II
I
III
Accuracy, Precision,
Percent Error, Precision of Measurement,
Significant Figures, &
Scientific Notation
101

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)

Agenda
 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

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

B. SI Units
l m t
T n
Kelvin or
Centigrade

Prefix megakilo-
BASE UNIT decicentimillimicronanopico-
B. SI Units
Symbol
M k
--d c m
 n p
Factor
10 6
10 3
10 0
10 -1
10 -2
10 -3
10 -6
10 -9
10 -12

C. Derived Units


1 cm 3 = 1 mL
3
3
1 dm 3 = 1 L




3
3

D =
M
V

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

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 = ?
D

M
V
WORK :
D = M
V
D = 25 g
29 mL
D = 0.87 g/mL

D. Problem Solving Example – Density
 An object has a volume of 825 cm 3 and a density of 13.6 g/cm 3 . Find its mass.
GIVEN: WORK :
3
3
3
3
D

M
V M = 11,200 g

II
I
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Let’s Experiment…
1.
2.
3.
Measure the level in the two graduated cylinders
Measure of the level in the beaker
Write your name on the chart at the front of the room and record the above measurements in the columns indicated




Actual Measurement in each is _8.3___
 How close to the actual measurement is our data?
 How close are our readings to one another?
What could account for the differences in your own measurements?
What could account for the differences between your readings and those of your classmates?

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

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

Audience Participation

B. Percent Error


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
% error
 experiment al
 literature literature

100 your value accepted value

 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
 Performing heart surgery

B. Percent Error
 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

1.40
g/mL

1.36
g/mL
1.36
g/mL

100
% error = 2.9 %

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

Let’s Experiment…
1.
2.
3.
4.
Measure the wooden block with the metric measuring stick
Bring measurement of the level in the two graduated cylinders
Bring measurement of the level in the beaker
Write your name on the chart at the front of the room and record the above measurements in the columns indicated

Lab Results
 Did we all come up with exactly the same numbers?
 Why or Why not?
 Which are most precise measurements?
 Why?
 Which are most accurate measurements?
 Why?
 What is the percent error?
 Perform the calculations

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)

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 and the parts can’t function properly so the equipment won’t run
 Too little and the parts bind up against each other which can cause damage

Precision of Measurement
 To calculate precision of measurement:
 Average the data
 Determine the range from lowest to 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 that original values

Precision of Measurement
Gap Between Piston

& Cylinder
Average (mean) = Total
No. of samples
4.24 μm / 7 = 0.61 μm
0.60 μm
0.62 μm
0.59 μm
0.60 μm
0.65 μm
0.60 μm
0.58 μm
Total = 4.24 μm
 Range = highest – lowest
0.65 μm – 0.58 μm = 0.07 μm
 Spread = Range / 2
0.07 μm / 2 = 0.035 μm
 Precision of Measurement =
Average +/- Spread
0.61 μm +/- 0.035 μm

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

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

Check for Understanding
 Accuracy – Correctness of data
 Precision – Consistency of results
 Percent Error – Comparison of experimental data to published data
 Precision of Measurement – Determining the spread from average value (tolerance)

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?

Independent Practice
 Accuracy and Precision Worksheet 1

C. 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

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

C. Significant Figures
 Indicate precision of a measurement

 Sig figs in a measurement include the known digits plus a final estimated digit
 Sig figs are also called significant digits
2.33 cm

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

C. Significant Figures
 Gory details and rules approach

C. Significant Figures
 All non-zero digits are significant.

Zeros between two non-zero digits are significant. -- 2.004 has 4 sf.
 Count all numbers EXCEPT:
 Leading zeros -- 0.0025
 Trailing zeros without a decimal point -- 2,500
( Trailing zeros are significant if and only if they follow a decimal as well )

C. Significant Figures

Zeros to the right of the decimal point are significant. 20.0 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. 210 has 3 sf.
 When a number ends in zero and has a decimal point, all digits to the left of the decimal pt. are significant. 110. has 3 sf.

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/s 2 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!!!!!

C. Significant Figures



Zeros that are not significant are still used
They are called “placeholders”
Example –
 528 0 ~ The zero tells us we have something in the thousands
 0.0
8 ~ The zeros tell us we have something in the hundredths

C. Significant Figures
4 sig figs
3 sig figs
3 sig figs
2 sig figs

Significant Figures - Basics
 Independent practice – Problem Set 1 link

C. Significant Figures
 Calculating with Sig Figs
 Multiplying / Dividing - The number with the fewest sig figs determines the number of sig figs in the answer.
3
3
4 SF 3 SF 3 SF

C. Significant Figures
 Calculating with Sig Figs
 Adding / Subtracting - The number with the fewest number of decimals determines the place of the last sig fig in the answer.
 If there are no decimals, go to least sig figs.

