l chedid 2008 significance in measurement measurements always involve a comparison. when you say...

L Chedid 2008L Chedid 2008

Significance in MeasurementSignificance in Measurement

Measurements always involve a Measurements always involve a comparisoncomparison.. When you say that a table is 6 feet long, When you say that a table is 6 feet long,

you're really saying that the table is six times you're really saying that the table is six times longer than an object that is 1 foot long. longer than an object that is 1 foot long.

The foot is a The foot is a unitunit; you measure the length of ; you measure the length of the table by comparing it with an object like a the table by comparing it with an object like a yardstick or a tape measure that is a known yardstick or a tape measure that is a known number of feet long. number of feet long.



The comparison always involves some The comparison always involves some uncertaintyuncertainty.. If the tape measure has marks every foot, and the If the tape measure has marks every foot, and the

table falls between the sixth and seventh marks, you table falls between the sixth and seventh marks, you can be certain that the table is longer than six feet can be certain that the table is longer than six feet and less than seven feet. and less than seven feet.

To get a better idea of how long the table actually is , To get a better idea of how long the table actually is , though, though, you will have to read between the scale you will have to read between the scale division marksdivision marks. .

This is done by estimating the measurement This is done by estimating the measurement to theto the nearest one tenth of the space between scale nearest one tenth of the space between scale divisions.divisions.



Which of the following Which of the following best describes the length best describes the length of the beetle's body in of the beetle's body in the picture to the left? the picture to the left?

a)a) Between 0 and 2 in Between 0 and 2 in

b)b) Between 1 and 2 inBetween 1 and 2 in

c)c) Between 1.5 and 1.6 inBetween 1.5 and 1.6 in

d)d) Between 1.54 and 1.56 inBetween 1.54 and 1.56 in

e)e) Between 1.546 and 1.547 Between 1.546 and 1.547 inin



The correct answerThe correct answer

is . . . is . . .

(d), between 1.54 and (d), between 1.54 and 1.56 inches1.56 inches



Measurements are often written as a Measurements are often written as a single single numbernumber rather than a rangerather than a range. . The beetle's length in the previous frame was The beetle's length in the previous frame was

between 1.54 and 1.56 inches long. between 1.54 and 1.56 inches long. The single number that best represents the The single number that best represents the

measurement is the center of the range, measurement is the center of the range, 1.551.55 inches.inches. When you write the measurement as a single number, When you write the measurement as a single number,

it's understood that the it's understood that the last figurelast figure (the second of the (the second of the two 5’s in this case) had to be estimated. Consider two 5’s in this case) had to be estimated. Consider measuring the length of the same object with two measuring the length of the same object with two different rulers. different rulers.



For each of the For each of the rulers, give the rulers, give the correct length correct length measurement for measurement for the steel pellet as a the steel pellet as a single number single number rather than a rangerather than a range



For the ruler on the left you should have For the ruler on the left you should have had . . .had . . .

1.4 in1.4 in For the ruler on the right, you should have For the ruler on the right, you should have

had . . .had . . .

1.47 in1.47 in



A A zerozero will occur in the last will occur in the last placeplace of a measurement if of a measurement if the measured value falls the measured value falls exactly on a scale division.exactly on a scale division.

For example, the temperature on For example, the temperature on the thermometer should be the thermometer should be recorded as recorded as 30.0°C..

Reporting the temperature as Reporting the temperature as 30°C would imply that the would imply that the measurement had been taken on measurement had been taken on a thermometer with scale marks a thermometer with scale marks 100°C apart! 100°C apart!


Significance in MeasurementSignificance in Measurement A temperature of 17.00°C A temperature of 17.00°C

was recorded with one of was recorded with one of the three thermometers to the three thermometers to the left. Which one was it?the left. Which one was it?A) the top oneA) the top oneB) the middle oneB) the middle oneC) the bottom oneC) the bottom oneD) either the top one, or the D) either the top one, or the middle onemiddle oneE) either the middle one, or E) either the middle one, or the bottom onethe bottom oneF) it could have been any of F) it could have been any of them them



The correct answer is . . .The correct answer is . . .DD

This is the best choice because . . .This is the best choice because . . . The top thermometer reads to 0.1 so 0.0The top thermometer reads to 0.1 so 0.011 can can

be estimatedbe estimated The middle thermometer reads to 0.2 so 0.0The middle thermometer reads to 0.2 so 0.022

can be estimatedcan be estimated In both cases, a reading of 17.0In both cases, a reading of 17.000 is possible is possible



Use the bottom of the Use the bottom of the meniscusmeniscus (the curved (the curved interface between air and interface between air and liquid) as a point of liquid) as a point of reference in making reference in making measurements of volume measurements of volume in a graduated cylinder, in a graduated cylinder, pipet, or buret. pipet, or buret.

