Percent Error, Precision of Measurement, 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

I. Units of Measurement

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

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

B. SI Units 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

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

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: V = 29 mL M = 25 g D = ? 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 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 g

II. Using Measurements

Let’s Experiment… 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 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 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 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 % error = 2.9 % 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 = 2.9 %

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 4.17% 1.90% 34.5%

Let’s Experiment… 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? 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 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 Gap Between Piston & Cylinder 0.60 μm 0.62 μm 0.59 μm 0.65 μm 0.58 μm Total = 4.24 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 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)

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 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

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/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

Counting Sig Fig Examples C. Significant Figures Counting Sig Fig Examples 1. 23.50 1. 23.50 4 sig figs 2. 402 2. 402 3 sig figs 3. 5,280 3. 5,280 3 sig figs 4. 0.080 4. 0.080 2 sig figs

Significant Figures - Basics Independent practice – Problem Set 1 link

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 sig figs determines the number of sig figs in the answer. (13.91g/cm3)(23.3cm3) = 324.103g 4 SF 3 SF 3 SF 324 g

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 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 3.75 mL + 4.1 mL 7.85 mL 224 g + 130 g 354 g 224 g + 130 g 354 g  7.9 mL  350 g

C. Significant Figures Practice Problems 2 SF 4 SF 2 SF 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  18.1 g 18.06 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

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 93,000,000,000 147,000,000,000,000 147,000,000,000 Size of an electron 2.8x10-15 meters 0.0000000000000028 Atoms in mole 602,000,000,000,000,000,000,000

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

Practice Problems Converting Decimal Numbers to Scientific Notation D. Scientific Notation Practice Problems Converting Decimal Numbers to Scientific Notation 1. 2,400,000 g 2. 0.00256 kg 3. 0.00007 km 4. 62,000 mm 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 D. Scientific Notation 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

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 Type on your calculator: 5.44 7 8.1 4 Calculating with Sci. Notation (5.44 × 107 g) ÷ (8.1 × 104 mol) = Type on your calculator: EXP EE EXP EE ENTER EXE 5.44 7 8.1 4 ÷ = 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 2 sig figs 7.35 x 10154 3 sig figs 6.080 x 1055 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?