Presentation on theme: "Digital Values Digital Measurements"— Presentation transcript:
1Digital Values Digital Measurements Integers only, “0” & “1” for computersOn or Off, Yes or No, In or Out, up or down …Dozen eggs is exactly 12, not 12 +/-1Biped has exactly 2 legs, tripod has 3NO fractions or partial values, just integersRelatively error free transcriptionCan apply automatic corrections, parity, ECCNO uncertainty, values are exactNature modeled digitally at atomic levelsQuantum numbers, energy levels, spin direction
2Analog Values Analog measurements, everyday norm Variable quantities, any value allowedIntensity of light and sound, level of painEveryday life is continuously variableWhat we weigh, sense of smell & hearingValues experienced are NOT fixedIf any value is OK, how to prevent errors?Precision & accuracy become important
3Number Notation Common symbols in text books 102 = 100,√25 = 5Calculators and computers (e.g. Excel) use other conventional symbols100 = 10^2 = 10E2 (Excel) = 10exp2 (Casio)25^0.5 = 25E0.5 = 25^(1/2) for square rootsyx also does ANY powers & roots
4Why use Exponents? Huge range of values in nature 299,792,458 meters/sec speed of light602,214,200,000,000,000,000,000 atoms/molemeters, wavelength of red lightelectron chargeMuch simpler to utilize powers of 103.00*108 meters/sec speed of light6.02*1023 atoms/mole6.25*10-7 meters for wavelength red light1.60*10-19 Coulombs for electron’s charge
5People like small numbers Tend to think in 3’sgood, better, best (Sears appliances)Small, medium, large (T-shirts, coffee serving)1-3 digit numbers easier to rememberTemperature, weight, volumeModifiers turn big back into small numbers2000 lb 1 ton, 5280 feet 1 mileKilograms, Megabytes, Gigahertz, picoliters (ink jet)
8Exponential or Scientific Notation keeps numbers relatively simple Decimal number identifying significant digitsExample: 5,050,520Exponent of 10 identifies overall magnitudeExample: 10^6 or E6 (denoting 1 million)Combined expression gives entire valuex (usual text book notation)*10^ (computers, Excel)*10exp6 (some calculators)E (alternative in Excel)
9Exponential Notation Notation method More Examples Leading digit (typically) before decimal pointSignificant digits (2-3 typical) after decimalPower of 10 after all significant digitsMore Examples1,234 = x 103 = 1.234E3 (Excel)= x 10-4 = 1.234E-46-7/8 inch hat size, in decimal notation6+7/8 = = inch decimal equivalent6.875, could also write E1 = 68.75E-19
10Exponential Notation 3100 x 210 = 651,000 In Scientific Notation: E3 x 2.10E2Coefficients handled as usual numbersx 2.10 6.51 with 3 significant digitsExponents add when values multipliedE3 (1,000) * E2 (100) = E5 (100,000)Asterisk (*) indicates multiplication in ExcelFinal answer is 6.51E5 = 6.51*10^5NO ambiguity of result or accuracy10
12Significant Figures Precision must be tailored for the situation Result cannot be more precise than input dataData has certain + uncertain aspectsCertain digits are known for sureFinal (missing) digit is the uncertain one2/3 cups of flour (intent is not )Fraction is exact, but unlimited precision not intendedContext says the most certain part is 0.6Uncertain part is probably the 2nd digitRecipe probably works with 0.6 to 0.7 cupsHow to get rid of ambiguity?
