Presentation on theme: "Digital Values Digital Measurements –Integers only, “0” & “1” for computers On or Off, Yes or No, In or Out, up or down … Dozen eggs is exactly 12, not."— Presentation transcript:
Digital Values Digital Measurements –Integers only, “0” & “1” for computers On or Off, Yes or No, In or Out, up or down … Dozen eggs is exactly 12, not 12 +/-1 Biped has exactly 2 legs, tripod has 3 –NO fractions or partial values, just integers Relatively error free transcription Can apply automatic corrections, parity, ECC NO uncertainty, values are exact –Nature modeled digitally at atomic levels Quantum numbers, energy levels, spin direction
Analog Values Analog measurements, everyday norm –Variable quantities, any value allowed Intensity of light and sound, level of pain –Everyday life is continuously variable What we weigh, sense of smell & hearing –Values experienced are NOT fixed If any value is OK, how to prevent errors? Precision & accuracy become important
Number Notation Common symbols in text books –10 2 = 100, –√25 = 5 Calculators and computers (e.g. Excel) use other conventional symbols –100 = 10^2 = 10E2 (Excel) = 10exp2 (Casio) –25^0.5 = 25E0.5 = 25^(1/2) for square roots –y x also does ANY powers & roots
Why use Exponents? Huge range of values in nature –299,792,458 meters/sec speed of light –602,214,200,000,000,000,000,000 atoms/mole – meters, wavelength of red light – electron charge Much simpler to utilize powers of 10 –3.00*10 8 meters/sec speed of light –6.02*10 23 atoms/mole –6.25*10 -7 meters for wavelength red light –1.60* Coulombs for electron’s charge
People like small numbers Tend to think in 3’s –good, better, best (Sears appliances) –Small, medium, large (T-shirts, coffee serving) 1-3 digit numbers easier to remember –Temperature, weight, volume –Modifiers turn big back into small numbers 2000 lb 1 ton, 5280 feet 1 mile Kilograms, Megabytes, Gigahertz, picoliters (ink jet)
SI metric prefix nomenclature
more SI prefixes (also on Blackboard web)
Exponential or Scientific Notation keeps numbers relatively simple Decimal number identifying significant digits –Example: 5,050,520 Exponent of 10 identifies overall magnitude –Example: 10^6 or E6 (denoting 1 million) Combined expression gives entire value – x 10 6 (usual text book notation) – *10^6 (computers, Excel) – *10exp6 (some calculators) – E6 (alternative in Excel)
Exponential Notation Notation method –Leading digit (typically) before decimal point –Significant digits (2-3 typical) after decimal –Power of 10 after all significant digits More Examples –1,234 = x 10 3 = 1.234E3 (Excel) – = x = 1.234E-4 6-7/8 inch hat size, in decimal notation –6+7/8 = = inch decimal equivalent –6.875, could also write E1 = 68.75E-1
Exponential Notation 3100 x 210 = 651,000 In Scientific Notation: 3.100E3 x 2.10E2 Coefficients handled as usual numbers – x 2.10 6.51 with 3 significant digits Exponents add when values multiplied –E3 (1,000) * E2 (100) = E5 (100,000) –Asterisk (*) indicates multiplication in Excel Final answer is 6.51E5 = 6.51*10^5 –NO ambiguity of result or accuracy
Exponential Notation Exponents subtract in division –E3 (1,000) / E2 (100) = E1 (10) –Forward slash (/) indicates division Computers multiply & divide FIRST –Example 1+2*3= 7, not 9 –Example (1+2)*3 = 9 –Work inside parenthesis always done first –Use (extra) parenthesis to avoid errors
Significant Figures Precision must be tailored for the situation –Result cannot be more precise than input data Data has certain + uncertain aspects –Certain digits are known for sure –Final (missing) digit is the uncertain one –2/3 cups of flour (intent is not ) Fraction is exact, but unlimited precision not intended Context says the most certain part is 0.6 Uncertain part is probably the 2 nd digit Recipe probably works with 0.6 to 0.7 cups How to get rid of ambiguity?
