making measurements

Making MeasurementsMaking Measurements

Precision vs AccuracyPrecision vs Accuracy AccuracyAccuracy : A measure of how close a measurement comes to the : A measure of how close a measurement comes to the

actual, accepted or true value of whatever is measured.actual, accepted or true value of whatever is measured.

ExampleExample: Over two trials, a student measures the boiling point of : Over two trials, a student measures the boiling point of ethanol and then calculates the average.ethanol and then calculates the average.

Trial No.Trial No. °C°C

11 79.279.2

22 78.878.8

AverageAverage 79.079.0

She then checks a chemistry handbook (CRC) to see how close her She then checks a chemistry handbook (CRC) to see how close her measurements are to the actual value.measurements are to the actual value.

Accepted Value = Accepted Value = 78.478.4 °C°C

Precision vs AccuracyPrecision vs Accuracy PrecisionPrecision : A measure of how close a series of measurements : A measure of how close a series of measurements

are to one another. This is best determined by the deviation of are to one another. This is best determined by the deviation of the data points.the data points.

ExampleExample: Two students independently determine the average : Two students independently determine the average boiling point of ethanol. boiling point of ethanol.

Trial No.Trial No. Student AStudent A Student BStudent B

11 79.279.2 79.779.7

22 78.878.8 76.976.9

AverageAverage 79.079.0 78.378.3

DeviationDeviation 0.40.4 2.82.8

Student A’s data is Student A’s data is less accurateless accurate but but more precisemore precise than student B’s. than student B’s.

Accepted Value = Accepted Value = 78.478.4 °C°C

Determining ErrorDetermining Error ErrorError : The difference between the accepted value and the : The difference between the accepted value and the

experimental value.experimental value.

FormulaFormula: Error = experimental value – accepted value: Error = experimental value – accepted value

Calculate the Calculate the % error% error for the data given below. for the data given below.

Accepted Value = 78.4 Accepted Value = 78.4 °C°C

Percent errorPercent error : The absolute value of the error divided by the accepted value, multiplied by 100. : The absolute value of the error divided by the accepted value, multiplied by 100.

FormulaFormula: % Error =: % Error = error error ׀׀ Accepted Accepted׀׀

valuevalue

X X 100100

Student A = 79.0 Student A = 79.0 °C°C

%E%E = = 0.8 %0.8 %

79.0-78.479.0-78.4 ׀׀׀׀ 78.78.

44

X X 100100

%E%E = =

Scientific NotationScientific Notation


Scientific NotationScientific Notation : An expression of numbers in the form : An expression of numbers in the form m x 10m x 10nn where m is where m is ≥≥ 1 and 1 and << 10 and 10 and nn is an integer.. is an integer..

Example #1Example #1: A single gram of hydrogen contains approximately : A single gram of hydrogen contains approximately

602 000 000 000 000 000 000 000 atoms

which can be rewritten in scientific notation as which can be rewritten in scientific notation as 6.02 6.02 x 10x 102323

ExampleExample #2#2: The mass of an atom of gold is: The mass of an atom of gold is

UsingUsing scientific notation scientific notation makes it easier to work with makes it easier to work with numbers that are very large and very small.numbers that are very large and very small.

0.000 000 000 000 000 000 000 327 gram

which can be rewritten in scientific notation as which can be rewritten in scientific notation as 3.27 3.27 x 10x 10-22-22


InIn scientific notation scientific notation, there is a coefficient and an exponent, or power., there is a coefficient and an exponent, or power.

6.02 x 106.02 x 102323

Coefficient

Exponent

6 02

For exampleFor example: When changing a large number to scientific notation, : When changing a large number to scientific notation, the decimal is moved left until one non-zero digit remains.the decimal is moved left until one non-zero digit remains.

000 000 000 000 000 000 000

The exponent value is determined by moving a decimal point.The exponent value is determined by moving a decimal point.

atoms


6 02

000 000 000 000 000 000 000

Remember to count the number of place values as you move the decimal. Remember to count the number of place values as you move the decimal.

23 23

Now drop the “trailing” zeros and add an “x 10”. Now drop the “trailing” zeros and add an “x 10”.

atomsplaceplace

ss


6 02

2323

Now drop the “trailing” zeros and add an “x 10”. Now drop the “trailing” zeros and add an “x 10”.

Place the 23 as an exponent of the “10”. Place the 23 as an exponent of the “10”.

x 10

Remember to count the number of place values as you move the decimal. Remember to count the number of place values as you move the decimal.

