Uncertainty in Measurements
Measurements always involve a comparison. When you say that a
table is 6 feet long, you're really saying that the table is six times longer
than an object that is 1 foot long. The foot is a unit; you measure the
length of the table by comparing it with an object like a yardstick or a tape
measure that is a known number of feet long.
The comparison always involves some uncertainty. If the tape
measure has marks every foot, and the table falls between the sixth and
seventh marks, you can be certain that the table is longer than six feet
and less than seven feet. To get a better idea of how long the table
actually is , though, you will have to read between the scale division
marks. This is done by estimating the measurement to the nearest one
tenth of the space between scale divisions.
Which of the following best describes the length of the beetle's
body in the picture to the left? Between 0 and 2 in
Between 1 and 2 in
Between 1.5 and 1.6 in
Between 1.54 and 1.56 in
Between 1.546 and 1.547 in
Uncertainty in Length Measurements
• Measurements are often written as a single number
rather than a range. The beetle's length in the previous
frame was between 1.54 and 1.56 inches long. The
single number that best represents the measurement is
the center of the range, 1.55 inches. When you write the
measurement as a single number, it's understood that
the last figure (the 5 in this case) had to be estimated.
Consider measuring the length of the same object with
two different rulers.
• Give the correct length measurement for the steel pellet
for each of the rulers, as a single number rather than a
range:
Left Ruler: ................ in
Right Ruler: .............. in
ANSWERS
• The left ruler has scale markings every inch,
so you must estimate the length of the pellet
to the nearest 1/10 of an inch. 1.4 in, 1.5 in, or
1.6 in would be acceptable answers.
• The right ruler has scale markings every 0.1
inches, so you must estimate the length of
the pellet to the nearest 0.01 inches. 1.46
inches is an acceptable answer.
Uncertainty in Length Measurements
• Give the correct length measurement for this
electronic component for each of the rulers,
as a single number rather than a range:
Blue Ruler:................ cm
White Ruler:.............. cm
The blue ruler has scale markings every
0.1 cm, so you must estimate the length
of the electronic component to the
nearest 0.01 cm. 1.85 cm, 1.86 cm, or 1.87
cm would be acceptable answers.
The white ruler has scale markings every 1 cm, so you must estimate the
length of the electronic component to the nearest 0.1 cm. 1.9 cm is an
acceptable answer.
Uncertainty in Temperature Measurements
• A zero will occur in the last
place of a measurement if
the the measured value fell
exactly on a scale division.
For example, the
temperature on the
thermometer just should be
recorded as 30.0°C.
Reporting the temperature
as 30°C would imply that the
measurement had been
taken on a thermometer with
scale marks 10°C apart!
Uncertainty in Temperature Measurements
• A temperature of 17.00°C
was recorded with one of
the three thermometers to
the left. Which one was it?
the top one
the middle one
the bottom one
either the top one, or the
middle one
either the middle one, or the
bottom one
it could have been any of
them
ANSWERS
• The top thermometer had scale markings
0.1°C apart; it could be read to the nearest
0.01°C.
• The middle thermometer has scale markings
ever 0.2°C, so it can be read to the nearest
0.02°C.
• The bottom thermometer has markings every
degree, and can be read to the nearest tenth
of a degree. Try again.
Uncertainty in Volume Measurements
• Use the bottom of the meniscus (the
curved interface between air and
liquid) as a point of reference in
making measurements of volume in
a graduated cylinder, pipet, or buret.
In reading any scale, your line of
sight should be perpendicular to the
scale to avoid 'parallax' reading
errors.
• The graduated cylinder on the left
has scale marks 0.1 mL apart, so it
can be read to the nearest 0.01 mL.
Reading across the bottom of the
meniscus, a reading of 5.72 mL is
reasonable (5.73 mL or 5.71 mL are
acceptable, too). Enter the volume
readings for the middle and right
cylinders below, assuming each
scale is in mL.
• Middle Cylinder Volume:................mL
• Right Cylinder Volume:................mL
ANSWERS
• The middle cylinder has graduations every mL, and can
be read to the nearest 0.1 mL. Since the meniscus
touches the mark, the reading should be recorded as 3.0
mL, NOT as 3 mL. If you read 3.1 mL, you were probably
reading across the top of the meniscus. Read at the
bottom of the meniscus.