Significant Figures “Sig Figs” = establish values of realistic influence –1cup sugar to 3 flour does not require exact ratio of –Unintended 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 cup Can’t be more Sig-Figs than least accurate measure –Final “Sig Fig” is “Uncertainty Digit” … least accurately known –adding gram sugar to 1.1 gram flour = 1.1 gram mixture
How to Interpret Sig-Figs (mostly common sense) All nonzero digits are significant –1.234 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
Handling 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 o C has only 1 significant figure –0.012 g has 2 significant figures – 1.51 nanometers (or meter), 3 sig figs Trailing zeroes that are to the right of a decimal point with numerical values are always significant: – mL has 3 significant figures –0.20 g has 2 significant figures –1.510 nanometers ( meters), 3 sig figs
More examples with zeros Leading zeros don’t count –Often just a scale factor ( = microgram) Middle zeros between numbers always count –1.001 measurement has 4 decades of accuracy Trailing zeros MIGHT count –YES if part of measured or defined value, –YES if placed intentionally, 7000 grains = 1 pound –NO if zeros to right of non-decimal point 1,000 has 1 sig-fig … but 1,000.0 has 5 sig-figs –NO if only to demonstrate scale Carl Sagan’s “BILLIONS and BILLIONS of stars” –Does NOT mean “BILLIONS” + 1 = 1,000,000,001
More Sig-Fig Examples Class interaction: how many sig figs below? Zeros between –60.8 has __ significant figures –39008 has __ sig-figs Zeros in front – has __ sig-figs – has __ sig-fig –0.012 has __ sig-figs Zeros at end –35.00 has __ sig-figs –8, has __ sig-figs –1,000 could be 1 or 4 … if 4 intended, best to write 1.000E4
More Sig-Fig Examples Zeros between –60.8 has 3 significant figures –39008 has 5 sig-figs Zeros in front – has 5 sig-figs – has 1 sig-fig –0.012 has 2 sig-figs Zeros at end –35.00 has 4 sig-figs –8, has 7 sig-figs –1,000 could be 1 or 4 … if 4 intended, best to write 1.000E4
Sig-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 figures –50,600 calories could be 3, 4, or 5 sig-figs Ambiguity is avoided using exponential notation to exactly define significant figures of 3, 4, or 5 by writing 50,600 calories as: –5.06 × 10E4 calories (3 significant figures) or –5.060 × 10E4 calories (4 significant figures), or – × 10E4 calories (5 significant figures). –Remember values right of decimal ARE significant
Exact Values Some 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 tripod Exact 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 calculation –2.54 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 !
more Sig-Fig Accounting Addition & Subtraction –Least Significant Figure determines outcome – = 2.01 (limited by 1.01) Multiplication & Division –Least Significant Figure determines outcome –1.01 x = 1.01 Round-Off –Calculators yield more sig-figs than justified –Must reduce answer to lowest sig-fig component
Sig-Fig Multiply & Divide Good first step to use scientific notation –Multiply * 5280 1.13E-1 * 5.280E3 Multiply the leading values, add the exponents Becomes E2 Sig.Fig. set by least precise input 5.96E2 –Divide 4995 by.0012 4.995E3 / 1.2E-3 Divide leading values, subtract the exponents Becomes E6 Sig.Fig. set by least precise input 4.2E6
Sig-Fig Addition & Subtraction First get the decimals (blue) to align –Take E3 same as 1,023.4 –Then add 1.0E-4same as –Then subtract 15.22same as –Do the math 1, –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 !
Partial Values Averages, fractions, yields –2/3 cups flour = …cups? –>2 digit precision inappropriate for cookies –See Mrs. Fields Cookie Recipe “superfluous accuracy” –unjustified or unwarranted level of detail –Precision needs to fit the situation “Rounding Off” to appropriate accuracy –Need rules to set the values
more Sig-Fig Accounting Round-Off –Calculations can yield more sig-figs than justified –Must reduce result to lowest sig-fig component Methodology (usual & customary rules) –If value beyond last sig-fig is ≥5, round UP For 3 sig-fig accuracy, becomes 5.26 –If value beyond last sig-fig is <5, round OFF For 3 sig-figs accuracy, becomes 5.25
Rounding Rules … Traditional Rule is Simplest When trailing digit is <5 round off –1.244 rounded to 3 digits 1.24 – rounded to 3 digits 1.24 When trailing digit is ≥5 round up –1.246 rounded to 3 digits 1.25 – rounded to 3 digits 1.25 Note lack of symmetry at “5” –5 is in the middle, but rounds up –Unintended bias is towards larger values
Rounding Rules … “Banker’s Rule” addresses bias When trailing digit is < 5 round off –1.244 rounded to 3 digits 1.24 When trailing digit is > 5 round up –1.246 rounded to 3 digits 1.25 What to do with a trailing “5” ? –Aim is equal opportunity, round up or down Try to avoid statistical bias in large data sets –“rule” is to look at digit preceding rounding Equal probability of odd or even value Arbitrary rule to round up if odd, down if even 17.8 also 17.8
Guidelines 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.
