Accuracy and Precision With Significant Figures. Accuracy Accuracy – how closely a measurement agrees with an accepted value. For example: –The accepted.

Accuracy and PrecisionAccuracy and Precision

With Significant FiguresWith Significant Figures

AccuracyAccuracy

AccuracyAccuracy – how closely a measurement – how closely a measurement agrees with an agrees with an acceptedaccepted value. value.

For example:For example:– The accepted density of zinc is 7.14 g/cmThe accepted density of zinc is 7.14 g/cm33

– Student A measures the density as 5.19 g/cmStudent A measures the density as 5.19 g/cm33

– Student B measures the density as 7.01 g/cmStudent B measures the density as 7.01 g/cm33

– Student C measures the density as 8.85 g/cmStudent C measures the density as 8.85 g/cm33

– Which student is most accurate?Which student is most accurate?

ErrorError

All measurements have some error.All measurements have some error.– Scientists attempt to reduce error by taking Scientists attempt to reduce error by taking

the same measurement many times.the same measurement many times.Assuming no bias in the instruments.Assuming no bias in the instruments.

BiasBias – A systematic (built-in) error that makes all – A systematic (built-in) error that makes all measurements wrong by a certain amount.measurements wrong by a certain amount.

– Examples:Examples:

A scale that reads “1 kg” when there is nothing on it.A scale that reads “1 kg” when there is nothing on it.

You always measure your height while wearing thick-You always measure your height while wearing thick-soled shoes.soled shoes.

A stopwatch takes half a second to stop after being A stopwatch takes half a second to stop after being clicked.clicked.

ErrorError

Error = experimental value – accepted valueError = experimental value – accepted value

Example: The accepted value for the specific heat of Example: The accepted value for the specific heat of water is 4.184 J/gºC. Mark measures the specific heat water is 4.184 J/gºC. Mark measures the specific heat of water as 4.250 J/gºC. What is Mark’s error?of water as 4.250 J/gºC. What is Mark’s error?

Error = exp.value – acc.valueError = exp.value – acc.value

Error = 4.250 J/gºC – 4.184 J/gºCError = 4.250 J/gºC – 4.184 J/gºC

Error = 0.066 J/gºCError = 0.066 J/gºC

Percent ErrorPercent Error

%Error = x 100%%Error = x 100%

Example: The accepted value for the molar mass of Example: The accepted value for the molar mass of methane is 16.042 g/mol. Jenny measures the molar methane is 16.042 g/mol. Jenny measures the molar mass as 14.994 g/mol. What is Jenny’s percent error?mass as 14.994 g/mol. What is Jenny’s percent error?

First, find the error:First, find the error:– Error = exp.value – acc.value = 14.994 g/mol – 16.042 g/molError = exp.value – acc.value = 14.994 g/mol – 16.042 g/mol– Error = -1.048 g/molError = -1.048 g/mol

Percent error = │error │/ acc.value x 100%Percent error = │error │/ acc.value x 100%

Percent error = (1.048 g/mol) / (16.042 g/mol) x 100%Percent error = (1.048 g/mol) / (16.042 g/mol) x 100%

Percent error = 0.06533 x 100%Percent error = 0.06533 x 100%

Percent error = 6.533%Percent error = 6.533%

│error │

acc.value

PrecisionPrecision

PrecisionPrecision – describes the closeness of a – describes the closeness of a set of measurements taking under the set of measurements taking under the same conditions.same conditions.

Good precision does Good precision does notnot mean that mean that measurements are accurate.measurements are accurate.

Accuracy and PrecisionAccuracy and Precision

Decent accuracy, but poor precision: the

average of the shots is on the bullseye, but

they are widely spread out. If this were a

science experiment, the methodology or

equipment would need to be improved.

Good precision, but poor accuracy. The

shots are tightly clustered, but they

aren’t near the bullseye. In an experiment this

represents a bias.

Good accuracy and good precision. If this

were a science experiment, we would

consider this data to be valid.

Accuracy and PrecisionAccuracy and Precision

Another way to think of accuracy and Another way to think of accuracy and precision:precision:– Accuracy means telling the truth…Accuracy means telling the truth…– Precision means telling the same story over Precision means telling the same story over

and over.and over.– They aren’t always the same thing.They aren’t always the same thing.