13Significant Figures“Sig Figs” = establish values of realistic influence1cup sugar to 3 flour does not require exact ratio ofUnintended accuracy termed “superfluous precision”Need to define actual measurement precision intended“Cup of flour” in recipe could be +/- 10% or 0.9 to 1.1 cupCan’t be more Sig-Figs than least accurate measureFinal “Sig Fig” is “Uncertainty Digit” … least accurately knownadding gram sugar to 1.1 gram flour = 1.1 gram mixture
14How to Interpret Sig-Figs (mostly common sense) All nonzero digits are significant1.234 g has 4 significant figures,1.2 g has 2 significant figures.“0” between nonzero digits significant:3.07 Liters has 3 significant figures.1002 kilograms has 4 significant figures
15Handling zeros in Sig-Figs Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point (overall magnitude):0.001 oC has only 1 significant figure0.012 g has 2 significant figures1.51 nanometers (or meter), 3 sig figsTrailing zeroes that are to the right of a decimal point with numerical values are always significant:mL has 3 significant figures0.20 g has 2 significant figures1.510 nanometers ( meters), 3 sig figs
16More examples with zeros Leading zeros don’t countOften just a scale factor ( = microgram)Middle zeros between numbers always count1.001 measurement has 4 decades of accuracyTrailing zeros MIGHT countYES if part of measured or defined value,YES if placed intentionally, 7000 grains = 1 poundNO if zeros to right of non-decimal point1,000 has 1 sig-fig … but 1,000.0 has 5 sig-figsNO if only to demonstrate scaleCarl Sagan’s “BILLIONS and BILLIONS of stars”Does NOT mean “BILLIONS” + 1 = 1,000,000,001
17More Sig-Fig Examples Class interaction: how many sig figs below? Zeros between60.8 has __ significant figures39008 has __ sig-figsZeros in fronthas __ sig-figshas __ sig-fig0.012 has __ sig-figsZeros at end35.00 has __ sig-figs8, has __ sig-figs1,000 could be 1 or 4 … if 4 intended, best to write 1.000E4
18More Sig-Fig Examples Zeros between Zeros in front Zeros at end 60.8 has 3 significant figures39008 has 5 sig-figsZeros in fronthas 5 sig-figshas 1 sig-fig0.012 has 2 sig-figsZeros at end35.00 has 4 sig-figs8, has 7 sig-figs1,000 could be 1 or 4 … if 4 intended, best to write 1.000E418
19Sig-Fig Exponential Notation A number ending with zeroes NOT to right of decimal point are not necessarily significant:190 miles could be 2 or 3 significant figures50,600 calories could be 3, 4, or 5 sig-figsAmbiguity is avoided using exponential notation to exactly define significant figures of 3, 4, or 5 by writing 50,600 calories as:× 10E4 calories (3 significant figures) or× 10E4 calories (4 significant figures), or× 10E4 calories (5 significant figures).Remember values right of decimal ARE significant
20Exact ValuesSome numbers are exact because they are known with complete certainty, or are defined by exact values:Many exact numbers are simple integers:12 inches per foot, 12 eggs per dozen, 3 legs to a tripodExact numbers are considered to have an infinite number of significant figures.Apparent significant figures in any exact number can be ignored when determining the number of significant figures in the result of a calculation2.54 cm per inch (exact)5/9 Centigrade/Fahrenheit degree (exact)5280 feet per mile (exact, based on definitions)The challenge is to remember which numbers are exact !
21more Sig-Fig Accounting Addition & SubtractionLeast Significant Figure determines outcome= (limited by 1.01)Multiplication & Division1.01 x = 1.01Round-OffCalculators yield more sig-figs than justifiedMust reduce answer to lowest sig-fig component
22Sig-Fig Multiply & Divide Good first step to use scientific notationMultiply * 5280 1.13E-1 * 5.280E3Multiply the leading values, add the exponentsBecomes E2Sig.Fig. set by least precise input 5.96E2Divide 4995 by 4.995E3 / 1.2E-3Divide leading values, subtract the exponentsBecomes E6Sig.Fig. set by least precise input 4.2E622
23Sig-Fig Addition & Subtraction First get the decimals (blue) to alignTake E3 same as 1,023.4Then add 1.0E-4 same asThen subtract same asDo the math ,Round to least decimal sig fig 1,008.2“spitting in the ocean” analogy … if you measure ocean volume by cubic meters or miles, adding a teaspoon is undetectable !23
24Partial Values Averages, fractions, yields “superfluous accuracy” 2/3 cups flour = …cups?>2 digit precision inappropriate for cookiesSee Mrs. Fields Cookie Recipe“superfluous accuracy”unjustified or unwarranted level of detailPrecision needs to fit the situation“Rounding Off” to appropriate accuracyNeed rules to set the values
25more Sig-Fig Accounting Round-OffCalculations can yield more sig-figs than justifiedMust reduce result to lowest sig-fig componentMethodology (usual & customary rules)If value beyond last sig-fig is ≥5, round UPFor 3 sig-fig accuracy, becomes 5.26If value beyond last sig-fig is <5, round OFFFor 3 sig-figs accuracy, becomes 5.25
