Chapter TwoChapter Two

Significant FiguresSignificant Figures

Accuracy vs. PrecisionAccuracy vs. Precision

►Most people think they mean the same Most people think they mean the same thingthing

► In science, however, they have two In science, however, they have two very distinct meaningsvery distinct meanings

AccuracyAccuracy

►Refers to the closeness of measurements Refers to the closeness of measurements to the correct or accepted value of the to the correct or accepted value of the quantity measured.quantity measured.

►Let’s say your lab group wants to Let’s say your lab group wants to measure the mass of a block of measure the mass of a block of aluminum. aluminum. You get out a balance.You get out a balance. You tare the balance.You tare the balance. You mass the block of aluminum…it weighs You mass the block of aluminum…it weighs

6.00 grams.6.00 grams.

Your teacher comes along…Your teacher comes along…

►And wants to verify your And wants to verify your measurement.measurement. She tares the balanceShe tares the balance She places the block of aluminum on itShe places the block of aluminum on it She gets 6.02 g.She gets 6.02 g. Not bad. VERY close. Not bad. VERY close.

►Guess you’re both pretty Guess you’re both pretty ACCURATE.ACCURATE. Assuming your balance is working Assuming your balance is working

correctly.correctly.

Now another group wants to Now another group wants to repeat your measurements…repeat your measurements…

►But it’s that other lab group, you But it’s that other lab group, you know, the ones who are always know, the ones who are always breaking things.breaking things.

►They get out a balance, but since they They get out a balance, but since they never pay attention to details, they never pay attention to details, they forget to tare the balance. forget to tare the balance.

►They each take turns massing the They each take turns massing the aluminum.aluminum.

Their results:Their results:

►Desmond got 5.40 gDesmond got 5.40 g►Mathilda got 5.41 gMathilda got 5.41 g► Jim-Bob got 5.38 gJim-Bob got 5.38 g►Cindylou got 5.41 gCindylou got 5.41 g

►Wow. All four got almost the exact Wow. All four got almost the exact same measurement. How accurate same measurement. How accurate are they?are they?

NOT VERY.NOT VERY.

►Remember, they forgot to tare the Remember, they forgot to tare the balance. balance. Their measurements were all close to each Their measurements were all close to each

other, but they’re all off a bit compared to other, but they’re all off a bit compared to what you and your teacher got. (and your what you and your teacher got. (and your teacher is NEVER wrong).teacher is NEVER wrong).

So they break a lot of glassware, and they’re So they break a lot of glassware, and they’re not very accurate.not very accurate.

But they are But they are PP RECISE.RECISE.

PrecisionPrecision

►Refers to the closeness of a set of Refers to the closeness of a set of measurements of the same quantity measurements of the same quantity made in the same way. made in the same way.

►These measurements are close to one These measurements are close to one another, but not close to the accepted another, but not close to the accepted value.value.

So what happens ifSo what happens if

►Your measurements are neither Your measurements are neither accurate nor precise?accurate nor precise?

►Well, you’ll probably fail the lab Well, you’ll probably fail the lab (JK) (JK)►There is SOME margin of error when There is SOME margin of error when

you’re doing an experiment.you’re doing an experiment. Of course you want the best results Of course you want the best results

possible.possible. When you write your lab report, consider When you write your lab report, consider

what could have gone wrong.what could have gone wrong.

Percentage ErrorPercentage Error

►We can calculate how accurate your We can calculate how accurate your experimental values are compared to experimental values are compared to the correct or accepted valuethe correct or accepted value

►Percentage error is calculated by Percentage error is calculated by subtracting the accepted value from subtracting the accepted value from the experimental value, dividing the the experimental value, dividing the difference by the accepted value, then difference by the accepted value, then multiplying by 100multiplying by 100

The FormulaThe Formula

► Percentage error =Percentage error = ValueValueexperimental experimental – Value– Valueacceptedaccepted

x 100x 100

ValueValueacceptedaccepted

Or, Or, what you got – what you shoulda gotwhat you got – what you shoulda got x x 100100

what you shoulda gotwhat you shoulda got

Why are there errors in Why are there errors in measurements?measurements?

