1. Percent Error, Precision of Measurement,
Accuracy, Precision,
Percent Error, Precision of Measurement,
Significant Figures, &
Scientific Notation
101

2. Learning Objectives
The Learners Will (TLW) collect data and make measurements with accuracy and precision, and will be able to calculate percent error and precision of measurement (TEKS 2.F)
TLW be able to express and manipulate quantities and perform math operations using scientific notation and significant figures (TEKS 2.G)

3. Agenda Why are Correct Measurement and Proper Units Important?
Part 1 – Units of Measurements
A. Number versus Quantity
B. Review SI Units
C. Derived Units
D. Problem Solving
Part 2 – Using Measurement
A. Accuracy vs. Precision
B. Percent Error
C. Precision of Measurement
D. Significant Figures
E. Scientific Notation
F. Using Both Scientific Notation & Significant Figures

4. I. Units of Measurement

5. How Calibrated are Your Eyes?
Measure the long side of the wooden block at your lab bench
Measure the level in the two graduated cylinders
Measure of the level in the beaker
Write your findings on the chart at the front of the room by recording the above measurements in the columns indicated

6. How Calibrated are Your Eyes?
Actual measurement of long side of wooden block is cm .
Actual Measurement in each cylinder and beaker is 7.0 mL .

7. Lab Results Did we all come up with exactly the same numbers?
Why or Why not?
Which measurements are close to one another?
How close did you come to actual measurement?
Which measurements matched correct one?

8. Causes of Measurement Errors / Differences
Equipment
Smaller is better – sm. grad. cyl. vs. beaker, sm. vs. lg. rulers
Condition – lg. grad. cyl. Clouded, residue in bottom, broken ruler
Humanness
Physical - eyesight
Care – focus, taking time to do job right
Attention to detail/instruction - measuring in correct system of units
Technique
Knowing how to properly use equipment
Cylinders sitting on table, bottom of meniscus
Random
Spillage, evaporation, air bubbles, sabotage

9. Why are Correct Measurement and Proper Units Important?
Sept. 30, NASA lost a $125 million Mars orbiter because a Lockheed Martin engineering team used English units of measurement for force (pounds) instead of metric (Newton), while the agency's team used the more conventional metric system for a key spacecraft operation. The units mismatch prevented navigation information from transferring between the Mars Climate Orbiter spacecraft team in at Lockheed Martin in Denver and the flight team at NASA's Jet Propulsion Laboratory in Pasadena, California.
After a 286-day journey, the probe fired its engine for the final time on September 23 to push itself into orbit. The engine fired, but the spacecraft came within 60 km (36 miles) of the planet due to the accumulation of small errors over all the firings and the spacecraft was lost in space.

10. A. Number vs. Quantity UNITS MATTER!!
Quantity = number + unit
UNITS MATTER!!

11. B. SI Units l meter m m kilogram kg v liters L t second s T K or C n
Quantity
Symbol
Base Unit
Abbrev. l
Length
meter m m
Mass
kilogram kg
Volume v
liters L t
second s
Time T
Kelvin or
Centigrade
K or C
Temp
Amount n
mole
mol

12. B. US Conventional Units
Quantity
Symbol
Base Unit
Abbrev.
Length l
feet ft m
Mass
pound lb
Volume v
gallons
gal t
second s
Time T
Rankine or
Fahrenheit
R or F
Temp
Amount n
mole
mol

13. B. SI Prefixes Prefix Symbol Factor mega- M 106 kilo- k 103 BASE UNIT
---
100
deci- d
10-1
centi- c
10-2
milli- m
10-3
micro- 
10-6
nano- n
10-9
pico- p
10-12

14. M V D = C. Derived Units Combination of base units. Volume (m3 or cm3)
height  width  length
1 cm3 = 1 mL
1 dm3 = 1 L
Density (kg/m3 or g/cm3)
mass per volume
D = M V

15. D. Problem-Solving Steps
1. Analyze - identify the given & unknown.
2. Plan - setup problem, use conversions.
3. Compute -cancel units, round answer.
4. Evaluate - check units, use sig figs.

