Download Accuracy, Precision, Percent Error, Significant

MEASUREMENT
Accuracy, Precision,
Percent Error, Precision of Measurement,
Significant Figures, &
I
II
III
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
I. Units of
Measurement
A. Number vs. Quantity
 Quantity = number + unit
UNITS MATTER!!
B. SI Units
Quantity
Length
Mass
Time
Temp
Amount
Symbol
Base Unit
Abbrev.
l
m
t
T
n
meter
kilogram
second
m
kg
s
K or C
mol
Kelvin or
Centigrade
mole
B. SI Units
Prefix
mega-
Symbol
M
Factor
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
C. Derived Units
 Combination of base units.
1 cm3 = 1 mL
3=1L
1
dm
3
3
 Volume (m or cm )
height  width  length
3
3
 Density (kg/m or g/cm )
mass per volume
M
D=
V
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.
D. Problem Solving Example – Density
 A liquid has a volume of 29 mL and a mass
of 25 g? What is the density?
GIVEN:
WORK:
V = 29 mL
M = 25 g
D=?
D=M
V
M
D
V
D=
25 g
29 mL
D = 0.87 g/mL
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:
WORK:
V = 825 cm3
M = DV
D = 13.6 g/cm3
M = 13.6 g/cm3 x
M=?
3
825cm
M
D
V
M = 11,200 g
II. Using
Measurements
I
II
III
Let’s Experiment…
1. Measure the level in the two
graduated cylinders
2. Measure of the level in the beaker
3. 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
ACCURATE = CORRECT
PRECISE = CONSISTENT
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
Let’s Play
The Accuracy? or Precision?
Game
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 
experimental  literature
your value
literature
 100
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
% error = 2.9 %
 100
B. Percent Error


1.
2.
3.
In groups of 2 calculate the percent error
Raise your hand when your team is done
Experimental Value = 5.75 g
Accepted Value = 6.00 g
Experimental Value = 107 ml
Accepted Value = 105 ml
Experimental Value = 1.54 g/ml
Accepted Value = 2.35 g/ml
Let’s Experiment…
1. Measure the wooden block with the
metric measuring stick
2. Bring measurement of the level in
the two graduated cylinders
3. Bring measurement of the level in
the beaker
4. 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
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

Average (mean) = 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 = 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:
5.5 L
5.8 L
5.0 L
5.6 L
4.8 L
5.2 L
 Determine Precision of
Measurement:
 Average:
L
 Range:
L
 Spread:
L
 Precision of Measurement
L+/-
L
Precision of Measurement – Practice in Pairs
 Determine Precision
 Determine Precision
of Measurement for:
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
 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
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
Trailing
zeros -- 0.0025
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/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!!!!!
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
C. Significant Figures
Counting Sig Fig Examples
1. 23.50
4 sig figs
2. 402
3 sig figs
3. 5,280
3 sig figs
4. 0.080
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.
(13.91g/cm3)(23.3cm3) = 324.103g
3 SF
4 SF
3 SF
324 g
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.
3.75 mL
+ 4.1 mL
7.85 mL  7.9 mL
224 g
+ 130 g
354 g  350 g
C. Significant Figures
Practice Problems
5. (15.30 g) ÷ (6.4 mL)
2 SF
4 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
Scientific Notation
I
II
III
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 1023
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 104
 Large # (>1)  positive exponent (104)
39458  3.9458 x 104
 Small # (<1)  negative exponent (10-4)
.39458  3.9458 x 10-4
 100 = 1. Used for whole numbers less than 10
3.9458  3.9458 x 100
D. Scientific Notation
Practice Problems Converting Decimal
Numbers to Scientific Notation
1. 2,400,000 g
2.4 
2. 0.00256 kg
2.56  10-3 kg
3.
0.00007 km
7  10-5 km
62,000 mm
6.2 
4.
6
10
4
10
g
mm
D. Scientific Notation
Practice Problems Converting Scientific
Notation to Decimal Numbers
5. 5.6 x
4
10
g
56,000 g
6. 3.45 x 10-2 L
0.0345 L
7. 1.986  107 m
19,860,000 m
8. 6.208  10-3 g
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 105 X 7.2 x 104
 Multiply the numbers (coefficients) –
example would be 10.08
 Add the exponents 5 + 4 = 9
 10.08 x 109  1.008 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
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 5 - 4 = 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
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
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 105 + 7.2 x 104
 Change smallest exponent to match larger one 
6.4 x 105 + .72 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 103 + 1.2 x 105
 On Your Own Try 6 x 10-3 + 3.9 x 10-2
D. Scientific Notation
 Let’s Practice Subtracting
 6.4 x 105 - 7.2 x 104
 Change smallest exponent to match higher one  6.4
x 105 - .72 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 103 - 1.2 x 105
 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 × 107 g) ÷ (8.1 × 104 mol) =
Type on your calculator:
5.44
EXP
EE
7
÷
8.1
EXP
EE
4
EXE
ENTER
= 671.6049383 = 670 g/mol
= 6.7 × 102 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 10154
6.080 x 1055
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?