1. 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

2. 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

3. 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

4. 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*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

5. 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)

6. SI metric prefix nomenclature6

7. more SI prefixes (also on Blackboard web)7

8. 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 (usual text book notation)
*10^ (computers, Excel)
*10exp6 (some calculators)
E (alternative in Excel)

9. Exponential Notation Notation method More Examples
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 103 = 1.234E3 (Excel)
= x 10-4 = 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 9

10. Exponential Notation 3100 x 210 = 651,000
In Scientific Notation: E3 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 10

11. 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 11

12. 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 2nd digit
Recipe probably works with 0.6 to 0.7 cups
How to get rid of ambiguity?

13. 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

14. 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

15. 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 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

16. 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

17. 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

18. More Sig-Fig Examples Zeros between Zeros in front Zeros at end
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 18

19. 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:
× 10E4 calories (3 significant figures) or
× 10E4 calories (4 significant figures), or
× 10E4 calories (5 significant figures).
Remember values right of decimal ARE significant

20. 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 !

21. more Sig-Fig Accounting
Addition & Subtraction
Least Significant Figure determines outcome
= (limited by 1.01)
Multiplication & Division
1.01 x = 1.01
Round-Off
Calculators yield more sig-figs than justified
Must reduce answer to lowest sig-fig component

22. 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  4.995E3 / 1.2E-3
Divide leading values, subtract the exponents
Becomes E6
Sig.Fig. set by least precise input  4.2E6 22

23. Sig-Fig Addition & Subtraction
First get the decimals (blue) to align
Take E3 same as 1,023.4
Then add 1.0E-4 same as
Then subtract same as
Do the math ,
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 ! 23

24. Partial Values Averages, fractions, yields “superfluous accuracy”
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

25. 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

26. 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

27. 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  also  17.8

28. 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.

29. 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

30. 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

31. 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”

32. 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

33. 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 33

34. 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. 34

35. From Chem. 15 Lab Manual Exercises Page 2, # 4J
( – ) x (3.248E4 – E3)
Solve what’s inside parenthesis FIRST
Initial value 1st parenthesis E-3
Subtract 2nd value E-3
Result after subtraction E-3
Round to least accurate E-3
Second Parenthesis Calculation
3.248E4 same as , E3
Subtract E3 same as , E3
Result after subtraction , E3
Round to low of 4 sig fig , E3
Multiply results from parenthesis calculations
* 27,890 = 
Multiplication accuracy limited to least sig figs = 3 in this case

36. 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?

37. 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.

38. Good accuracy This example shows good accuracy, but low precision

39. 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).

40. Target analogy This example has high precision, but poor accuracy

41. Accuracy versus Precision

42. Barley, original standard for “Grain”42

43. Standard Deviation 43

44. 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 44

45. 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) 45

46. Gordon Moore’s Law transistors in a CPU doubles every 18-24 months46

47. 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 , or E3
Add to or E3
Sum = , or E3

48. 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

49. 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
17.75  also  17.8

50. 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 50