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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
– 3.00*108 meters/sec speed of light
– 6.02*1023 atoms/mole
– 6.25*10-7 meters for wavelength red light
– 1.60*10-19 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
– 5.05052 x 106
(usual text book notation)
– 5.05052*10^6
(computers, Excel)
– 5.05052*10exp6 (some calculators)
– 5.05052E6
(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 = 1.234 x 103 = 1.234E3 (Excel)
– 0.0001234 = 1.234 x 10-4 = 1.234E-4
• 6-7/8 inch hat size, in decimal notation
– 6+7/8 = 6+0.875 = 6.875 inch decimal equivalent
– 6.875, could also write 0.6875E1 = 68.75E-1
Exponential Notation
• 3100 x 210 = 651,000
• In Scientific Notation: 3.100E3 x 2.10E2
• Coefficients handled as usual numbers
– 3. 100 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 0.66666666667)
• Fraction is exact, but unlimited precision not intended
• Context says the most certain part is 0.6
• Uncertain part is probably the 2nd 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 0.3333333
– 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 .000001 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 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 Examples
Class 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
• 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) =
62.13711922 mph
– If 100 km/hr limit is exact (e.g. 100.00000 …)
• An exact value leads to infinite precision 62.13711922 …
• Mathematically correct, but impractical for speedometers
– If 100 km/hr limit is NOT exact (e.g. 99.5 - 100.4)
• 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 98.6oF
• Convert to Celsius: (98.6– 32) oF * (5oC/9oF) = 37.0oC
– Accepted (customary) value in Europe is 37oC
• Convert to Fahrenheit (37oC * 9oF/5oC) + 32oF = 99oF
• 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 Manual
Exercises 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
– Result after subtraction
0.0014265
1.4265 E-3
–
0.00143
1.43
Round to least accurate
E-3
Second Parenthesis Calculation
– 3.248E4
same as
32,480
– Subtract 4.58983E3 same as
•
E-3
32.48
E3
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 analogy
This example has high precision, but poor accuracy
Accuracy versus Precision
Barley, original standard for “Grain”
BARLEY
Laura
Julie
Elise
Nancy
Kan
Jacob
Annie
Oye
Connie
Average
Maximum
Miimum
Range (Max-Min)
Range % of Average
Number of
Grains
100
100
100
100
100
100
100
100
100
100.00
100.00
100.00
0.00
Grams
of Weight
4.040
3.362
3.483
3.957
3.880
3.764
3.473
3.829
3.759
3.73
4.04
3.36
0.68
grams
per Grain
0.0404
0.0336
0.0348
0.0396
0.0388
0.0376
0.0347
0.0383
0.0376
0.0373
0.0404
0.0336
0.0068
18%
grains
per pound
11,238
13,504
13,035
11,473
11,701
12,062
13,072
11,857
12,078
12,224
13,504
11,238
2,266
Standard Deviation
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 18-24 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
• Add to
5.678
or
• Sum =
1,239.678
or
1.234 E3
.005678 E3
1.23978 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
– 0.12300 has 5 significant figures
• “0” in front of decimal NOT significant
– 0.00000000123 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. 1000 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
– 1.2460111 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
• 17.75  17.8
also 17.85  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