• The right cylinder has graduations every 0.1 mL, and
can be read to the nearest 0.01 mL. Since the meniscus
is just below the halfway mark between 0.3 and 0.4, the
reading should be recorded as 0.34 mL (although
readings of 0.35 mL or 0.33 mL are acceptable). If you
read 0.37 or 0.38, you were probably reading across the
top of the meniscus. Read at the bottom of the
meniscus.
Exact Numbers
•
•
•
•
Numbers obtained by counting have no uncertainty unless the count is very
large. For example, the word 'sesquipedalian' has 14 letters. "14 letters" is
not a measurement, since that would imply that we were uncertain about
the count in the ones place. 14 is an exact number here.
Very large counts often do have some uncertainty in them, because of
inherent flaws in the counting process or because the count fluctuates. For
example, the number of human beings in the state of Maryland would be
considered a measurement because it can not be determined exactly at the
present time.
Numbers obtained from definitions have no uncertainty unless they have
been rounded off. For example, a foot is exactly 12 inches. The 12 is not
uncertain at all. A foot is also exactly 30.48 centimeters from the definition
of the centimeter. The 8 in 30.48 is not uncertain at all. But if you say 1 foot
is 30.5 centimeters, you've rounded off the definition and the rounded digit
is uncertain.
Which of the following quantities can be determined exactly? (Select all that
are NOT measurements.)
The number of light switches in the room you're sitting in now
The number of ounces in one pound
The number of stars in the sky
The number of inches per meter
The number of red blood cells in exactly one quart of blood
ANSWERS
• Anything that can be easily counted is exact.
The number of light switches in the room
you're sitting in now is exact, for
example.Any defined quantity is exact. The
number of ounces in one pound is exactly
16. The number of inches per meter must be
exact since there are exactly 30.48
centimeters in a foot, exactly 12 inches in a
foot, and exactly 100 centimeters in a meter.
• Stars in the sky and red blood cells in a
given volume of blood can be counted, but
the counts are so large that there will
inevitably be some uncertainty in the final
result.
What are Significant Digits?
All of the digits up to and including the estimated digit are
called significant digits. Consider the following
measurements. The estimated digit is red:
Measurement
142.7 g
103 nm
2.99798 x 108 m
Number of
Significant
Digits
4
3
6
Distance between Markings
onMeasuring Device
1g
10 nm
0.0001 x 108 m
QUESTIONS
A sample of liquid has a measured volume of 23.01 mL.
Assume that the measurement was recorded properly.
1. How many significant digits does the
measurement have?
Enter your answer here:…………..
2. Suppose the volume measurement was made with
a graduated cylinder. How far apart were the scale
divisions on the cylinder, in mL?
A) 10 mL B)1 mL C) 0.1 mL D) 0.01 mL
E) not enough information
3. Which of the digits in the measurement is
uncertain?
A) "2" B) "3" C)"0" D)"1"
E)not enough information
ANSWERS
1. This measurement had FOUR significant figures.
The 2, 3, and 0 are certain; the 1 is uncertain.
Significant digits include all of the figures up to and
including the first uncertain digit.
2. Since the hundredths place was uncertain, the
graduated cylinder must have had markings 0.1 mL
apart.
3. The last figure is the uncertain one; writing the
number as 23.01 mL means: My best estimate of the
volume is 23.01 mL, but it could have been 23.00 mL
or maybe 23.02 mL.
QUESTIONS
A piece of steel has a measured mass of 1.0278 g.
Assume that the measurement w as recorded properly.
1. How many significant digits does the
measurement have?
Enter your answer here: ………….
2. How far apart were the marks on the scale the
mass was read from, in g?
A) 1 g B) 0.1 g C) 0.01 g D) 0.001 g E) 0.0001 g
F) 0.00001 g G) not enough information
3. Which of the digits in the measurement is
uncertain?
A) "1" B) "0" C) "2" D) "7" E) "8"
F) not enough information
ANSWERS
1. This measurement had FIVE significant
figures. The 1, 0, 2, and 7 are certain; the 8 is
uncertain. Significant digits include all of the
figures up to and including the first uncertain
digit.
2. Since the ten thousandths place was
uncertain, the scale must have had markings
0.001 g (1 mg) apart.
3. The last figure is the uncertain one;
writing the number as 1.0278 g means: My
best estimate of the volume is 1.0278 g, but it
could have been 1.0277 g or maybe 1.0279.