You have just converted standard notation to scientific notation You have just converted standard notation to scientific notation

602 000 000 000 000 000 000 000

6.02 x 1023

atomsplaceplace

ss


3 27 gram

When changing a very small number to scientific notation, the When changing a very small number to scientific notation, the decimal is moved to the right until it passes a non-zero number.decimal is moved to the right until it passes a non-zero number.

0 000 000 000 000 000 000 000 - 22 - 22

placesplaces Again, count the number of place values the decimal moved. Again, count the number of place values the decimal moved.


3 27 gram


0 000 000 000 000 000 000 000 - 22 - 22

placesplaces Again, count the number of place values the decimal moved. Again, count the number of place values the decimal moved.

Now drop the “leading” zeros and add an “x 10”. Now drop the “leading” zeros and add an “x 10”.

x 10



3 27- 22 - 22 placesplaces Again, count the number of place values the decimal moved. Again, count the number of place values the decimal moved.


gram

x 10



3 27

Again, count the number of place values the decimal moved. Again, count the number of place values the decimal moved.

- - 2222


Finally, place the -22 as an exponent of the “10”. Finally, place the -22 as an exponent of the “10”.

Again you have converted standard notation to scientific notation Again you have converted standard notation to scientific notation

0.000 000 000 000 000 000 000 327

3.27 x 10-22

gram


Change the following number to proper scientific notation.Change the following number to proper scientific notation.

0.000 000 12

Change the following number to proper standard notation.Change the following number to proper standard notation.

4.3 x 105

= 1.2 x 10= 1.2 x 10--

77

= 430 000= 430 000

Normally, scientific notation is not used for measurements that Normally, scientific notation is not used for measurements that produce an exponent of +1 or -1.produce an exponent of +1 or -1.

Examples Examples ::

0.32 = 3.2 x 100.32 = 3.2 x 10-1-1 (not usually done) (not usually done)

89.1 = 8.91 x 1089.1 = 8.91 x 1011 (not usually done) (not usually done)

Practice problemsPractice problems

Recording MeasurementsRecording Measurements

Assume you are using a thermometer and want to Assume you are using a thermometer and want to record a temperature. To do this, you must first record a temperature. To do this, you must first determine the instrument’s precision, or I.P.determine the instrument’s precision, or I.P.

The precision of an instrument is equal The precision of an instrument is equal to the smallest division on the to the smallest division on the instrument’s scale.instrument’s scale.

The I.P. is = 0.1 The I.P. is = 0.1 °C°C

Pictured to the right is a thermometer Pictured to the right is a thermometer scale. What is the scale’s precision?scale. What is the scale’s precision?

°C25

24

23

22

Smallest division


The next thing to do is record the temperature and unit. The next thing to do is record the temperature and unit.

How would you record the temperature How would you record the temperature shown?shown?

If you recorded the temperature as If you recorded the temperature as 24.3 24.3 °C°C then you were then you were notnot being as precise as you being as precise as you could be with the scale given.could be with the scale given.

°C25

24

23

22

If you look carefully at the top of the thermometer’s If you look carefully at the top of the thermometer’s fluid you will see it rises a bit higher than fluid you will see it rises a bit higher than 24.3 24.3 °C°C.. You might now record the temperature asYou might now record the temperature as

24.31 24.31 °C °C or or 24.32 °C24.32 °C


°C25

24

23

22

24.32 °C24.32 °C

The first three digits in our measurements are The first three digits in our measurements are known with certainty. However, the last digit was known with certainty. However, the last digit was estimated and involves some uncertainty.estimated and involves some uncertainty.

All measurements consist of known digits and All measurements consist of known digits and oneone estimated digit. Together, they are called estimated digit. Together, they are called significant digitssignificant digits..

Estimated Estimated digitdigit

Certain or Certain or known digitsknown digits

Significant digitsSignificant digits


We will now express our measurement as We will now express our measurement as

°C25

24

23

22

Error in measurement may be represented by a Error in measurement may be represented by a tolerance interval.tolerance interval.

Machines used in manufacturing often set tolerance Machines used in manufacturing often set tolerance intervals, or ranges in which product measurements intervals, or ranges in which product measurements will be tolerated before they are considered flawed.will be tolerated before they are considered flawed.

To determine the tolerance interval in a measurement, To determine the tolerance interval in a measurement, add and subtract (add and subtract (±) one-half of the precision of the ±) one-half of the precision of the measuring instrument to the measurement.measuring instrument to the measurement.

24.32 °C 24.32 °C ± 0.05 °C (±T.I.)± 0.05 °C (±T.I.)

