Measurement Analog and Digital –Both are natural and common in real world Exponential Notation...

Measurement• Analog and Digital

– Both are natural and common in real world

• Exponential Notation

– Decimal coefficient + Power of 10 multiplier

• Significant Figures

– Accuracy and precision

• Rounding Off

– Rules for rounding up and down

• Dimensional Analysis

– Mind the units being converted

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


– Both are natural and common in real world• Exponential Notation

– Decimal coefficient + Power of 10 multiplier• Significant Figures

– Accuracy and precision• Rounding Off

– Rules for rounding up and down• Dimensional Analysis

– Mind the units being converted• Accuracy and Precision

– Average, max, min, deviation, standard deviation

Number Notation

• Common symbols in text books

– 102 = 100,

– √25 = 5

• Calculators and computers (e.g. Excel) use other conventional symbols

– 100 = 10^2 or 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 Notationkeeps 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

– Single digit (typically) before decimal point

– Significant digits (2-3 typical) after decimal

– Power of 10 after the 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, also OK is 0.6875E1 = 6.875E0 = 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







• Rounding Off



– Mind the units

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:

– 0.001 oC has only 1 significant figure

– 0.012 g has 2 significant figures

– 1.51 nanometers (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, 1.001

– 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 ExamplesHow 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


• Zeros at end


– 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.

• Most 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.

• Apparently significant figures in any exact number must be ignored when determining the number of significant figures in the result of a calculation

– 2.54 cm per inch (exact, NOT 3 sig figs)

– 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

• Multiplication & Division

– Least Significant Figure determines outcome

– 1.01 x 1.0000001 = 1.01

• Round-Off

– Calculations can 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 (or 596.64)

• 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 (4,162,500 zeros=magnitude)

• 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 !







• Rounding Off



– Mind the units

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

jj• Mostly single digits in

recipe, no tolerances on measurements, so what is intended degree of precision?

• 1 cup flour could be 0.5 cup (rounded up to 1) to 1.4 cup (rounded down to 1)

• ½ cup sugar (0.5 cup) could be 0.45 to 0.54 cup per rounding rules

• 1/3 cup chocolate chips may be exactly 0.3333333333333333 cups, probably not.

• Context (& common sense) must be used to interpret precision of values seen in daily life.

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 <4 round down/off

– 1.244 rounded to 3 digits 1.24


• When trailing digit is ≥5 round up



• 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


• When trailing digit is > 5 round up


• 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

Rounding & Sig-Figs NOT exact

• Several papers illustrate the issues

– See 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

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 8.6

– 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.

Don’t round off until the endExample shows common error

Good NOT so goodA Initial value 20 miles / gallon 20B reference value 5,280 feet / mile 5,200C=A*B interim result 105,600 feet / gallon 104,000

D reference value (exact) 4.00 quart / gallon 4.00E reference value 1.06 quart / liter 1.00F=C*E/D Interim result 27,984 feet / liter 26,000

G reference value (exact) 12 inches / foot 12H reference value (exact) 2.54 cm / inch 2.5I reference value (exact) 100 cm/meter 100J reference value (exact) 1,000 km / meter 1,000K=F*G*H / (I*J) final result 8.530 km / liter 7.800

Rounded to 3 sig fig 8.53 7.80Error 8.55%

Moral: round at the end, not along the way !

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”







• Rounding Off



– Mind the units

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

• 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 healthy people negates need for higher sig fig.

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

Addition & Subtraction(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 (perhaps thousands) 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

Big Slip Video

• Is this example precise or accurate?

• Experiment repeated by Myth Busters

– Performed experiment into a lake

• Hitting water less damaging than dirt

• Show Video ….

Ancient Dimensions … the "Cubit"Defined as distance between elbow (upright on table) to farthest fingertip

Pyramid of Giza, height when built 280 Cubits

Noah's Ark length, Biblical reference 300 Cubits

Our Class Inch Deviation Variance, orPerson measured inch/cubit from Average Deviation^2

1 Penka 18.0 0.40 0.1602 Danielle 15.0 3.40 11.5603 Jason 18.5 0.10 0.0104 Rey 18.0 0.40 0.1605 Mohammed 20.0 1.60 2.5606 Dinesh 19.5 1.10 1.2107 James 18.5 0.10 0.0108 Jose 19.0 0.60 0.3609 Michael 20.5 2.10 4.410

10 Olga 17.0 1.40 1.960

Sum of Squares 22.400

Average 18.4 N = number 10Maximum 20.5 N-1 9Miimum 15.0 Sum / (N-1) 2.5Range 5.5 Std Dev = sqrt 1.6