26Rounding Rules … Traditional Rule is Simplest When trailing digit is <5 round off1.244 rounded to 3 digits 1.24rounded to 3 digits 1.24When trailing digit is ≥5 round up1.246 rounded to 3 digits 1.25rounded to 3 digits 1.25Note lack of symmetry at “5”5 is in the middle, but rounds upUnintended bias is towards larger values
27Rounding Rules … “Banker’s Rule” addresses bias When trailing digit is < 5 round off1.244 rounded to 3 digits 1.24When trailing digit is > 5 round up1.246 rounded to 3 digits 1.25What to do with a trailing “5” ?Aim is equal opportunity, round up or downTry to avoid statistical bias in large data sets“rule” is to look at digit preceding roundingEqual probability of odd or even valueArbitrary rule to round up if odd, down if even17.75 also 17.8
28Guidelines for using calculators Don’t round off too soon, do it at end of calculation(5.00 / 1.235) (6.35 / 4.0)=1st division results in 3 sig-figs, last division results in 2 sig-figs.3 numbers added should result in 1 digit after the decimal. Thus, the correct rounded final result should be 8.6. This final result has been limited by the accuracy in the last division.Warning: carrying all digits through to the final result before rounding is critical for many mathematical operations in statistics. Rounding intermediate results when calculating sums of squares can seriously compromise the accuracy of the result.
29Rounding & Sig-Figs NOT exact Several papers illustrate the issuesWikipedia articleRounding issues tend to be academicProf. Mulliss, Univ. of Toledo OhioTried millions of calculations to test the rulesAdd-Subtract simple rule ≈ 100% accurateMultiply-Divide standard rule ≈ 46% accurateMultiply-Divide (Std Rule+1) ≈ 59% accurateMult-Divide best-case rules ≈ 90% accurate
30Metric-English Conversions Convert 10.0 inches to centimeters10.0 inch * 2.54 cm/inch = 25.4 cmPrecision is 3 sig figs, input & outputBut …. Inches are bigger units of measure3rd significant figure for inches is 2-½ x larger !Inches not the same size as centimeters!A tolerance setting problem for international companiesOften add one more sig-fig to inches when converting
31Take Away Message Rounding & Sig-Figs not infallible It’s a math model, numbers on a pageReality may be different (hopefully not by much)Units of measure may not have same magnitudeUtility is to make results more rationalAvoids a conclusion not justified by the inputNumerical methods fail when pushed too farNature is not the problemOur use of numbers and rules are the issueWalt Kelly in “Pogo” had it right, “we’ve met the enemy … and it’s us”
32Dimensional Analysis Making the units come out right Useful strategy to avoid calculation errorsRelies on “cancellation of dimensions”If sec^2 instead of sec/sec cancel, something got invertedShould always put dimensions on initial formulasGood NewsEasy to doAvoids silly answers with wrong dimensions.Bad NewsDoes not insure right physical relationshipsNo guarantee of right answer … but units OK
33Dimensional Analysis Speed Limit 100 km/hr vs. miles/hr (100 km/hr *1000 m/km *100 cm/m) / (2.54 cm/inch*12 inch/foot*5280 foot/mile) = mphIf 100 km/hr limit is exact (e.g …)An exact value leads to infinite precision …Mathematically correct, but impractical for speedometersIf 100 km/hr limit is NOT exact (e.g )3 sig fig limit sets speed at 62.1 mph2 sig fig limit sets speed at 62 mph1 sig fig sets speed limit at 60 mph33
34Dimensional Analysis Human Body Temperature Accepted healthy value in USA is 98.6oFConvert to Celsius: (98.6– 32) oF * (5oC/9oF) = 37.0oCAccepted (customary) value in Europe is 37oCConvert to Fahrenheit (37oC * 9oF/5oC) + 32oF = 99oFResult is 2 sig-figs, and an apparent temperature riseWhat happened… are Europeans bodies hotter?2 digit sig-fig on a larger unit of measure (oC), vs 3 sig figs on smaller degree (oF) is inconsistent.Europeans might argue that variability between health people negates need for higher sig fig.34
35From Chem. 15 Lab Manual Exercises Page 2, # 4J ( – ) x (3.248E4 – E3)Solve what’s inside parenthesis FIRSTInitial value 1st parenthesis E-3Subtract 2nd value E-3Result after subtraction E-3Round to least accurate E-3Second Parenthesis Calculation3.248E4 same as , E3Subtract E3 same as , E3Result after subtraction , E3Round to low of 4 sig fig , E3Multiply results from parenthesis calculations* 27,890 = Multiplication accuracy limited to least sig figs = 3 in this case
36Accuracy and Precision Accuracy is the degree of conformity of a measured or calculated quantity to its actual (true) value.Precision, also called reproducibility or repeatability, is the degree to which further measurements or calculations show the same or similar results.A measurement can be accurate but not precise; precise but not accurate; neither; or both.A result is valid if it is both accurate and preciseRelated terms are error (random variability) and bias (non-random or directed effects) caused by a consistent and possibly unrelated factor.Show water slide video … is he accurate or precise?