Accuracy and PrecisionAccuracy and Precision

Four teams (A, B, C, and D) set out to measure the radius of the Earth. Four teams (A, B, C, and D) set out to measure the radius of the Earth. Each team splits into four groups (1, 2, 3, and 4) who compile their data Each team splits into four groups (1, 2, 3, and 4) who compile their data separately, then they get back together and compare measurements. Their separately, then they get back together and compare measurements. Their data are presented below:data are presented below:

Team A

Team B

Team CTeam D

Group 1 Group 2 Group 3 Group 3 Averages

6330.2 km 6880.3 km 6940.3 km 6752.0 km

6112.2 km 6099.5 km6105.2 km 6130.7 km

6741.1 km 6912.9 km6015.1 km 5810.0 km

6366.3 km 6400.1 km6038.7 km 6380.0 km

6613.2 km

6111.9 km

6369.8 km

6296.3 km

Team A

Team B

Team CTeam D

Group 1 Group 2 Group 3 Group 3 Averages

6330.2 km 6880.3 km 6940.3 km 6752.0 km

6112.2 km 6099.5 km6105.2 km 6130.7 km

6741.1 km 6912.9 km6015.1 km 5810.0 km

6366.3 km 6400.1 km6038.7 km 6380.0 km

6613.2 km

6111.9 km

6369.8 km

6296.3 km

Which team’s data were most precise?Which team’s data were most precise?– Team B’s data was most precise, because their measurements Team B’s data was most precise, because their measurements

were very consistent.were very consistent.

Which team’s data were most accurate?Which team’s data were most accurate?– We can’t say yet, because we don’t know the accepted value for We can’t say yet, because we don’t know the accepted value for

the radius of Earth.the radius of Earth.

The accepted value is 6378.1 km.The accepted value is 6378.1 km.– % Error of Team A = 3.686%% Error of Team A = 3.686%– % Error of Team B = 4.174%% Error of Team B = 4.174%– % Error of Team C = 0.130%% Error of Team C = 0.130%– % Error of Team D = 1.283%% Error of Team D = 1.283%

Team C was the most accurate team, even though their data Team C was the most accurate team, even though their data weren’t the most precise.weren’t the most precise.

Significant FiguresSignificant Figures

An Easy Method to Avoid An Easy Method to Avoid Producing Misleading ResultsProducing Misleading Results

A thought problemA thought problem

Suppose you had to find the density of a Suppose you had to find the density of a rock. Density = mass / volumerock. Density = mass / volume– You measure the rock’s mass as 45.59 gYou measure the rock’s mass as 45.59 g– You measure the rock’s volume as 9.3 cmYou measure the rock’s volume as 9.3 cm33

– When you type 45.59 / 9.3 into the calculator, When you type 45.59 / 9.3 into the calculator, you get 4.902150538 g/cmyou get 4.902150538 g/cm33

– Should you really write all those digits in your Should you really write all those digits in your answer, ORanswer, OR

– Is the precision of your answer limited by your Is the precision of your answer limited by your measurements?measurements?

A thought problemA thought problem

The calculator’s answer is The calculator’s answer is misleadingmisleading..– You don’t You don’t reallyreally know the rock’s density with know the rock’s density with

that much precision.that much precision.– It’s scientifically dishonest to claim that you It’s scientifically dishonest to claim that you

do.do.– Your answer must be rounded to the most Your answer must be rounded to the most

precise (but still justifiable) value.precise (but still justifiable) value.

How do scientists round numbers to avoid How do scientists round numbers to avoid giving misleading answers?giving misleading answers?

A thought problemA thought problem

Scientists use the concept of significant Scientists use the concept of significant figures to give reasonable answers.figures to give reasonable answers.– We will use sig.figs. in class to practice good We will use sig.figs. in class to practice good

science.science.

If a scientist divided 45.59 g by 9.3 cmIf a scientist divided 45.59 g by 9.3 cm33, he , he or she would report the answer as 4.9 g/cmor she would report the answer as 4.9 g/cm33..– Let’s find out why.Let’s find out why.

It’s Easy and Fast!It’s Easy and Fast!

Only two rules:Only two rules:– One for adding and One for adding and

subtracting.subtracting.– One for multiplying One for multiplying

and dividing.and dividing.

When Adding or SubtractingWhen Adding or Subtracting

Note the precision of the measurementsNote the precision of the measurements– Nearest 0.1? 0.01? 0.001?Nearest 0.1? 0.01? 0.001?

The result should have as many decimal The result should have as many decimal places as the measured number with the places as the measured number with the smallest number of decimal places.smallest number of decimal places.

For ExampleFor Example

5.51 grams + 8.6 grams5.51 grams + 8.6 grams– Round answer to nearest tenth of a gram.Round answer to nearest tenth of a gram.– Calculator gives: 14.11 gCalculator gives: 14.11 g– You write: 14.1 g You write: 14.1 g

For ExampleFor Example

52.09 mL – 49 mL52.09 mL – 49 mL– Round answer to nearest milliliter.Round answer to nearest milliliter.– Calculator gives: 3.09 mLCalculator gives: 3.09 mL– You write: 3 mLYou write: 3 mL

When Multiplying or DividingWhen Multiplying or Dividing

You must You must countcount significant figures significant figures (sig.figs.).(sig.figs.).– The result should have as many sig.figs. as The result should have as many sig.figs. as

the measured number with the least number the measured number with the least number of sig.figs.of sig.figs.

Counting Sig. Figs.Counting Sig. Figs.

All digits are significant All digits are significant EXCEPT:EXCEPT:– Zeroes preceding a decimal Zeroes preceding a decimal

fraction andfraction and– Zeroes at the end of a number Zeroes at the end of a number

that has that has nono decimal point. decimal point.