16. D. Problem Solving Example – Density
A liquid has a volume of 29 mL and a mass of 25 g? What is the density?
GIVEN:
V = 29 mL
M = 25 g
D = ?
WORK:
D = M V
D = g
29 mL
D = g/mL

17. D. Problem Solving Example – Density
An object has a volume of 825 cm3 and a density of 13.6 g/cm3. Find its mass.
GIVEN:
V = 825 cm3
D = 13.6 g/cm3
M = ?
WORK:
M = DV
M = 13.6 g/cm3 x cm3
M = 11,200 g

18. II. Using Measurements

19. A. Accuracy vs. Precision
Accuracy - how close a measurement is to the accepted value (published, target)
Precision - how close a series of measurements are to each other
ACCURATE = CORRECT
PRECISE = CONSISTENT

20. A. Accuracy vs. Precision
PRECISE – a golfer hits 20 balls from the same spot out of the sand trap onto the fringe of the green. Each shot is within 5 inches of one another. Wow – that’s CONSISTENT
ACCURATE – the golfer’s 20 shots aren’t very accurate, because they need to be much closer to the hole so she can score easily – that would be CORRECT

21. B. Percent Error your value accepted value
Indicates accuracy of a measurement obtained during an experiment as compared to the literature * value (* may be called accepted, published, reference, etc.)
Error is the difference between the experimental value and the accepted value (or vice versa). Can be written with or without absolute sign – negative indicates one value is lower than the other.
your value
accepted value

22. For our purposes a percent error of
< 3% is considered accurate
In the real world, percent error can be larger or smaller.
Considering the following areas that need much smaller percents of error
Landing an airplane – 0%
Performing heart surgery – 0%

23. B. Percent Error % error = 2.90 %
A student determines the density of a substance to be 1.40 g/mL. Find the % error if the accepted value of the density is 1.36 g/mL.
% error = 2.90 %

24. B. Percent Error In groups of 2 calculate the percent error
Raise your hand when your team is done
1. Experimental Value = 5.75 g
Accepted Value = 6.00 g
2. Experimental Value = 107 ml
Accepted Value = 105 ml
3. Experimental Value = 1.54 g/ml
Accepted Value = 2.35 g/ml
Which are accurate?
4.17% (can be – 4.17%) Not accurate > 3%
1.90% Accurate < 3%
34.5% (can be – 34.5%) Not Accurate > 3%

25. Audience Participation
Let’s Play
The Accuracy? or Precision? Game

26. C. Precision of Measurement
Even the best crafts people and finest manufacturing equipment can’t measure the exact same dimensions every time
Precision of Measurement determines the spread from average value (tolerance)

27. Precision of Measurement
“Tolerance” is used constantly in manufacturing and repair work
Example – parts for autos, pumps, other rotating equipment can have a small amount of space between them.
Too much space… the parts can’t function properly so the equipment won’t run
Too little space… the parts bind (rub) up against each other which can cause damage

28. Precision of Measurement
To calculate precision of measurement:
Average the data by adding to find total of data and dividing by the number of items of data
Determine the range (difference) by subtracting the smallest (lowest) value from the greatest (highest) value
Divide the range by 2 to determine the spread
Precision of measurement is expressed as the average value +/- the spread
Smaller the spread the more accurate and precise the measurement
You may have a spread that has 1 more significant figure or place value than original values. Include units for all.

29. Precision of Measurement
Average = Total
No. of samples
4.24 μm / 7 = 0.61 μm
Range = highest – lowest
0.65 μm – 0.58 μm = 0.07 μm
Spread = Range / 2
0.07 μm / 2 = μm
Precision of Measurement =
Average +/- Spread
0.61 μm +/ μm
Gap Between Piston & Cylinder 0.60 μm 0.62 μm 0.59 μm 0.65 μm 0.58 μm Total = 4.24 μm

30. Precision of Measurement – Let’s Practice Together
Given the following volume measurements:
5.5 L
5.8 L
5.0 L
5.6 L
4.8 L
5.2 L
Total _____
Determine Precision of Measurement:
Average: L
Range: L
Spread: L
Precision of Measurement
L + / - L

31. Precision of Measurement – Practice in Pairs
Determine Precision of Measurement for:
6.25 m
6.38 m
6.44 m
6.80 m
Determine Precision of Measurement for:
80.6 g
81.3 g
80.5 g
80.8 g
80.2 g
81.1 g

32. Check for Understanding
Accuracy – Correctness of data
Precision – Consistency of results with one another
Percent Error – Comparison of experimental data to published data. < 3% accurate
Precision of Measurement – Shows the spread from average value (tolerance). Smaller the spread more precise the measurement to what you want to obtain.

33. Check for Understanding
How can you ensure accuracy and precision when performing a lab?
What is the percent error when lab data indicates the density of molasses is 1.45 g/ml and Perry’s Handbook for Chemical Engineering shows 1.47 g/ml?

34. Independent Practice
Accuracy and Precision Worksheet 1

35. Significant Figures
As we experienced first hand from our lab, obtaining accurate and precise measurements can be tricky
Some instruments read in more detail than others
If we have to eyeball a measurement we can each read something different, or we can make an error in estimating
You should trust only the least accurate measure
Enter what scientists & engineers call Significant Figures

36. C. Significant Figures Measuring… Sig Figs and the Role of Rounding
TeacherTube Video Clip – Why Are Significant Figures Significant?