UnitUnitSignificant Significant figuresfigures

Tolerance Tolerance intervalinterval


For exampleFor example: How would you read the temperature shown to the right?: How would you read the temperature shown to the right?

Always round the experimental measurement or Always round the experimental measurement or result to the same decimal place as the uncertainty.result to the same decimal place as the uncertainty.

73.2 °C 73.2 °C ± 0.1 °C (±T.I.)± 0.1 °C (±T.I.)

°C 76

75

74

73

72

71

70

It would be confusing (and perhaps dishonest) to It would be confusing (and perhaps dishonest) to suggest that you knew the digit in the hundredths suggest that you knew the digit in the hundredths (or thousandths) place when you admit that you’re (or thousandths) place when you admit that you’re unsure of the tenth’s place.unsure of the tenth’s place.

It should be read as…It should be read as…

This 3This 3rdrd digit digit was rounded was rounded

downdown

to match the to match the place value of place value of

the T.I.the T.I.


Time Time (min)(min)

°C °C (T.I. = ± 0.05°C)(T.I. = ± 0.05°C)

11 19.0519.05

22 21.2521.25

33 25.8525.85

44 31.1031.10

55 34.0034.00

Average =Average = 26.2526.25

NoteNote: It is not necessary to write the tolerance : It is not necessary to write the tolerance interval for each measurement of a series of interval for each measurement of a series of measurements made using the same instrument.measurements made using the same instrument.

For exampleFor example: If you record a series : If you record a series of temperature measurements in a of temperature measurements in a data table, then you need only state data table, then you need only state the tolerance interval once.the tolerance interval once.

Keep in mindKeep in mind there are many ways of there are many ways of showing uncertainty in measurement. showing uncertainty in measurement. This is why you must indicate the This is why you must indicate the source of the uncertainty, such as…source of the uncertainty, such as…

Tolerance Interval (T.I.)Tolerance Interval (T.I.)

Standard deviation (± SD)Standard deviation (± SD)

Standard error (± SE)Standard error (± SE)

± ± SD =SD = ± ± 5.675.67

± ± SE =SE = ± ± 2.532.53

% E =% E = 5.4 %5.4 %Percent error (%E)Percent error (%E)

Accepted value = Accepted value =

27.74 °C27.74 °C

StatisticStatistic What it isWhat it is Statistical interpretationStatistical interpretation SymbSymbolol

AverageAverageAn estimate of the An estimate of the "true" value of the "true" value of the

measurementmeasurementThe central value The central value XX

Standard Standard deviationdeviation

A measure of the A measure of the "spread" in the "spread" in the

datadata

You can be reasonably sure (about You can be reasonably sure (about 70% sure) that if you repeat the 70% sure) that if you repeat the same measurementsame measurement one more one more

time, that time, that next measurementnext measurement will will be less than one standard be less than one standard

deviation away from the average. deviation away from the average.

SDSD

Standard Standard errorerror

An estimate in the An estimate in the uncertainty in the uncertainty in the

average of the average of the measurementsmeasurements

You can be reasonably sure (about You can be reasonably sure (about 70% sure) that if you do the 70% sure) that if you do the entire entire experimentexperiment again with the same again with the same

number of repetitions, the number of repetitions, the average value from the new average value from the new

experimentexperiment will be less than one will be less than one standard error away from the standard error away from the

average value from this average value from this experiment. experiment.

SESE


Significant FiguresSignificant Figures

The rules for recognizing significant figures are as follows:The rules for recognizing significant figures are as follows:

Significant figuresSignificant figures are all the digits that can be known precisely in a measurement, plus a last estimated digit. are all the digits that can be known precisely in a measurement, plus a last estimated digit.

Determining Significant FiguresDetermining Significant Figures

√√ Zeros within a number are always significant.Zeros within a number are always significant.

√√ Zeros that do nothing but set the decimal Zeros that do nothing but set the decimal pointpointare not are not

significant.significant.

√√ Trailing zeros that aren’t needed to hold a Trailing zeros that aren’t needed to hold a decimaldecimalpoint are point are

significant.significant.

Both Both 43084308 and and 40.0540.05 contain four sig. figs. contain four sig. figs.

570 000570 000 and and 0.0100.010 and and 310310 contain two sig. figs. contain two sig. figs.

Both Both 4.004.00 and and 0.03200.0320 contain three sig. figs. contain three sig. figs.

In our example of In our example of 0.001800.00180, a decimal is , a decimal is presentpresent..