Counting Significant Digits
Moving the Decimal Point Doesn't Change
Significant Figures
Usually one can count significant digits simply by
counting all of the digits up to and including the
estimated digit. It's important to realize, however, that
the position of the decimal point has nothing to do with
the number of significant digits in a measurement. For
example, you can write a mass measured as 124.1 g as
0.1241 kg. Moving the decimal place doesn't change the
fact that this measurement has FOUR significant figures.
Suppose a mass is given as 127 ng. That's 0.127 µg, or
0.000127 mg, or 0.000000127 g. These are all just
different ways of writing the same measurement, and all
have the same number of significant digits: three.
Counting Significant Digits
Moving the Decimal Point Doesn't Change
Significant Figures
If significant digits are all digits up to and including the
first estimated digit, why don't those zeros count? If
they did, you could change the amount of uncertainty in
a measurement that significant figures imply simply by
changing the units. If 0.00125 L has 3 significant digits,
you know the uncertainty is about 1 part in 100 (1 part
in 125, to be exact). If it had 6 significant digits, the
uncertainty would only be about 1 part in a million. By
not counting those leading zeros, you ensure that the
measurement has the same number of figures (and the
same relative amount of uncertainty) whether you write
it as 0.00125 L, 1.25 mL, or 1250 µL.
QUESTIONS
Fill in the blanks in the following table.
Measurement
Number of
Significant
Figures
1. 0.000341 kg = 0.341 g = 341 mg
2. 12 mg = 0.000012 g = 0.000000012 kg
3. 0.01061 Mg = 10.61 kg = 10610 g
………………
………………
………………
ANSWERS
1. 0.000341 kg = 0.341 g = 341 mg. 341 mg
clearly has 3 significant figures, and so must
the same measurement written in kg or g.
2. 12 µg = 0.000012 g = 0.000000012 kg. 12
µg clearly has 2 significant figures, and so
must the same measurement written in g or
kg.
3. 0.01061 Mg = 10.61 kg = 10610 g. 10.61 kg
clearly has 4 significant figures, and so must
the same measurement written in g or Mg.
Counting Significant Digits
•
•
•
•
A Procedure for Counting Significant Digits
How can you avoid counting zeros that serve
merely to locate the decimal point as
significant figures? Follow this simple
procedure:
Move the decimal point so that it is just to
the right of the first nonzero digit, as you
would in converting the number to scientific
notation.
Any zeros the decimal point moves past are
not significant, unless they are sandwiched
between two significant digits.
All other figures are taken as significant.
Counting Significant Digits
Any zeros that vanish when you convert a measurement to scientific
notation were not really significant figures. Consider the following
examples.
Converted to
Scientific
Significant
Measurement Notation
Figures
0.01234 kg
1.234 x 10-2 kg
4
Leading zeros (0.01234 kg)
just locate the decimal point.
They're never significant.
0.012340 kg
1.2340 x 10-2 kg
5
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
significant.
0.000011010 m 1.1010 x 10-5 m
5
Again, the leading zeros
vanish but the trailing zero
doesn't.
4
Ditto.
0.3100 m
3.100 x 10-1 m
Counting Significant Digits
Measure
ment
Converted to
Scientific
Notation
321,010,000 3.2101 x 106
miles
miles
84,000 mg
8.4 x 104 mg
Signific
ant
Figures
5 (at Ignore commas. Here, the
least) 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.
2 (at The decimal point moves past
least) the zeros (84,000 mg) in the
conversion. They should not be
counted as significant.
Counting Significant Digits
Measure
ment
Converted to
Scientific
Notation
32.00 mL 3.200 x 101 mL
302.120
lbs
3.02120 x 102
lbs
Signific
ant
Figures
4
The decimal point didn't move
past those last two zeros. They
are significant.
6
The decimal point didn't move
past the last zero (302.120 lbs, so
it is significant. 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 significant as well. All of the
figures in this measurment are
significant.
QUESTIONS
Fill in the blanks in the following table.
Number of
Measurement Significant Figures
a) 1. 0.010010 g ………………….
b) 10.00 g
………………….
c) 1010010 g
…………………..
ANSWERS
a) 0.010010 g has 5 significant figures. The trailing zero is
significant; the leading zeros are not.
b) 10.00 g has 4 significant figures. The trailing zeros are
significant. The decimal point does not move past them when the
number is converted to scientific notation.
c) 1010010 g has at least 6 significant figures. The trailing zero(s)
can not be counted as significant.
When are Zeros Significant?