37. C. Significant Figures Recording Sig Figs (sf)
Indicate precision of a measurement
Recording Sig Figs (sf)
Sig figs in a measurement include the known digits plus a final estimated digit
Sig figs are also called significant digits
2.33 cm

38. C. Significant Figures
The Pacific/Atlantic Rule to identify significant figures
Let’s go over a few examples together
Then we’ll practice independently

39. C. Significant Figures
Short version & Gory details and rules approach

40. C. Significant Figures
Exact Numbers do not limit the # of sig figs in the answer.
Counting numbers: 12 students
Exact conversions: 1 m = 100 cm
“1” in any conversion: 1 in = 2.54 cm
Constants – such as gravity 9.8 m/s2 or speed of light 3.00 m/s
Number that is part of an equation (for example area of triangle 1/2bh)
So, sig fig rules do not apply in these cases!!!!!

41. C. Significant Figures All non-zero digits are significant.
Zeros between two non-zero digits are significant has 4 sf.
Count all numbers EXCEPT:
Leading zeros
Trailing zeros without a decimal point -- 2,500 (Trailing zeros are significant if and only if they follow a decimal as well)

42. C. Significant Figures
Count the number of significant figures starting from the left-most non-zero digit. Watch for decimals.
Include zeros to the right of non-zero numbers
0.08 (1) (3)
Don’t include zeros to the left of decimal, unless preceded by non-zeros
0.2 (1) (4)

43. C. Significant Figures
Zeros to the right of the decimal point are significant has 3 sf.
A bar placed above a zero indicates all digits including one with bar and those to the left of it are significant has 3 sf only has 3 s.f.
When a number ends in zero and has a decimal point, all digits to the left of the decimal pt. are significant has 3 sf.

44. C. Significant Figures Zeros that are not significant are still used
They are called “placeholders”
Example –
5280 ~ The zero tells us we have something in the thousands
0.08 ~ The zeros tell us we have something in the hundredths

45. C. Significant Figures
The numbers in exponents when using scientific notation DO NOT count
5.60 x sf
9.2 x sf
Round up if number immediately to the right of last sig fig is > 5 round down if < 5
Final answer has 2 sf  110
Final answer has 4 sf 

46. Counting Sig Fig Examples
C. Significant Figures
Counting Sig Fig Examples
4 sig figs
3 sig figs
3. 5,280
3. 5,280
3 sig figs
2 sig figs

47. Significant Figures - Basics
Independent practice – Problem Set 1
link

48. C. Significant Figures (13.91g/cm3)(23.3cm3) = 324.103g 3 SF 4 SF 3 SF
Calculating with Sig Figs
Multiplying / Dividing - The number with the fewest number of sig figs determines the number of sig figs in the answer.
(13.91g/cm3)(23.3cm3) = g
4 SF
3 SF
3 SF
324 g

49. C. Significant Figures 3.75 mL + 4.1 mL 7.85 mL 3.75 mL + 4.1 mL
Calculating with Sig Figs
Adding / Subtracting - The number with the fewest number of places determines the place of the last sig fig in the answer.
If there are no decimals on one or more of the numbers, go to least sig figs.
3.75 mL mL
7.85 mL
3.75 mL mL
7.85 mL
224 g
+ 130 g
354 g
224 g
+ 130 g
354 g
 7.9 mL
 350 g

50. C. Significant Figures Practice Problems 2 SF 4 SF 2 SF
5. (15.30 g) ÷ (6.4 mL)
4 SF
2 SF
= g/mL
 2.4 g/mL
2 SF g g
 18.1 g
18.06 g

51. C. Significant Figures More Rules….
Be sure you maintain the proper units
For sig fig the exponential numbers used in scientific notation don’t count towards number of s.f.
For example – you can’t add centimeters and kilometers without converting them to the same scale first
1 m = 100 cm
4.5 cm + 10 m = 4.5 cm cm
= cm  1005 cm

52. C. Significant Figures
When adding and subtracting numbers in scientific notations:
You must change them so that they all have the same exponent (usually best to change to largest exponent)
Then add or subtract
Round answer appropriately according to proper significant figure rules
Put answer in correct scientific notation

53. C. Significant Figures
When multiplying numbers in scientific notations:
Multiply coefficients, then add the exponents
When dividing numbers in scientific notations:
Divide coefficients, then subtract the exponents
For Both
Round answer appropriately according to proper significant figure rules
Put answer in correct scientific notation

54. C. Significant Figures Handling Multiple Step Problems
Either round using correct sig figs for each distinct math operation along the way or
At the end of entire calculation round to least sig figs of all numbers present
Exceptions per Hsu in Essential Physics textbook
The rule is suspended when the result will be part of another calculation.
For intermediate results, one extra significant figure should be carried to minimize errors in subsequent calculations.

