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

I

II

III

Accuracy, Precision,

Percent Error, Precision of Measurement,

Significant Figures, &

Scientific Notation

101

MEASUREMENT

Learning Objectives

The Learners Will (TLW) collect data and make measurements with accuracy and precision, and will be able to calculate percent error

TEKS 2F

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 Base Unit Abbrev.

Length

Mass

Time

Temp

meterkilogramsecondKelvin

mkgsK

Amount mole mol

Symbol

l

m

tTn

B. SI Units

mega- M 106

deci- d 10-1

centi- c 10-2

milli- m 10-3

Prefix Symbol Factor

micro- 10-6

nano- n 10-9

pico- p 10-12

kilo- k 103

BASE UNIT --- 100

C. Derived Units

Combination of base units.

Volume (m3 or cm3) height width length

D = MV

1 cm3 = 1 mL1 dm3 = 1 L

Density (kg/m3 or g/cm3)mass per volume

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

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 825cm3

M = 11,200 gV

MD

D. Problem Solving Example – Density

A liquid has a density of 0.87 g/mL. What volume is occupied by 25 g of the liquid?

GIVEN:

D = 0.87 g/mL

V = ?

M = 25 g

WORK:

V = M D

V = 25 g

0.87 g/mLV = 29 mLV

MD

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II. Using Measurements

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 ______ 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

100literature

literaturealexperimenterror %

your value accepted value

For our purposes a percent error of

< 4% 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.

100g/mL 1.36

g/mL 1.36g/mL 1.40error %

% 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. 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


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 precise the measurement You may have a spread that has 1 more significant

figure that original values


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 = 0.415 μm

Average (mean) = Total

No. of samples

0.415 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


L + / - L

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

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)


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


Measuring… Sig Figs and the Role of Rounding

TeacherTube Video Clip – Why Are Significant Figures Significant?


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


The Pacific/Atlantic Rule to identify significant figures

Let’s go over a few examples together Then we’ll practice independently


Gory details and rules approach

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)


All non-zero digits are significant.


Zeros to the right of the decimal point are significant. 20.0 has 3 sf.

A bar placed above a zero indicates all digits 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 FiguresExact Numbers do not limit the # of sig figs in the answer.

Counting numbers: 12 studentsExact conversions: 1 m = 100 cm“1” in any conversion: 1 in = 2.54 cmSo, sig fig rules do not apply in these three cases!!!!!


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

4. 0.080

3. 5,280

2. 402

1. 23.50


Counting Sig Fig Examples

1. 23.50

2. 402

3. 5,280

4. 0.080

4 sig figs

3 sig figs

3 sig figs

2 sig figs

Significant Figures - Basics

Independent practice – Problem Set 1

link


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

324 g

4 SF 3 SF 3 SF


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

224 g

+ 130 g

354 g 7.9 mL 350 g

3.75 mL

+ 4.1 mL

7.85 mL

224 g

+ 130 g

354 g


5. (15.30 g) ÷ (6.4 mL)

Practice Problems

= 2.390625 g/mL

18.1 g

6. 18.9 g

- 0.84 g18.06 g

4 SF 2 SF

2.4 g/mL2 SF


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


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


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


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

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?


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


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


1. 2,400,000 g

2. 0.00256 kg

3. 0.00007 km

4. 62,000 mm

Practice Problems Converting Decimal Numbers to Scientific Notation

2.4 106 g

2.56 10-3 kg

7 10-5 km

6.2 104 mm


Practice Problems Converting Scientific Notation to Decimal Numbers

5. 5.6 x 104 g

6. 3.45 x 10-2

L

7. 1.986 107 m

8. 6.208 10-3 g

56,000 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


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


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


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


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


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 first1 m = 100 cm4.5 cm + 10 cm = 4.5 cm + 1000 cm

= 1004.5 cm


Now you try it…. Group Practice on Scientific Notations section of Worksheet 1

Independent Practice Multiplication/Division Problem Set link



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



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


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


Calculating with Sci. Notation

(5.44 × 107 g) ÷ (8.1 × 104 mol) =

5.44 EXPEXP

EEEE÷÷

EXPEXP

EEEE ENTERENTER

EXEEXE7 8.1 4

= 671.6049383 = 670 g/mol= 6.7 × 102 g/mol

Type on your calculator:

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 2 sig figs

7.35 x 10154 3 sig figs

6.080 x 1055 4 sig figs


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


the same exponent (usually best to change to largest exponent)

Then add or subtract Round answer appropriately according to


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


Independent Practice Problem Set 4 – Sig Figs and Sc. Not.

link


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


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?

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

Documents

units of measurement

g slide

volume slide

consistent slide

correct slide

f slide

game slide

precision of measurement