37Accuracy Degree of error in achieving the established measurement goal The Cubit average value has not changed much since biblical times at about 18 inches so it has remained relatively accurate over hundreds of years.
38Good accuracy This example shows good accuracy, but low precision
39PrecisionHow well multiple measurements agree with one another to provide a consistent value. (e.g. tight grouping, low dispersion, “all together” series of events).The “cubit” is not a very precise measure of distance, since it varies between observers using the same definition. No two people are the same, so length data is dispersed. (e.g. inconsistent individual measurements).
40Target analogy This example has high precision, but poor accuracy
44Standard Deviation, why bother? Range a poor indicator of accuracyOne bad measurement controls the rangeAveraging scheme redefines errorRMS (root mean squared) is common toolMoves error to an average value basisSuppresses random error contribution44
45Non-Linear representations Exponential Growth (or decline)Changes associated with exponent of 10 valueExample: exp of 2100x, exp of 31000xMoore’s Law, Chain Reaction of UraniumLogarithmic ScaleSome differences too large to put on a linear scaleHearing, visual acuity, earthquakes, concentration of ionsLogarithm scale “compresses” scales“decibel” for sound, “pH” for acid concentrationRichter scale for earthquakesRichter 9 (SF 1906) is 1000x that of Richter 6 (mild shake)45
46Gordon Moore’s Law transistors in a CPU doubles every 18-24 months 46
47Summary, Exponential Notation Number represented by decimal + exponentExample: 1,234 = 1.234*10^3 or 1.234E3Multiplication:multiply decimals, add exponents6*10^6 x 2*10^2 = 12*10^8 = 1.2*10^9 (or 1.2E9)Division:Divide decimals, subtract denominator exponent from numerator6*10^6 / 2*10^2 = 3*10^3 (or 3E3)Addition & SubtractionLine up numbers by decimal (and same exponent) before addingAdd , or E3Add to or E3Sum = , or E3
48Summary, Significant Figures All nonzero digits are significant1.234 has 4 significant figures,“0” between nonzero digits significant:1002 has 4 significant figures“0” after decimals always significanthas 5 significant figures“0” in front of decimal NOT significanthas 3 significant figures“0” after non-zero digit MAY be significant1,000 could be 1, 2, 3, or 4 significant figures4 if an exact number, e.g grams per kilogramDepends on context, better to write in exponentials
49Summary, Rounding Rules When trailing digit is <5 round off (truncate)1.244 rounded to 3 digits 1.24When trailing digit is ≥5 round uprounded to 3 digits 1.25Lack of symmetry at “5”Unintended bias is towards larger values“Banker’s rule”: look at digit preceding roundingEqual probability of odd or even valueArbitrary rule to round up if odd, down if even17.75 also 17.8
50Summary, Dimensional Analysis Relies on “cancellation of dimensions”7 days/week * 52 week/year = 364 days/yearAlways put dimensions on initial formulasList starting and ending (desired) dimensionsConversion dimensions between start and endMultiply or divide to eliminate unwanted dimensionsWriting dimensions avoids squared vs cancelledUse exact values when practicalAvoids sig-fig confusionRound off answer only after all calculationsRounding too soon can multiply uncertainty error50