Here is a “trick” that can help you with significant figures.Here is a “trick” that can help you with significant figures.


QuestionQuestion: How many sig. figs. are in : How many sig. figs. are in 0.00180 0.00180 ??

Step 1: Check to see if the number has a decimal. If Step 1: Check to see if the number has a decimal. If yesyes, think , think

“ “presentpresent.” If .” If nono, think “, think “absentabsent.”.”

Step 2: Note that Step 2: Note that PresentPresent starts with a “ starts with a “PP” and so does ” and so does Pacific.Pacific.

Pacific Ocean

(decimal present)


Step 3: Now place the number inside the U.S.A. pictured below.Step 3: Now place the number inside the U.S.A. pictured below.

Step 4:Step 4: Draw an arrow from the Draw an arrow from the Pacific OceanPacific Ocean through the number through the number

until you encounter a non-zero digit.until you encounter a non-zero digit.

Pacific Ocean

(decimal present)

0.001800.00180

√√ Rule:Rule: All digits to the right of the arrow tip are significant. All digits to the right of the arrow tip are significant.

In our example, In our example, 0.001800.00180 has has threethree significant figures. significant figures.


Pacific Ocean

(decimal present)

Atlantic Ocean

(decimal absent)

In our example of In our example of 403 200403 200, a decimal is , a decimal is absentabsent..

New QuestionNew Question: How many sig. figs. are in : How many sig. figs. are in 403 200403 200 ??

Step 1: Check to see if the number has a decimal. If Step 1: Check to see if the number has a decimal. If yesyes, think , think

“ “presentpresent.” If .” If nono, think “, think “absentabsent.”.”

Step 2: Note that Step 2: Note that AbsentAbsent starts with an “ starts with an “AA” and so does ” and so does Atlantic.Atlantic.


Pacific Ocean

(decimal present)

Atlantic Ocean

(decimal absent)

403 200403 200

Step 3: Now place the number inside the U.S.A. pictured below.Step 3: Now place the number inside the U.S.A. pictured below.

Step 4:Step 4: Draw an arrow from the Draw an arrow from the Atlantic OceanAtlantic Ocean through the number through the number

until you encounter a non-zero digit.until you encounter a non-zero digit.

√ √ Rule:Rule: All digits to the left of the arrow tip are significant. All digits to the left of the arrow tip are significant.

In our example, In our example, 403 200403 200 has has fourfour significant figures. significant figures.

How many sig. figs are in each of the following measurements?How many sig. figs are in each of the following measurements?


280.00 280.00

2.8000 x 10 2.8000 x 10 22

4.5 x 10 4.5 x 10 22

fivefive

fivefive

twotwo

450450

0.00030.0003

2.00 x 10 2.00 x 10 --

44

twotwo

oneone

threthreee

100.0030100.0030 seveseven n

0.000300.00030 twotwo

Now change this measurement into scientific notation.Now change this measurement into scientific notation.

How would you record the following measurement?How would you record the following measurement?


It should have been recorded as 150.00 It should have been recorded as 150.00 mm ± mm ± 0.05 mm 0.05 mm

mm 149 150 200

It should have been written as 1.5000 x 10 It should have been written as 1.5000 x 10 22 ± ± 0.05 mm0.05 mm

Note that the three zeros after the numeral Note that the three zeros after the numeral 55 must be retained in order to uphold the precision of the must be retained in order to uphold the precision of the measurement.measurement.

How many significant figures does How many significant figures does 1.5000 x 101.5000 x 1022 contain? contain?

It contains It contains five.five.

Let’s apply what you have learned.Let’s apply what you have learned.

Calculating with Significant FiguresCalculating with Significant Figures

This is analogous to saying that a chain cannot This is analogous to saying that a chain cannot be stronger than its weakest link.be stronger than its weakest link.

In general,In general, a calculated answer cannot be more precise than the a calculated answer cannot be more precise than the least precise measurement from which it is calculated.least precise measurement from which it is calculated.


Round the answer to the same number of Round the answer to the same number of decimal places (not digits) as the measurement decimal places (not digits) as the measurement with the least number of decimal places.with the least number of decimal places.

Here is the rule for Here is the rule for AddingAdding or or SubtractingSubtracting significant figures significant figures

Here is the rule for Here is the rule for Multiplying Multiplying oror Dividing Dividing significant figures significant figures

Round the answer to the same number of Round the answer to the same number of significant figures as the measurement with significant figures as the measurement with the least number of significant figures.the least number of significant figures.