In reading any scale, your In reading any scale, your line of sight should be line of sight should be perpendicular to the scale perpendicular to the scale to avoid 'parallax' reading to avoid 'parallax' reading errors. errors.



Determine the Determine the volume readings for volume readings for the two cylinders to the two cylinders to the right, assuming the right, assuming each scale is in mL.each scale is in mL.


Numbers obtained by Numbers obtained by countingcounting have no have no uncertaintyuncertainty unless the count is very large. unless the count is very large. For example, the word For example, the word 'sesquipedalian''sesquipedalian' has has

14 letters. 14 letters. "14 letters" is not a measurement, since that "14 letters" is not a measurement, since that

would imply that we were uncertain about the would imply that we were uncertain about the count in the ones place. count in the ones place.

14 is an exact number here. 14 is an exact number here.

Significance of MeasurementSignificance of Measurement



Numbers obtained from Numbers obtained from definitionsdefinitions have no have no uncertaintyuncertainty unless they have been rounded off. unless they have been rounded off. For example, a foot is exactly 12 inches. The 12 For example, a foot is exactly 12 inches. The 12 is not is not

uncertain at alluncertain at all. . A foot is also exactly 30.48 centimeters from the A foot is also exactly 30.48 centimeters from the

definition of the centimeter. The 8 in 30.48 definition of the centimeter. The 8 in 30.48 is not is not uncertain at alluncertain at all. .

But if you say 1 foot is 30.5 centimeters, you've But if you say 1 foot is 30.5 centimeters, you've rounded off the definition and rounded off the definition and the rounded digit is the rounded digit is uncertainuncertain..



You should have picked choices . . .You should have picked choices . . .

1, 2 and 41, 2 and 4 Choices 3 and 5 are incorrect because Choices 3 and 5 are incorrect because

both are counts are very large.both are counts are very large. Recall that very large counts have uncertainty Recall that very large counts have uncertainty

because of because of inherent flaws in the counting inherent flaws in the counting processprocess..



All of the digits up to and including the All of the digits up to and including the estimated digit are calledestimated digit are called significant significant digitsdigits. . Consider the following measurements. The Consider the following measurements. The

estimated digit is in gold: estimated digit is in gold: Measurement Number of Distance Between Markings

Significant Digits on Measuring Device

142.142.77 g g 4 4 1 g 1 g

101033 nm nm 3 3 10 nm 10 nm

2.99792.99798 8 x 10x 1088 m m 6 6 0.0001 x 10 0.0001 x 1088 m m



A sample of liquid has a measured volume A sample of liquid has a measured volume of 23.01 mL. Assume that the of 23.01 mL. Assume that the measurement was recorded properly.measurement was recorded properly.

How many significant digits does the How many significant digits does the measurement have? 1, 2, 3, or 4?measurement have? 1, 2, 3, or 4?

The correct answer is . . .The correct answer is . . .

44



Suppose the volume measurement was Suppose the volume measurement was made with a graduated cylinder. How far made with a graduated cylinder. How far apart were the scale divisions on the apart were the scale divisions on the cylinder, in mL? 10 mL, 1 mL, 0.1 mL, or cylinder, in mL? 10 mL, 1 mL, 0.1 mL, or 0.01 mL?0.01 mL?


0.1 mL0.1 mL



Which of the digits in the measurement is Which of the digits in the measurement is uncertain? The “2,” “3,” “0,” or “1?”uncertain? The “2,” “3,” “0,” or “1?”


11



Usually one can count significant digits Usually one can count significant digits simply by countingsimply by counting all of the digits up to and all of the digits up to and including the estimated digitincluding the estimated digit.. It's important to realize, however, that the position of It's important to realize, however, that the position of

the decimal point has nothing to do with the number the decimal point has nothing to do with the number of significant digits in a measurement. of significant digits in a measurement.

For example, you can write a mass measured as For example, you can write a mass measured as 124.1 g124.1 g as as 0.1241 kg0.1241 kg. .

Moving the decimal place doesn't change the fact that Moving the decimal place doesn't change the fact that this measurement has this measurement has FOURFOUR significant figures. significant figures.



Suppose a mass is given as 127 ng.Suppose a mass is given as 127 ng. That's 0.127 µg, or 0.000127 mg, or That's 0.127 µg, or 0.000127 mg, or

0.000000127 g. 0.000000127 g. These are all just different ways of writing the These are all just different ways of writing the

same measurement, and all have the same same measurement, and all have the same number of significant digits: number of significant digits: THREETHREE. .



If significant digits are all digits up to and If significant digits are all digits up to and including the first estimated digit, why don't including the first estimated digit, why don't those those zeroszeros count? count? If they did, you could change the amount of If they did, you could change the amount of

uncertainty in a measurement that significant figures uncertainty in a measurement that significant figures imply imply simply by changing the unitssimply by changing the units. .

