Learn to calculate percent error, use scientific notation, determine significant figures, and perform dimensional analysis accurately for precise metrics. Explore examples and rules for improving accuracy in calculations.

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DATA ANALYSIS

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  1. DATA ANALYSIS Percent Error Scientific Notation Using Significant Figures Using the Metric System Metric Conversion dimensional analysis Accuracy and Precision Graphing Techniques

  2. Calculating Percent Error % Error = lexperimental value –accepted valuel* X 100% accepted or actual value Use the density of water experiment to check % error • always use absolute value for this calculation

  3. Calculating Percent Error EXAMPLE – A student determines the density of a piece of wood to be .45g/cm. The actual value is .55g/cm. What is the student’s percent error? .45 - .55 X 100% = .10 x 100% = 18% .55 .55

  4. Calculating Percent Error • EXAMPLE – In the lab you determine that the melting point of a substance is 55 C. the accepted melting point is 53 C. What is you percent error in this experiment? 55 – 53 x 100% = 2 x 100% = 3.8% 53 53

  5. Scientific Notation • Scientific Notation: Easy way to express very large or small numbers • A x 10x • A – number with one non-zero digit before decimal • x -exponent- whole number that expresses the number decimal places • if x is (-) then it is a smaller • if x is (+) than it is larger

  6. Calculating in Scientific notation • Multiplying- • Multiple the numbers • Add the exponents • (2.0 x 104) (4.0 x 103) = 8.0 x 107

  7. Dividing • divide the numbers • subtract the numerator exponent from the denominator exponent • 9.0 x 107 • 3.0 x 105 3.0 x 102

  8. Add or subtract • get the exponents of all # to be the same • calculate as stated • make sure the final answer is in correct scientific notation form • 7.0 x 10 4 + 3.0 x 10 3 = • 7.0 x 10 4 + .3 x 10 4 = 7.3 x 10 4 70000 + 3000 = 73000= 7.3 x104

  9. Using Significant Figures (Digits) • value determined by the instrument of measurement plus one estimated digit • reflects the precision of an instrument • example – if an instrument gives a length value to the tenth place – you would estimate the value to the hundredths place

  10. 1 2 3 4 5 Significant figures (sig figs) How many numbers mean anything When we measure something, we can (and do) always estimate between the smallest marks.

  11. Significant figures (sig figs) object 1 2 3 4 5 mm The better marks the better we can estimate. Scientist always understand that the last number measured is actually an estimate

  12. 1. all non-zero # are Sig fig- 314g 3sf 12,452 ml 5sf 2. all # between non-zero # are sig fig 101m 3sf 6.01mol 3sf 36.000401s 8s 3. place holders are not sf 0.01kg 1sf

  13. 4. zeros to the right of a decimal are sig fig if 3.0000s 5sf Preceded by non-zero 0.002m 1sf 13.0400m 6sf 5. Zero to right of non-zero w/o decimal point 600m 1sf are not sig fig 600.m 3sf 600.0 m 4sf 600.00 m 5sf

  14. RULES FOR USING SIGNIFICANT FIGURES • use the arrow rule to determine the number of significant digits • decimal present all numbers to right of the first non zero are significant (draw the arrow from left to right) ----------> 463 3 sig. digits ----------> 125.78 5 sig. digits ----------> .0000568 3 sig. digits ----------> 865 000 000. 9 sig. digits

  15. RULES FOR USING SIGNIFICANT FIGURES • use the arrow rule to determine the number of significant digits • decimal not present < -------- all numbers to the left of the first non zero are significant(draw arrow from right to left) 246 000 <---------- 3 sig. digits 400 000 000 <---------- 1 sig. digit

  16. Significant figures http://www.youtube.com/watch?v=puvE8hF6zrY

  17. Rules of Significant Figures Pacific: 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. Atlantic: If the decimal point is absent start counting from the right to left until encountering the first nonzero digit. All digits are significant.

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

  19. Use appropriate rules for rounding • If the last digit before rounding is less than 5 it does not change ex. 343.3 to 3 places  343 1.544 to 2 places  1.54 • If the last digit before rounding is greater than 5 – round up one ex. 205.8 to 3 places  206 10.75 to 2 places  11

  20. 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.

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

  22. 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.

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

  24. use fewest number of decimal places rule for addition and subtraction 1) 2) 3) 4) 24.05 5.6 237.52 88 123.770 28 - 21.4 - 4.76 0.46 8.75 10.2 7 _________ ______ _______ ______

  25. Use least number of significant figuresrule for multiplication and division • 23.7 x 6.36 2) .00250 x 14 3) 750. / 25 4) 15.5 / .005

  26. II. Reliability of Measurement • ACCURACY – how close a measured value is to the accepted value • PRECISION – how close measurements are to one another - if measurements are precise they show little variation * Precise measurements may not be accurate

  27. Precision- refers to how close a series of measurements are to one another; precise measurements show little variation over a series of trials but may not be accurate. • LESS THAN .1 IS PRECISE • Oscar performs an experiment to determine the density of an unknown sample of metal. He performs the experiment three times: • 19.30g/ml • 19.31g/ml • 19.30g/ml • Certainty is +/- .01 Are his results precise?

  28. Accuracy – refers to how close a measured value is to an (theoretical) accepted value. • The metal sample was gold( which has a density of 19.32g/ml) • Certainty is +/- .01 • Are his results accurate? Need to calculate percent error. • 5% OR LESS IS ACCURATE

  29. Oscar finds the volume of a box 2.00cm3 (ml) • It is really 3.00ml -is it precise? To know if it is precise you need more trials • Accurate? Percent error Experimental – Actual X 100% = Actual 2-3 3 X 100 = 33.3%

  30. Activity: basket and paper clip 1. Throw 3 paper clips at basket 2. Measure the distance from the basket to determine accuracy and precision 3.distance to center of basket- trial-1 trial-2 trial-3 4. Calculate % error 5. were you precise or accurate- or both?