369.76369.76 369.76369.76

Example addition problem:Example addition problem:


12.5212.52 349.0349.0 8.248.24 ++ ++

Step 1Step 1: Stack the numbers and : Stack the numbers and align them by decimal locationalign them by decimal location

Step 2Step 2: Add the numbers: Add the numbers

Step 3Step 3: Locate the measurement : Locate the measurement with the least number of digits with the least number of digits to the right of the decimal pointto the right of the decimal point

The first measurement (349.0 meters) has the least number of digits (one) to the right of the decimal point.

Step 4Step 4:The answer must be :The answer must be rounded to one digit after the rounded to one digit after the decimal point.decimal point.

369.8 or 3.698 x 10369.8 or 3.698 x 1022

Example multiplication problem:Example multiplication problem:


2.10 m2.10 m 0.70 m0.70 m

xx

1.47 m1.47 m22

The first measurement (0.70) has smallest number of significant figures (two).

Step 3Step 3: The answer must be : The answer must be rounded to the same number rounded to the same number of significant figures as the of significant figures as the measurement with the least measurement with the least number of significant figuresnumber of significant figures

1.5 m1.5 m22

Step 1Step 1: Multiply the numbers: Multiply the numbers

Step 2Step 2: Locate the measurement : Locate the measurement with the least number of with the least number of significant figures.significant figures.

1.47 m1.47 m22

Example Example additionaddition problem in scientific notation problem in scientific notation

Calculating with Scientific NotationCalculating with Scientific Notation

EvaluateEvaluate 5.2 x 10 5.2 x 10-2-2

+ 1.82 x 10+ 1.82 x 10-3-3

5.2 x 105.2 x 10-2-2

+ 0.182 x 10+ 0.182 x 10-2-2

5.382 x 105.382 x 10-2-2

Example Example subtractionsubtraction problem in scientific notation problem in scientific notation

EvaluateEvaluate 7.0 x 10 7.0 x 1055 - 5.2 x 10 - 5.2 x 1044

7.0 x 107.0 x 1055

- 0.52 x 10- 0.52 x 1055

6.48 x 106.48 x 1055

Change to same Change to same exponentsexponents

Change to same Change to same exponentsexponents

6.5 x 106.5 x 1055

5.4 x 105.4 x 10-2-2

1010thth’s place’s place



Example Example multiplicationmultiplication problem in scientific notation problem in scientific notation

Calculating with Scientific NotationCalculating with Scientific Notation

Example Example divisiondivision problem in scientific notation problem in scientific notation

x 10 x 10 11

EvaluateEvaluate

2 x 10 2 x 10 -3-3

x 3.6 x 10 x 3.6 x 10 44

EvaluateEvaluate

4.7 x 10 4.7 x 10 22

1.2 x 10 1.2 x 10 77

Add the Add the exponentsexponents

Multiply the Multiply the coefficientscoefficients

Subtract the Subtract the exponentsexponents

Divide the Divide the coefficientscoefficients

x 10 x 10 55 2.6 x 10 2.6 x 10 44

7 x 10 7 x 10 11

Round to 1 sig. fig.Round to 1 sig. fig.

7.2 7.2

= 0.255319148= 0.255319148

Round to Round to 2 sig. figs.2 sig. figs.

ProblemProblem


5.43 0.023 5.43 0.023

Scientific Scientific notationnotation

Rounded Rounded answeranswer

Calculator Calculator answeranswer

236.0869565 236.0869565 240240 2.4 x 102.4 x 1022

47.2 47.2 5050 5 x 105 x 1011 236 236 xx 0.2 0.2

9250.3 9250.3 92509250 9.25 x 109.25 x 1033 0.300 + 9 2500.300 + 9 250

226.84 226.84 226.8226.8 2.268 x 102.268 x 1022 236.04 - 9.2 236.04 - 9.2

66999.96466999.964 6700067000 6.7 x 106.7 x 1044 6.70 x 10 6.70 x 10 44 - 3.6 x 10 - 3.6 x 10 -2-2**

0.24480.2448 0.240.24 2.4 x 102.4 x 10-1-12.04 x 10 2.04 x 10 -3-3 xx 1.2 x 10 1.2 x 10 22

(2 sf)(2 sf)

(1 sf)(1 sf)

(1 p)(1 p)

(0.1 p)(0.1 p)

(1 p)(1 p)

(2 sf)(2 sf)

* Change to 67000 x 10* Change to 67000 x 100 0 – 0.036 x 10– 0.036 x 1000

making measurements

Documents

small number

number of place values

exponent value

scientific notationx

large number

actual value

decimal point

absolute value