If "True Value" = 18.0 inches, historical value

"error" for class is 2.2% (True- Measured)/true

Statistical Terms• Mean value

– Sum of measurements divided by their number (18.4)• Mode

– The most common or “popular” value (18.5 twice)• Range

– The largest versus smallest measured value (5.5)• Deviation (from the average)

– How each measurement differs from average• Standard Deviation

– A mathematical way to minimize influence of “flyers”

– Our result was 1.6, a lot less than range of 5.5

– “RMS” (Root Mean Square) used a lot in Engineering

Reviewing Cubit Data

• Our class data is a useful example

• Accuracy was good, agrees with history

– Historical value 18 inches

• Precision not as good, a lot of variation

– Size varies greatly between individuals

– Average is more consistent

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

Standard Deviation

End of lecture on Sig Figs

Deviation & Variation

• Variance is the average of the squared differences between data points and the mean. Variance is tabulated in units squared.

• Standard deviation is the square root of the sum of variances, and measures the spread of data about the mean, with the same units.

• Said more formally, the standard deviation is the root mean square (RMS) deviation of values from their arithmetic mean.

Non-Linear representations• Exponential Growth (or decline)

– Changes associated with exponent of 10 value

– Example: exp of 2100x, exp of 31000x

• 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 Original Idea

Moore’s Law applies elsewhere

Moore’s Law

• Moore's Law describes an important trend in the history of computer hardware: that the number of transistors that can be inexpensively placed on an integrated circuit is increasing exponentially, doubling approximately every two years.[1] The observation was first made by Intel co-founder Gordon E. Moore in a 1965 paper.[2][3][4] The trend has continued for more than half a century and is not expected to stop for a decade at least and perhaps much longer.[5]

More exact rounding rules• If digit dropped > 5, the last retained digit increases by one.

– For example, 12.6 is rounded to 13 (no change from simple rule)

• If the digit dropped is < 5, the last remaining digit left as-is.

– For example, 12.4 is rounded to 12 (no change from simple rule)

• If the digit dropped = 5, and following digit not zero, the last remaining digit is increased by one (happens often).

– For example, 12.51 is rounded to 13

• If the digit dropped = 5 and followed by all zeroes, the last remaining digit increases by one if odd, left as is when even

– For example, 11.5 is rounded to 12, 12.5 is rounded to 12.

• This rule means if the digit to be dropped is 5 followed only by zeroes, the result is always rounded up to an even digit.

• This “5 with all trailing zeros” rule not applicable very often. The rationale is to avoid bias in rounding: half of the time we round up (odd), half the time we round down (even).

Rounding Problems• Consider the value “1 inch”

– Has 1 significant figure, perhaps already rounded

– Original value could have been 0.51 to 1.49

– Range of uncertainty also about “1”, same as value!

• Consider conversion to centimeters

– 1 inch * 2.54 cm/inch = 2.54 cm

– Rounding to 1 sig-fig yields “3 cm”

• Problem is that cm is >2x larger than inch

– Original uncertainty about ½ inch in either direction

– ½ inch is 2.54/2 = 1.27 cm uncertainty

– Uncertainty therefore exceeds rounded amount !

Final disposition models & analogies

Grains from Tuesday labRye Barley 1 Barley 2 Wheat

1 0.02870 0.027 0.040 0.0352 0.01873 0.029 0.031 0.0573 0.02876 0.027 0.029 0.0214 0.02877 0.029 0.031 0.0385 0.02878 0.027 0.032 0.0296 0.02880 0.021 0.028 0.0567 0.02882 0.029 0.029 0.0428 0.02884 0.018 0.028 0.0359 0.02886 0.036 0.031 0.026

10 0.02889 0.040 0.013 0.045Average 0.02780 0.028 0.029 0.038

Max 0.02889 0.040 0.040 0.057Min 0.01873 0.018 0.013 0.021

Range 0.01016 0.022 0.027 0.036% of Avg 37% 78% 92% 94%

Rounding Rules … “McMurry’s Rule” in text, pg. 21

• When trailing digit is < 5 round off


• When trailing digit is ≥ 6 round up


• What to do with a trailing “5” ?

– If non-zero digits follow rounded value, round UP

• 5.664525 rounded to 4 sig.fig 5.665

– If no digits follow round off point, round OFF

• 5.664500 rounded to 4 sig.fig 5.664

Measurement Analog and Digital –Both are natural and common in real world Exponential Notation...

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Transcript of Measurement Analog and Digital –Both are natural and common in real world Exponential Notation...