55. One More Rule…. Be sure you maintain the proper units
For example – you can’t add centimeters and kilometers without converting them to the same scale first
1 m = 100 cm
4.5 cm + 10 m = 4.5 cm cm = cm

56. C. Significant Figures Your Turn….
Independent Practice on Problem Set 2 – Basic Math Operations
Link

57. 348.432 rounded to 2 decimal places  348.43
PoT - Rounding
Round to more conservative measurements / numbers.
Practice Set or teacher will instruct you on how many places to round to
Rounding – round up of > 5, round down if < 5
Round to nearest whole number
(look to number next to place you are rounding to)
 1946
rounded to 2 decimal places 

58. Scientific Notation

59. How Big is Big? How Small is Small?
Write out the decimal number for the distance from earth to the sun in:
miles meters kilometers
Using decimal numbers write the size of an electron in meters
Use decimal numbers to write how many atoms are in a mole
Distance from earth to sun 93 Million miles Billion Meters 147 Million kms
93,000,000, ,000,000,000, ,000,000,000
Size of an electron 2.8x meters
Atoms in mole 602,000,000,000,000,000,000,000

60. How Big is Big? How Small is Small?
Distance from earth to sun
93 Million miles Billion Meters 147 Million kms
93,000, ,000,000, ,000,000
Size of an electron in meters
Atoms in mole
602,000,000,000,000,000,000,000

61. D. Scientific Notation Why did Scientists create Scientific Notation?
To make it easier to handle really big or really small numbers use base 10
For example ~ Avogadro’s Number for number of particles in a mole
602,000,000,000,000,000,000,000
or 6.02 x 1023
Which would you rather write?

62. D. Scientific Notation
Scientific Notation can also help determine number of significant figures
Answers questions
Are last zeros significant?
Are zeros to left of decimal significant after all?
Example
216,500 (4) x 105 (5)

63. D. Scientific Notation Converting into Scientific Notation:
Move decimal until there’s 1 digit and only one digit to its left. This number is called a coefficient.
68000 
Must be a whole, non-zero number from 1 – 9
6… not 68…. Or .68

64. D. Scientific Notation Places decimal is moved = the exponent of 10
68000  moved 4 places
= 6.8 x 104
Number >1  positive exponent (5)  x 105
Number <1  negative exponent (-1)
 x 10-1
100 = 1. Used for whole numbers less than 10
 x 100

65. Practice Problems Converting Decimal Numbers to Scientific Notation
D. Scientific Notation
Practice Problems Converting Decimal Numbers to Scientific Notation
1. 2,400,000 g kg km
,000 mm
2.4  106 g
2.56  10-3 kg
7  10-5 km
6.2  104 mm

66. D. Scientific Notation Converting from Scientific Notation:
Positive exponent… move to the right the number of places you have in exponent
4.682 x 106  4,682,000
Negative exponent…. move decimal to the left the number of places in exponent
9.73 x 10-5 

67. Practice Problems Converting Scientific Notation to Decimal Numbers
D. Scientific Notation
Practice Problems Converting Scientific Notation to Decimal Numbers
x 104 g
x 10-2 L
 107 m
 10-3 g
56,000 g L
19,860,000 m g

68. Independent Practice
Practice Set 1 – Decimal numbers to Scientific Notation
Practice Set 2 – Scientific Notation to decimal numbers
link

69. D. Scientific Notation
When multiplying numbers in scientific notations:
Multiply the numbers (coefficients)
Add the exponents
When dividing numbers in scientific notations:
Divide the numbers (coefficients)
Subtract the exponents
Round answer appropriately according to proper significant figure rules
Put answer in correct scientific notation

70. D. Scientific Notation Let’s Practice Multiplying
1.4 x 105 X 7.2 x 104
Multiply the numbers (coefficients) – example would be 10.08
Add the exponents = 9
10.08 x 109  x 1010
As Group Now Try 7 x 103 x 8.2 x 10-5
On Your Own Try 6 x 10-3 x 3.9 x 10-2

71. D. Scientific Notation Let’s Practice Dividing 1.4 x 105 ÷ 7.2 x 104
Divide the numbers (coefficient) – example would be .194
Subtract the exponents = 1
.194 x 101  1.94 x 100
As Group Now Try 7 x 103 ÷ 8.2 x 106
On Your Own Try 6 x 103 ÷ 3.9 x 10-2