7.9 mL

350 g

C. Significant Figures
5. (15.30 g) ÷ (6.4 mL)
4 SF 2 SF
= 2.390625 g/mL

2.4 g/mL
2 SF
6. 18.9 g
- 0.84 g
18.06 g

18.1 g

C. Significant Figures
 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 + 1000 cm
= 1004.5 cm  1005 cm

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

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

C. Significant Figures
 Exception to Rule
 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.

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

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Scientific Notation

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 147 Billion Meters 147 Million kms
 93,000,000 147,000,000,000 147,000,000
 Size of an electron 2.8x10
-15 meters
0.0000000000000028
 Atoms in mole 602,000,000,000,000,000,000,000

D. Scientific Notation
 Why did Scientists create Scientific
Notation?


To make it easier to handle really big or really small numbers
For example ~ Avogadro’s Number for number of particles in a mole
 602,000,000,000,000,000,000,000 or 6.02 x 10 23
Which would you rather write?

D. Scientific Notation
 Converting into Scientific Notation:
 Move decimal until there’s 1 digit to its left. This number is called a coefficient.
 68000  6.8000
 Must be a whole number from 1 – 9

6… not 68…. Or .6

D. Scientific Notation
 Places moved = the exponent of 10
 68000  6.8000 moved 4 places
 = 6.8 x 10 4
 Large # ( > 1)
 positive exponent (10 4 )
39458  3.9458 x 10 4
 Small # ( < 1)
 negative exponent (10 -4 )
.39458  3.9458 x 10 -4
 10 0 = 1. Used for whole numbers less than 10
3.9458  3.9458 x 10 0

D. Scientific Notation
Practice Problems Converting Decimal
Numbers to Scientific Notation
1. 2,400,000
 g 2.4

10 6
 g
2. 0.00256 kg 2.56

10 -3 kg
3.
4.
0.00007 km
62,000 mm
7

10 -5 km
6.2

10 4 mm

D. Scientific Notation
Practice Problems Converting Scientific
Notation to Decimal Numbers
5. 5.6 x 10 4 g 56,000 g
6. 3.45 x 10 -2 L
7. 1.986

10 7 m
8. 6.208

10 -3 g
0.0345 L
19,860,000 m
0.006208 g

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

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

D. Scientific Notation
 Let’s Practice Multiplying
 1.4 x 10 5 X 7.2 x 10 4
 Multiply the numbers (coefficients) – example would be 10.08
 Add the exponents 5 + 4 = 9
 10.08 x 10 9  1.
008 x 10 10


As Group Now Try 7 x 10 3 x 8.2 x 10 -5
On Your Own Try 6 x 10 -3 x 3.9 x 10 -2

D. Scientific Notation
 Let’s Practice Dividing
 1.4 x 10 5 ÷ 7.2 x 10 4
 Divide the numbers (coefficient) – example would be .194
 Subtract the exponents 5 4 = 1
 .194 x 10 1  1.
94 x 10 0


As Group Now Try 7 x 10 3 ÷ 8.2 x 10 6
On Your Own Try 6 x 10 3 ÷ 3.9 x 10 -2

D. Scientific Notation
 Calculating with Sci. Notation the
“Old Fashion Way” without a
Graphing Calculator …
(5.44 × 10 7 g) ÷ (8.1 × 10 4 mol) =
5.44 g = 0.67 (or 6.7 x 10 -1 ) x 10 3 =
8.1 mol
= 6.7 x 10 2 g/mol

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 cm = 4.5 cm + 1000 cm
= 1004.5 cm

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

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

D. Scientific Notation
 Let’s Practice Adding
 6.4 x 10 5 + 7.2 x 10 4
 Change smallest exponent to match larger one 
6.4 x 10 5 + .
72 x 10 5
 Add the numbers (coefficient) and carry along the exponents
 7 .12 x 10 5
 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 10 3 + 1.2 x 10 5
On Your Own Try 6 x 10 -3 + 3.9 x 10 -2

D. Scientific Notation
 Let’s Practice Subtracting
 6.4 x 10 5 7.2 x 10 4
 Change smallest exponent to match higher one  6.4 x 10 5 .72 x 10 5
 Subtract the numbers (coefficient) and carry along the exponents
 5 .68 x 10 5
 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 10 3 1.2 x 10 5
On Your Own Try 6 x 10 -3 3.9 x 10 -2

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

D. Scientific Notation
 Calculating with Sci. Notation
(5.44 × 10 7 g) ÷ (8.1 × 10 4 mol) =
Type on your calculator:
5.44
EXP
EE
7 ÷ 8.1
EXP
EE
4
EXE
ENTER
= 671.6049383 = 670 g/mol
= 6.7 × 10 2 g/mol

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 10 -4
7.35
x 10 154
6.080
x 10 55
2 sig figs
3 sig figs
4 sig figs

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…

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

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

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

Check for Understanding





Accuracy – Correctness of data
Precision – Consistency of results
Percent Error – Comparison of experimental data to published data
Significant Figures – Indicate the precision of measurement
Scientific Notation – Used by scientists to more easily write out very big or very small numbers

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?