  31. Using the Metric System A. Why do scientists use the metric system? • The metric system was developed in France in 1795 - used in all scientific work because it has been recognized as the world wide system of measurement since 1960. • SI system is from the French for Le Systeme International d’Unites. • The metric system is used in all scientific work because it is easy to use. The metric system is based upon multiples of ten. Conversions are made by simply moving the decimal point.

  32. Base Units (Fundamental Units)Used in This Class QUANTITY NAME SYMBOL _______________________________________________ Length meter m ----------------------------------------------------------------------------- Mass kilogram kg ------------------------------------------------------------------------------- Amount of Substance mole mol ------------------------------------------------------------------------------- Time second s _______________________________________________

  33. Derived Units • Base Units – independent of other units • Derived Units – combination of base units Examples • density  g/L (grams per liter) • volume  m x m x m = meters cubed • Velocity  m/s (meters per second

  34. SI Prefixes Prefix Symbol Multiplication Factor Term Pico p (0.000 000 000 001) one trillionth Nano n (0.000 000 001) one billionth Micro u (0.000 001) one millionth Milli m (0.001) one thousandth Centi c (0.01) one hundredth Deci d (0.1) one tenth ------------------------------------------------------------------------------- One Unit 1 one ------------------------------------------------------------------------------

  35. SI Prefixes Prefix Symbol Multiplication Factor Term Deka da 10 ten Hecto h 100 one hundred Kilo k 1000 one thousand Mega M 1 000 000 one million Giga G 1 000 000 000 one billion Tera T 1 000 000 000 000 one trillion

  36. Metric Units Used In This Class QUANTITY NAME SYMBOL Length meter m centimeter cm millimeter mm kilometer km Mass gram g kilogram kg centigram cg milligram mg Volume liter (liquid) L (l) milliliter* (liquid) mL (ml) cubic centimeter* (solid) cm3

  37. Metric Units Used In This Class Density grams/milliliter (liquid) g/mL grams/cubic centimeter (solid) g/cm3 grams/liter (gas) g/L Time second s minute min hour h • volume measurement for a liquid and a solid ( 1 mL = 1 cm3) These are equivalents.

  38. Measuring Temperature PROPERTY FAHRENHEIT CELSIUS KELVIN Water boils 212.0 100 373 Body temperature 98.6 37 310 Water freezes 32.0 0 273 Absolute zero -460 -273 0 Equalities (Fahrenheit and Celsius) F = 9/5 C + 32 C = 5/9 (F – 32) (Celsius and Kelvin) K = C + 273 C = K – 273

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

  40. Factor label method /Dimensional analysis Use equalities to problem solve converting units. quantity desired = quantity given x conversion factor ( ratio/ equality)

  41. Equalities You Need To Know 1 km = 1000 m 1 m = 100 cm 1 m = 1000 mm 1L = 1000 mL 1kg = 1000g 1 g = 100cg 1 g = 1000 mg

  42. use equality sheet - convert equalities to scientific notation. • underline what you are looking for • circle given • multiplication sign-draw line • unit given on bottom • unit desired on top • 1 goes with the prefix

  43. Remember- reciprocals – 1/10-3 is equal to 103 or 1000 1/103 is equal to 10 -3 or .001 • Convert 300 s to ms • ms = 300 s x 1ms • 1x10-3 s = • 300 x 103 ms (1000) or 300,000 ms

  44. Factor label method /Dimensional analysis Use equalities to problem solve converting units. quantity desired = quantity given x conversion factor (equality) A-given unit B-desired unit C-given unit  A x B C B C must equal 1 use equality sheet

  45. Four-step approach When using the Factor-Label Method it is helpful to follow a four-step approach in solving problems: 1.What is question – How many sec in 56 min 2. What are the equalities- 1 min = 60 sec 3. Set up problem (bridges) 56 min 60 sec 1 min 4. Solve the math problem -multiple everything on top and bottom then divide 56 x 60 / 1

  46. Graphing • graph – a visual representation of data that reveals a pattern • Bar- comparison of different items that vary by one factor • Circle – depicts parts of a whole • Line graph- depicts the intersection of data for 2 variables • Independent variable- factor you change • Dependent variable – the factor that is changed when independent variable changes

  47. Graphing • Creating a graph- must have the following points • Title graph • Independent variable – on the X axis – horizontal- abscissa • Dependent variable – on Y axis – vertical- ordinate • Must label the axis and use units • Plot points • Scale – use the whole graph • Draw a best fit line- do not necessarily connect the dots and it could be a curved line.

  48. GRAPHING • Interpreting a graph • Slope- rise Y2 –Y1 • Run X2 –X1 • relationship • direct – a positive slope • inverse- a negative slope • equation for a line – y = mx + b • m-slope • b – y intercept • extrapolate-points outside the measured values- dotted line • interpolate- points not plotted within the measured values-dotted line

  49. GRAPHING • Creating a graph- must have the following points 1.Title graph 2. Independent variable –on the X axis–horizontal- abscissa 3. Dependent variable – on Y axis – vertical- ordinate 4. Must label the axis and use units 5. Plot points 6. Scale – use the whole graph 7. Draw a best fit line- do not necessarily connect the dots and it could be a curved line.

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