• Say you measured 15 mLSay you measured 15 mL The 10’s place is certain and the 1’s place is estimated so you The 10’s place is certain and the 1’s place is estimated so you

have have 2 significant digits2 significant digits

• If the units to are changed to L, the number is written as If the units to are changed to L, the number is written as 0.015 L0.015 L

If the 0’s were significant, then just by rewriting the number with If the 0’s were significant, then just by rewriting the number with different units, suddenly you’d have different units, suddenly you’d have 4 significant digits4 significant digits



By By notnot counting those leading zeros, you counting those leading zeros, you ensure that the measurement has the same ensure that the measurement has the same number of figures (and the same relative number of figures (and the same relative amount of uncertainty) whether you write it as amount of uncertainty) whether you write it as 0.015 L or 15 mL0.015 L or 15 mL



Determine the number of significant Determine the number of significant digits in the following series of digits in the following series of numbers:numbers:

0.000341 kg = 0.341 g = 341 mg0.000341 kg = 0.341 g = 341 mg

12 µg = 0.000012 g = 0.000000012 kg12 µg = 0.000012 g = 0.000000012 kg

0.01061 Mg = 10.61 kg = 10610 g 0.01061 Mg = 10.61 kg = 10610 g



You should have answered as follows:You should have answered as follows:33

22

44



How can you avoid counting zeros that serve How can you avoid counting zeros that serve merely to merely to locate the decimal pointlocate the decimal point as as significant figures? Follow this simple significant figures? Follow this simple procedure:procedure: Move the decimal point so that it is Move the decimal point so that it is just to the rightjust to the right of of

the first nonzero digit, as you would in converting the the first nonzero digit, as you would in converting the number to scientific notation. number to scientific notation.

Any zeros the decimal point Any zeros the decimal point moves pastmoves past are not are not significant, unless they are sandwiched between two significant, unless they are sandwiched between two significant digits. significant digits.

All other figuresAll other figures are taken as significant. are taken as significant.



Any zeros that Any zeros that vanishvanish when you convert a when you convert a measurement to scientific notation were not measurement to scientific notation were not really significant figures. Consider the really significant figures. Consider the following examples:following examples:

0.01234 kg0.01234 kg

1.234 x 101.234 x 10-2-2 kg kg

4 sig figs4 sig figsLeading zeros (0.01234 kg) just locate the decimal point. They're Leading zeros (0.01234 kg) just locate the decimal point. They're never significant.never significant.



0.012340 kg1.2340 x 10-2 kg

5 sig figsNotice that you didn't have to move the decimal point past the Notice that you didn't have to move the decimal point past the trailing zero (0.012340 kg) so it doesn't vanish and so is considered trailing zero (0.012340 kg) so it doesn't vanish and so is considered significant.significant.

0.000011010 m1.1010 x 10-5 m

5 sig figsAgain, the leading zeros vanish but the trailing zero doesn't.



0.3100 m3.100 x 10-1 m

4 sig figsOnce more, the leading zeros vanish but the trailing zero doesn't.Once more, the leading zeros vanish but the trailing zero doesn't.

321,010,000 miles321,010,000 miles3.2101 x 103.2101 x 1088 miles miles

5 sig figs 5 sig figs Ignore commas. Here, the decimal point is moved past the trailing zeros (321,010,000 miles) in the conversion to scientific notation. They vanish and should not be counted as significant. The first zero (321,010,000 miles) is significant, though, because it's wedged between two significant digits.



84,000 mg

8.4 x 104 mg

2 sig figsThe decimal point moves past the zeros (84,000 mg) in the The decimal point moves past the zeros (84,000 mg) in the conversion. They should not be counted as significant.conversion. They should not be counted as significant.

32.00 ml

3.200 x 101 ml

4 sig figsThe decimal point didn't move past those last two zeros.



302.120 lbs

3.02120 x 102 lbs

6 sig figsThe decimal point didn't move past the last zero, so it is significant. The decimal point didn't move past the last zero, so it is significant. The decimal point did move past the 0 between the two and the The decimal point did move past the 0 between the two and the three, but it's wedged between two significant digits, so it's three, but it's wedged between two significant digits, so it's significant as well. significant as well.

All of the figures in this measurement are significantAll of the figures in this measurement are significant



When are When are zeroszeros significant? significant? From the previous slide, you know that From the previous slide, you know that

whether a zero is significant or not whether a zero is significant or not depends depends on just where iton just where it appearsappears. .

Any zero that serves Any zero that serves merely to locate the merely to locate the decimal pointdecimal point is not significant. is not significant.



