Measurement Exponential Notation –Decimal coefficient + Power of 10 multiplier Significant Figures –Accuracy and precision Rounding Off –Rules for rounding.

Measurement• 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 Jaguar 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

– 1,000 * 100 = 100,000

– 10 3 * 10 2 = 10 5

– 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

Measurement

• Exponential Notation




• 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 or 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), 4 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 has ___ sig figs

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.

• When using an exact number in a calculation, the idea of significant figures for that item is 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 !

Measurement





• 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 <5 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

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 !

Measurement





• 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 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

• (e.g. CA auto in Europe, Mexico, or Canada)

– (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 posted limit exact (e.g. 100.00000 …)

• An exact value leads to infinite precision 62.13711922 …

• Mathematically correct, but impractical for speedometers

– If 100 km/hr posted speed limit is NOT exact

• 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.

• 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).

Cubits in Class

• Historical unit of measure was “cubit”

– Distance between elbow and middle finger

– Biblical refrerences (Noah’s ark)

– Used by pyramid builders in Egypt

• What is precision and accuracy?

– Let’s take some data !

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

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

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

• Missing target in water less damaging than hitting dirt if you miss the wading pool!

• Show Video ….

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

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

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.

http://en.wikipedia.org/wiki/Root_mean_square

http://en.wikipedia.org/wiki/Arithmetic_mean

Standard Deviation

Measurement Exponential Notation –Decimal coefficient + Power of 10 multiplier Significant Figures –Accuracy and precision Rounding Off –Rules for rounding.

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