72. D. Scientific Notation
Calculating with Sci. Notation the “Old Fashion Way” without a Graphing Calculator…
(5.44 × 107 g) ÷ (8.1 × 104 mol) =
5.44 g = 0.67 (or 6.7 x 10-1) x 103 = 8.1 mol
= 6.7 x 102 g/mol

73. D. Scientific Notation One more rule….
Be sure you maintain the proper units
For example – you can’t add centimeters and kilometers without converting them to the same scale first
1 m = 100 cm
4.5 cm + 10 m = 4.5 cm cm = cm

74. D. Scientific Notation
Now you try it…. Group Practice on Scientific Notations section of Worksheet 1
Independent Practice
Multiplication/Division Problem Set
link

75. D. Scientific Notation 5.44 7 8.1 4 = 671.6049383 = 670 g/mol
Calculating with Sci. Notation
(5.44 × 107 g) ÷ (8.1 × 104 mol) =
Change Calculator Mode to SCI & Degrees
Type on your TI-84 calculator (2nd button, comma key-EE)
On your TI-30 calculator (2nd button, EE or use 10x key)
(or use shift red ee key at bottom). Casio use EXP key or 10x
2nd EE
2nd EE
5.44 7
8.1 4
ENTER ÷ =
= 670 g/mol
= 6.7 × 102 g/mol

76. D. Scientific Notation
When adding and subtracting numbers in scientific notations:
You must change them so that they all have the same exponent (usually best to change to smaller exponent to that of larger)
Then add or subtract numbers (coefficients)
Round answer appropriately according to proper significant figure rules
Put answer in correct scientific notation

77. D. Scientific Notation Let’s Practice Adding 6.4 x 105 + 7.2 x 104
Change smallest exponent to match larger one 
6.4 x x 105
Add the numbers (coefficient) and carry along the exponents
7.12 x 105
Rule still applies you must have one digit to left of decimal, so you may need to adjust exponent
As Group Now Try 4 x x 105
On Your Own Try 6 x x 10-2

78. D. Scientific Notation Let’s Practice Subtracting
6.4 x x 104
Change smallest exponent to match higher one  6.4 x x 105
Subtract the numbers (coefficient) and carry along the exponents
5.68 x 105
Rule still applies you must have one digit to left of decimal, so you may need to adjust exponent
As Group Now Try 4 x x 105
On Your Own Try 6 x x 10-2

79. Scientific Notation Group Practice – Problem III.4 on problem set 1
Addition/Subtraction Problem Set
link

80. E. Using Both Scientific Notation & Significant Figures
When you have numbers that contain both a number (coefficient) and scientific notation, ONLY the number (coefficient) determines the number of significant figures – not the exponent
It is actually easier to count sig figs if you convert to scientific notation (eliminates leading or trailing zeros – although you need to watch out for zero to far right in decimal numbers)
4.5 x sig figs
7.35 x sig figs
6.080 x sig figs

81. E. Using Both Scientific Notation & Significant Figures
Significant Figures and Scientific Notation can be confusing enough when dealt with individually….
It really gets exciting when we mix the two….
But take heart – there are some helpful rules to follow…

82. E. Using Both Sig Figs and Sc Not
When adding and subtracting numbers in scientific notations:
You must change them so that they all have the same exponent (usually best to change to largest exponent)
Then add or subtract
Round answer appropriately according to proper significant figure rules
Put answer in correct scientific notation

83. E. Using Both Sig Figs and Sc Not
When multiplying numbers in scientific notations:
Multiply coefficients, then add the exponents
When dividing numbers in scientific notations:
Divide coefficients, then subtract the exponents
For Both
Round answer appropriately according to proper significant figure rules
Put answer in correct scientific notation

84. E. Using Both Scientific Notation & Significant Figures
Independent Practice
Problem Set 4 – Sig Figs and Sc. Not.
link

85. Check for Understanding
Accuracy – Correctness of data
Precision – Consistency of results
Percent Error – Comparison of experimental data to published data. Less than < 3%
Precision of Measurement – spread from average value (tolerance). Smaller spread, more precision
Significant Figures / Rounding – Indicate the precision of measurement
Scientific Notation – Used to more easily write out very big or very small numbers

86. Check for Understanding
How can you ensure accuracy and precision when performing a lab?
What is the percent error when lab data indicates the density of molasses is 1.45 g/ml and Perry’s Handbook for Chemical Engineering shows 1.47 g/ml?
What are the Sig Fig Rules or the Pacific/Atlantic approach?
What are the Scientific Notation Rules?