For ExampleFor Example

0.0045 has 2 significant figures, BUT0.0045 has 2 significant figures, BUT

1.0045 has 1.0045 has 55 significant figures. significant figures.– Can you see the difference?Can you see the difference?

For ExampleFor Example

45.50 has 4 sig.figs.45.50 has 4 sig.figs.– while 45.5000 has 6 sig.figs.while 45.5000 has 6 sig.figs.– but 0.0005 has only 1 sig.fig.but 0.0005 has only 1 sig.fig.

Numbers With No Decimals are Numbers With No Decimals are AmbiguousAmbiguous

Does 5000 mL mean Does 5000 mL mean exactlyexactly 5000? 5000?– Maybe...maybe not.Maybe...maybe not.

So 5000, 500, 50, and 5 are all assumed So 5000, 500, 50, and 5 are all assumed to have to have oneone significant figure. significant figure.

If a writer means If a writer means exactlyexactly 5000 mL, he or 5000 mL, he or she must write 5000. mL or 5.000x10she must write 5000. mL or 5.000x1033 mL mL

How many sig.figs. in each How many sig.figs. in each number?number?

2000 mL2000 mL0.2 mL0.2 mL20.00 mL20.00 mL20 mL20 mL52.50 mL52.50 mL0.0900 mL0.0900 mL0.0042 mL0.0042 mL1.0000 mL1.0000 mL4.0 cm4.0 cm40 mm40 mm40. mm40. mm0.0040 mm0.0040 mm

Now let’s do some math!Now let’s do some math!

5.0033 g + 1.55 g5.0033 g + 1.55 g– Answer rounded to nearest hundredth of a Answer rounded to nearest hundredth of a

gram.gram.– Answer: 6.55 gAnswer: 6.55 g

Do you need to count sig.figs.?Do you need to count sig.figs.?– No.No. Not in this problem. Not in this problem.

Try this one...Try this one...

4.80 mL – 0.0015 mL4.80 mL – 0.0015 mL– Answer rounded to nearest hundredth of a Answer rounded to nearest hundredth of a

milliliter.milliliter.– Answer: 4.80 mLAnswer: 4.80 mL

You might say that 0.0015 mL is You might say that 0.0015 mL is insignificantinsignificant compared to 4.80 mL compared to 4.80 mL

Another one...Another one...

5.0033 g / 5.0 mL5.0033 g / 5.0 mL– Answer must have 2 sig.figs.Answer must have 2 sig.figs.– Answer: 1.0 g/mLAnswer: 1.0 g/mL

Did you have to count sig.figs.?Did you have to count sig.figs.?– YesYes. Because you are dividing, you must . Because you are dividing, you must

count sig.figs.count sig.figs.

One more...One more...

50.0 cm x 0.04000 cm50.0 cm x 0.04000 cm– Answer must have 3 sig.figs.Answer must have 3 sig.figs.– Answer: 2.00 cmAnswer: 2.00 cm22

Did you have to count sig.figs.?Did you have to count sig.figs.?– YesYes!!

A few special casesA few special cases

How many minutes are in 3.55 hours?How many minutes are in 3.55 hours?– 1 hour = 60 minutes, so...1 hour = 60 minutes, so...– 3.55 hours = 3.55 x 60 minutes = ???3.55 hours = 3.55 x 60 minutes = ???– How many sig.figs. in answer?How many sig.figs. in answer?

Conversion factors Conversion factors do not limitdo not limit sig.figs. sig.figs.– There are There are exactlyexactly 60 minutes in 1 hour. 60 minutes in 1 hour.– Only instruments and equipment do!Only instruments and equipment do!

Answer = 213 minutesAnswer = 213 minutes

A few special casesA few special cases

How many sig.figs. are in the number How many sig.figs. are in the number 4.50x104.50x1033??– Answer: 3 sig.figs.Answer: 3 sig.figs.– In scientific notation, the 10 and the exponent In scientific notation, the 10 and the exponent

are are notnot considered significant. considered significant.– But But allall of the digits in the base of the digits in the base areare sig.figs. sig.figs.

What do you do in this situation?What do you do in this situation?400. m x 50.0 m400. m x 50.0 m– Answer should have 3 sig.figs.Answer should have 3 sig.figs.– Calculator gives: 20 000 mCalculator gives: 20 000 m22

– You write: ???You write: ???– Can’t write 20 000 mCan’t write 20 000 m22

Only 1 sig.fig.Only 1 sig.fig.– Can’t write 20 000. mCan’t write 20 000. m22

Has 5 sig.figs.Has 5 sig.figs.– Can’t write 200.00Can’t write 200.00

Not the right answer!Not the right answer!

Solution: Either write the number in scientific Solution: Either write the number in scientific notation:notation:– 2.00x102.00x1044 m m22

Or write the number with a bar over the last sig.fig.:Or write the number with a bar over the last sig.fig.:– 20 000 m20 000 m33