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
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 - PowerPoint PPT Presentation
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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
– 102 = 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
– yx 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
– 0.000000625 meters, wavelength of red light
– 0.0000000000000000001602 electron charge• Much simpler to utilize powers of 10
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 oC has only 1 significant figure
– 0.012 g has 2 significant figures
– 1.51 nanometers (or 0.00000000151 meter), 3 sig figs
• Trailing zeroes that are to the right of a decimal point with numerical values are always significant:
– 0.0230 mL has 3 significant figures
– 0.20 g has 2 significant figures
– 1.510 nanometers (0.000000001510 meters), 3 sig figs
More examples with zeros• Leading zeros don’t count
– Often just a scale factor (0.000001 = 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, 1.0010
– 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 ExamplesClass interaction: how many sig figs below?
• Zeros between
– 60.8 has __ significant figures
– 39008 has __ sig-figs
• Zeros in front
– 0.093827 has __ sig-figs
– 0.0008 has __ sig-fig
– 0.012 has __ sig-figs
• Zeros at end
– 35.00 has __ sig-figs
– 8,000.000 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
– 0.093827 has 5 sig-figs
– 0.0008 has 1 sig-fig
– 0.012 has 2 sig-figs
• Zeros at end
– 35.00 has 4 sig-figs
– 8,000.000 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
– 5.0600 × 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
– 1.01 + 1.00000001 = 2.01 (limited by 1.01)
• Multiplication & Division
– Least Significant Figure determines outcome
– 1.01 x 1.0000001 = 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 0.113 * 5280 1.13E-1 * 5.280E3
• Multiply the leading values, add the exponents
• Becomes 5.96640E2
• 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 4.1625E6
• Sig.Fig. set by least precise input 4.2E6
Sig-Fig Addition & Subtraction
• First get the decimals (blue) to align
– Take 1.0234E3 same as 1,023.4
– Then add 1.0E-4 same as + 0.0001
– Then subtract 15.22 same as - 15.22
– Do the math 1,008.1803
– 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 = 0.66666666666666 …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, 5.255123 becomes 5.26
– If value beyond last sig-fig is <5, round OFF
• For 3 sig-figs accuracy, 5.254459 becomes 5.25
Rounding Rules … Traditional Rule is Simplest
• When trailing digit is <5 round off
– 1.244 rounded to 3 digits 1.24
– 1.2449999 rounded to 3 digits 1.24
• When trailing digit is ≥5 round up
– 1.246 rounded to 3 digits 1.25
– 1.2460111 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
• 17.75 17.8 also 17.85 17.8
Guidelines for using calculators
• Don’t round off too soon, do it at end of calculation
– (5.00 / 1.235) + 3.000 + (6.35 / 4.0)
– 4.04858 + 3.000 + 1.5875 = 8.630829
– 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
– 3rd 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
• 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 (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.
From Chem. 15 Lab ManualExercises Page 2, # 4J
(0.0048965 – 0.00347) x (3.248E4 – 4.58983E3)
• Solve what’s inside parenthesis FIRST
– Initial value 1st parenthesis 0.0048965 4.8965 E-3
– Subtract 2nd value 0.00347 3.47 E-3
– Result after subtraction 0.0014265 1.4265 E-3
– Round to least accurate 0.00143 1.43 E-3
• Second Parenthesis Calculation
– 3.248E4 same as 32,480 32.48 E3
– Subtract 4.58983E3 same as 4,589.83 - 4.58983 E3
– Result after subtraction 27,890.17 27.89017 E3
– Round to low of 4 sig fig 27,890 27.89 E3
• Multiply results from parenthesis calculations
– 0.00143 * 27,890 = 39.88270 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 analogyThis example has high precision, but poor accuracy
Accuracy versus Precision
Barley, original standard for “Grain”
BARLEY Number of Grams grams grainsGrains of Weight per Grain per pound