Rounding & Sig-Figs NOT exact Several papers illustrate the issues –Wikipedia article Rounding issues tend to be academic –Prof. Mulliss, Univ. of Toledo Ohio Tried millions of calculations to test the rules Add-Subtract simple rule ≈ 100% accurate Multiply-Divide standard rule ≈ 46% accurate Multiply-Divide (Std Rule+1) ≈ 59% accurate Mult-Divide best-case rules ≈ 90% accurate
Metric-English Conversions Convert 10.0 inches to centimeters –10.0 inch * 2.54 cm/inch = 25.4 cm –Precision is 3 sig figs, input & output –But …. Inches are bigger units of measure –3 rd significant figure for inches is 2-½ x larger ! Inches not the same size as centimeters! A tolerance setting problem for international companies Often add one more sig-fig to inches when converting
Take Away Message Rounding & Sig-Figs not infallible –It’s a math model, numbers on a page –Reality may be different (hopefully not by much) –Units of measure may not have same magnitude Utility is to make results more rational –Avoids a conclusion not justified by the input –Numerical methods fail when pushed too far Nature is not the problem –Our use of numbers and rules are the issue –Walt Kelly in “Pogo” had it right, “we’ve met the enemy … and it’s us”
Dimensional Analysis Making the units come out right –Useful strategy to avoid calculation errors Relies on “cancellation of dimensions” –If sec^2 instead of sec/sec cancel, something got inverted –Should always put dimensions on initial formulas Good News –Easy to do –Avoids silly answers with wrong dimensions. Bad News –Does not insure right physical relationships –No guarantee of right answer … but units OK
Dimensional 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) = mph –If 100 km/hr limit is exact (e.g …) An exact value leads to infinite precision … Mathematically correct, but impractical for speedometers –If 100 km/hr limit is NOT exact (e.g ) 3 sig fig limit sets speed at 62.1 mph 2 sig fig limit sets speed at 62 mph 1 sig fig sets speed limit at 60 mph
Dimensional Analysis Human Body Temperature –Accepted healthy value in USA is 98.6 o F Convert to Celsius: (98.6– 32) o F * (5 o C/9 o F) = 37.0 o C –Accepted (customary) value in Europe is 37 o C Convert to Fahrenheit (37 o C * 9 o F/5 o C) + 32oF = 99 o F Result is 2 sig-figs, and an apparent temperature rise –What happened… are Europeans bodies hotter? –2 digit sig-fig on a larger unit of measure ( o C), vs 3 sig figs on smaller degree ( o F) is inconsistent. Europeans might argue that variability between health people negates need for higher sig fig.
From Chem. 15 Lab Manual Exercises Page 2, # 4J ( – ) x (3.248E4 – E3) Solve what’s inside parenthesis FIRST –Initial value 1 st parenthesis E-3 –Subtract 2 nd value E-3 –Result after subtraction E-3 – Round to least accurate E-3 Second Parenthesis Calculation –3.248E4 same as 32, E3 –Subtract E3 same as 4, E3 –Result after subtraction 27, E3 –Round to low of 4 sig fig 27, E3 Multiply results from parenthesis calculations – * 27,890 = 39.9 –Multiplication accuracy limited to least sig figs = 3 in this case
Accuracy 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 precise Related 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?
Accuracy 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.
Good accuracy This example shows good accuracy, but low precision
Precision How 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).
Target analogy This example has high precision, but poor accuracy
Accuracy versus Precision
Barley, original standard for “Grain”
Standard Deviation, why bother? Range a poor indicator of accuracy –One bad measurement controls the range Averaging scheme redefines error –RMS (root mean squared) is common tool –Moves error to an average value basis –Suppresses random error contribution
Non-Linear representations Exponential Growth (or decline) –Changes associated with exponent of 10 value –Example: exp of 2 100x, exp of 3 1000x Moore’s Law, Chain Reaction of Uranium Logarithmic Scale –Some differences too large to put on a linear scale Hearing, visual acuity, earthquakes, concentration of ions –Logarithm scale “compresses” scales “decibel” for sound, “pH” for acid concentration Richter scale for earthquakes –Richter 9 (SF 1906) is 1000x that of Richter 6 (mild shake)
Gordon Moore’s Law transistors in a CPU doubles every months
Summary, Exponential Notation Number represented by decimal + exponent –Example: 1,234 = 1.234*10^3 or 1.234E3 Multiplication: –multiply decimals, add exponents –6*10^6 x 2*10^2 = 12*10^8 = 1.2*10^9 (or 1.2E9) Division: –Divide decimals, subtract denominator exponent from numerator –6*10^6 / 2*10^2 = 3*10^3 (or 3E3) Addition & Subtraction –Line up numbers by decimal (and same exponent) before adding Add 1,234 or E3 Add to or E3 Sum = 1, or E3
Summary, Significant Figures All nonzero digits are significant –1.234 has 4 significant figures, “0” between nonzero digits significant: –1002 has 4 significant figures “0” after decimals always significant – has 5 significant figures “0” in front of decimal NOT significant – has 3 significant figures “0” after non-zero digit MAY be significant –1,000 could be 1, 2, 3, or 4 significant figures 4 if an exact number, e.g grams per kilogram Depends on context, better to write in exponentials
Summary, Rounding Rules When trailing digit is <5 round off (truncate) –1.244 rounded to 3 digits 1.24 When trailing digit is ≥5 round up – rounded to 3 digits 1.25 Lack of symmetry at “5” –Unintended bias is towards larger values –“Banker’s rule”: look at digit preceding rounding Equal probability of odd or even value Arbitrary rule to round up if odd, down if even 17.8 also 17.8
Summary, Dimensional Analysis Relies on “cancellation of dimensions” –7 days/week * 52 week/year = 364 days/year –Always put dimensions on initial formulas List starting and ending (desired) dimensions Conversion dimensions between start and end Multiply or divide to eliminate unwanted dimensions –Writing dimensions avoids squared vs cancelled Use exact values when practical –Avoids sig-fig confusion Round off answer only after all calculations –Rounding too soon can multiply uncertainty error