From the previous frame, you know that whether a zero is significant
or not depends on just where it appears. Any zero that serves merely
to locate the decimal point is not significant. All of the possibilities
are covered by the following rules:
Rule
Examples
(Significant figures are red)
1. Zeros sandwiched between two
significant digits are always
significant.
1.0001 km
2501 kg
140.009 Mg
2. Trailing zeros to the right of the
3.0 m
decimal point are always significant. 12.000 µm
1000.0 µm (trailing zero is
significant by rule 2; others by
rule 1.)
When are Zeros Significant?
3. Leading zeros are never
significant.
0.0003 m
0.123 µm
0.0010100 µm (trailing zeros are
significant by rule 2;
sandwiched zeros by rule 1.)
4. Trailing zeros that all appear to the 3000 m
LEFT of the decimal point can not be 1230 µm
assumed to be significant.
92,900,000 miles
QUESTIONS
Fill in the blanks in the following table.
Minimum
Minimum
Number of
Number of
Measurement Significant Figures Measurement Significant Figures
a)
1010.010 g .......................
d) 32010.0 g ........................
b) 0.00302040 g .......................
e) 0.01030 g ........................
c)
f)
101000 g .......................
100 g ........................
ANSWERS
a) 1010.010 g has 7 significant figures. The trailing zero is right of
the decimal point and is significant; the zeros sandwiched between
the ones are also significant.
b) 0.00302040 g has 6 significant figures. The trailing zero are
significant, since it is to the right of the decimal point. The leading
zeros are not significant. The zeros that are sandwiched between
nonzero digits are significant.
c) 101000 g has at least 3 significant figures. The trailing zero(s) can
not be counted as significant.
d) 32010.0 g has 6 significant figures. The trailing zero is significant
because it is to the right of the decimal point; all of the other zeros
are sandwiched between two significant figures, so they're
significant, too.
e) 0.01030 g has 4 significant figures. The trailing zero(s) are
significant because they are to the right of the decimal point. Zero(s)
that are to the left of the first nonzero digit are NOT significant.
f) 100 g has at least 1 significant figures. The trailing zero(s) can not
be counted as significant.
Rounding Off
Often a recorded measurement that contains more than one uncertain digit
must be rounded off to the correct number of significant digits. For example,
if the last 3 figures in 1.5642 g are uncertain, the measurement should be
written as 1.56 g, so that only ONE uncertain digit is displayed.
Rules for Rounding Off Measurements
1. All digits to the right of the first uncertain digit have to be eliminated.
Look at the first digit that must be eliminated.
2. If the digit is greater than or equal to 5, round up.
1.35343 g rounded to 2 figures is 1.4 g.
1090 g rounded to 2 figures is 1.1 x 103 g.
2.34954 g rounded to 3 figures is 2.35 g.
3.
If the digit is less than 5, round down.
1.35343 g rounded to 4 figures is 1.353 g.
1090 g rounded to 1 figures is 1 x 103 g.
2.34954 g rounded to 5 figures is 2.3495 g.
Try these
a)
b)
c)
2.43479 rounded to 3 figures is
...........
b)1.756243 rounded to 4 figures is ...........
9.973451 rounded to 4 figures is
...........
Answers
a) 2.43479 should be rounded to 2.43. The fourth
figure is a 4, so the number is rounded down.
b) 1.756243 should be rounded to 1.756. The fifth
figure is a 2, so the number is rounded down.
c) 9.973451 should be rounded to 9.973. The fifth
figure is a 4, so the number is rounded down.
Counting Significant Digits for a Series of
Measurements
• Suppose you weigh a penny several times
and obtain the following masses.
• 2.5019 g
2.5023 g
2.5030 g
2.5037 g
2.5043 g
2.5009 g
• How many significant figures should the
average mass be reported to?
Counting Significant Digits for a Series of
Measurements
• The average can't be more precise than any of the
individual measurements. The exact average is
2.5026833333333333333333... g. Clearly this is not an
appropriate way to report the average, since it implies
more precision than any of the masses being averaged
actually have. Masses taken from the balance could be
estimated to the nearest tenth of a milligram (0.0001 g)
so the average can't be more precise than 2.5027 g.
• The last figure is the one and only uncertain figure.
Remember the definition of significant figures: all digits
up to and including the first uncertain digit. Look at the
penny weights above. Which digits are uncertain? The
last two places change from measurement to
measurement. If we want to write an average so that the
last figure recorded is the first (and only) uncertain
figure, it's best to write the average as 2.503 g.