All of the possibilities are covered by All of the possibilities are covered by the following rules:the following rules:

1. Zeros 1. Zeros sandwichedsandwiched between two between two significant digits are significant digits are always significantalways significant. .

1.0001 km 1.0001 km 2501 kg 2501 kg 140.009 Mg140.009 Mg

5, 4, 6



2.2. TrailingTrailing zeros to the right of the zeros to the right of the decimal decimal point are point are always significantalways significant. .

3.0 m 3.0 m 12.000 µm 12.000 µm 1000.0 µm1000.0 µm

2, 5, 5

3. 3. LeadingLeading zeros are zeros are never significantnever significant. . 0.0003 m 0.0003 m 0.123 µm 0.123 µm 0.0010100 µm0.0010100 µm

1, 3, 5



4. Trailing zeros that all appear to the 4. Trailing zeros that all appear to the LEFTLEFT of the decimal point of the decimal point are not are not assumed to assumed to be significantbe significant. .

3000 m 3000 m 1230 µm 1230 µm 92,900,000 miles 92,900,000 miles

1, 3, 3


Significance in MeasurementSignificance in Measurement Rule 4 covers an Rule 4 covers an ambiguousambiguous case. If the zero case. If the zero

appears at the end of the number but to the appears at the end of the number but to the left of the decimal point, we really can't tell if left of the decimal point, we really can't tell if it's significant or not. Is it just locating the it's significant or not. Is it just locating the decimal point or was that digit actually decimal point or was that digit actually measured?measured? For example, the number For example, the number 3000 m3000 m could have could have 1, 2, 3, 1, 2, 3,

or 4 significant figuresor 4 significant figures, depending on whether the , depending on whether the measuring instrument was read to the nearest 100, measuring instrument was read to the nearest 100, 10, or 1 meters, respectively. You just can't tell from 10, or 1 meters, respectively. You just can't tell from the number alone. the number alone.

All you can safely say is that All you can safely say is that 3000 m3000 m has has at at least 1least 1 significant figuresignificant figure. .



The writer of the measurement should use scientific The writer of the measurement should use scientific notation to remove this ambiguity. For example, if notation to remove this ambiguity. For example, if 3000 m was measured to the nearest meter, the 3000 m was measured to the nearest meter, the measurement should be written as 3.000 x 10measurement should be written as 3.000 x 1033 m. m.



How many significant figures are there in How many significant figures are there in each of the following measurements?each of the following measurements?

1010.010 g1010.010 g

32010.0 g32010.0 g

0.00302040 g0.00302040 g

0.01030 g0.01030 g

101000 g101000 g

100 g100 g

7, 6, 6, 4, 3, 17, 6, 6, 4, 3, 1



Rounding OffRounding Off Often a recorded measurement that contains Often a recorded measurement that contains

more than one uncertain digit must be more than one uncertain digit must be rounded offrounded off to the correct number of to the correct number of significant digits. significant digits.

For example, if the last 3 figures in 1.5642 g For example, if the last 3 figures in 1.5642 g are uncertain, the measurement should be are uncertain, the measurement should be written as 1.56 g, so that only written as 1.56 g, so that only ONEONE uncertain uncertain digit is displayed.digit is displayed.



Rules for rounding off measurements:Rules for rounding off measurements:1.1. All digits to the right of the first uncertain All digits to the right of the first uncertain

digit digit have to be eliminatedhave to be eliminated. Look at the first . Look at the first digit that must be eliminated. digit that must be eliminated.

2.2. If the digit is If the digit is greater than or equal to 5greater than or equal to 5, , round up. round up.

• 1.35343 g rounded to 2 figures is 1.4 g.1.35343 g rounded to 2 figures is 1.4 g.• 1090 g rounded to 2 figures is 1.1 x 101090 g rounded to 2 figures is 1.1 x 1033 g. g.• 2.34954 g rounded to 3 figures is 2.35 g.2.34954 g rounded to 3 figures is 2.35 g.



3.3. If the digit is If the digit is less than 5less than 5, round down. , round down. • 1.35343 g rounded to 4 figures is 1.353 g.1.35343 g rounded to 4 figures is 1.353 g.• 1090 g rounded to 1 figures is 1 x 101090 g rounded to 1 figures is 1 x 1033 g. g.• 2.34954 g rounded to 5 figures is 2.3495 g.2.34954 g rounded to 5 figures is 2.3495 g.

Try these:Try these:

2.43479 rounded to 3 figures 2.43479 rounded to 3 figures 1,756,243 rounded to 4 figures 1,756,243 rounded to 4 figures

9.973451 rounded to 2 figures9.973451 rounded to 2 figures



The correct answers The correct answers are . . .are . . .

2.43

1.756 x 106

1.0 x 101

l chedid 2008 significance in measurement measurements always involve a comparison. when you say...

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