► It could be the skill of the person doing It could be the skill of the person doing the measuringthe measuring

► It could the conditions under which It could the conditions under which you are measuringyou are measuring

► It could even be the measuring It could even be the measuring instruments you are usinginstruments you are using

►All measuring devices have varying All measuring devices have varying degrees of precision.degrees of precision.

BalancesBalances

BeakersBeakers

Graduated CylindersGraduated Cylinders

Significant FiguresSignificant Figures

► In science, we report measurements in In science, we report measurements in terms of terms of significant figuressignificant figures

►Significant Figures consist of all digits Significant Figures consist of all digits known with certainty, plus one more known with certainty, plus one more that is estimated.that is estimated.

►When you look at a measured When you look at a measured quantity, you must determine which quantity, you must determine which digits are significant.digits are significant.

Sig Fig RulesSig Fig Rules

All non zero digits are All non zero digits are significantsignificant

► 1.298 m1.298 m 44

► 65.455 g65.455 g 55

► 43,221.2 ft43,221.2 ft 66

► 2.1 L2.1 L 22

Trapped zeros are significantTrapped zeros are significant

►202.22 m202.22 m 55

►1,001 ft1,001 ft 44

►4.501 s4.501 s 44

Leading zeros are Leading zeros are NEVERNEVER significantsignificant

► Leading zeros are zeros at the Leading zeros are zeros at the beginning of beginning of a numbera number. It doesn’t matter where the . It doesn’t matter where the decimal is….they are decimal is….they are nevernever significant. significant.

► 0.00011 m0.00011 m 22

► 0.02567 ft0.02567 ft 44

► 0.000001 g0.000001 g 11

Zeros a the end of a numberZeros a the end of a number

►With a decimal are With a decimal are alwaysalways significant significant

► 2.000 g2.000 g 44

► 23.270 m23.270 m 55

► 202.1100 L202.1100 L 77

Zeros at the end of a numberZeros at the end of a number

►Without a decimal are Without a decimal are nevernever significant significant

► 23,000 ft23,000 ft 22

► 2,002,000 mi2,002,000 mi 44

► 1,000,000 km1,000,000 km 11

Adding MeasurementsAdding Measurements

► When adding and subtracting measurements, When adding and subtracting measurements, your answer should be rounded to the same your answer should be rounded to the same decimal place as the least precise numberdecimal place as the least precise number

► 6.071 g + 2.22 g + 1.1 g =6.071 g + 2.22 g + 1.1 g = 9.391 g9.391 g For our answer to be in correct sig figs, we round For our answer to be in correct sig figs, we round

to the decimal place of the least precise to the decimal place of the least precise measurement (1.1 g) . Our answer would round tomeasurement (1.1 g) . Our answer would round to

9.4 g9.4 g

Subtracting MeasurementsSubtracting Measurements

►When adding and subtracting When adding and subtracting measurements, your answer should be measurements, your answer should be rounded to the same decimal place as rounded to the same decimal place as the least precise numberthe least precise number

►64.22 g – 12.5611 g = 51.6589 g64.22 g – 12.5611 g = 51.6589 g►The least precise number here is 64.22 The least precise number here is 64.22

g, so our answer rounds to 2 decimal g, so our answer rounds to 2 decimal placesplaces

►51.66 g51.66 g

Multiplying MeasurementsMultiplying Measurements

►When multiplying or dividing, your When multiplying or dividing, your answer will round to the same number answer will round to the same number of significant figures as the least of significant figures as the least precise measurement (least number of precise measurement (least number of sig figs).sig figs).

►22.11 m x 1.678 m x 0.122 m =22.11 m x 1.678 m x 0.122 m = 4.52627076 m4.52627076 m The least precise measurement is 0.122 The least precise measurement is 0.122

m, so our answer will round to 3 sig figsm, so our answer will round to 3 sig figs►4.53 m4.53 m

Get used to ‘em, they’ll be Get used to ‘em, they’ll be following you all year following you all year