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CHEMISTRY CHAPTER 2, SECTION 3.
USING SCIENTIFIC MEASUREMENTS
Accuracy and Precision
Accuracy refers to the
closeness of measurements to
the correct or accepted value of
the quantity measured.
Precision refers to the closeness
of a set of measurements of the
same quantity made in the same
way. (Another definition: how
exact a measurement is.)
Accuracy and Precision
Example: 5 measurements of the density
of water made by two groups.
density (g/cm3)
Group 1
Group 2
1
1.08
0.92
2
1.10
1.09
3
1.11
1.03
4
1.09
0.98
5
1.08
0.95
Average
1.09
0.99
Which is more accurate? More precise?
Percentage error is calculated by
subtracting the accepted value from
the experimental value, dividing the
difference by the accepted value,
and then multiplying by 100.
Percentage error =
Valueexperimental -Valueaccepted
Valueaccepted
× 100
Sample Problem: A student measures
the mass and volume of a substance and
calculates its density as 1.40 g/mL. The
correct, or accepted, value of the density
is 1.30 g/mL. What is the percentage error
of the student’s measurement?
Percentage error =
Valueexperimental -Valueaccepted
Valueaccepted
× 100
1.40 g/ mL -1.30 g/ mL

 100  7.7%
1.30 g / mL
Note that units will cancel out.
Percentage error =
Valueexperimental -Valueaccepted
Valueaccepted
× 100
Note that if the measured value is
lower than the accepted value, the
percentage error will be a negative
number.
Error in Measurement
Some error or uncertainty
always exists in any
measurement.
skill of the measurer
conditions of measurement
measuring instruments
Significant Figures
Significant figures in a
measurement consist of all the
digits known with certainty plus
one final digit, which is
somewhat uncertain or is
estimated.
Reporting
Measurements
Using Significant
Figures
Rules for Significant Figures
Any digit that is not zero counts.
Any zero between two digits that are not zero
counts.
Numbers greater than 1: don’t count zeroes
at the right, unless they are after the decimal
point or are followed by a decimal point.
examples:
7,200 - 2 significant figures
7,200. - 4 significant figures
7,200.0 - 5 significant figures
Numbers less than 1: don’t count
zeroes at the left (before the first
digit that is not zero). Zeroes to the
right are counted.
examples:
.0038 – 2 significant figures
0.0038 – 2 significant figures
0.00380 – 3 significant figures
How many significant figures?
1.23
0.00123
350.
7060
100
2.0
0.0200
How many significant figures?
1.23
3
0.00123
3
350.
3
7060 3
100
2.0
0.0200
1
2
3
All digits (not including the 10x) are
significant when a number is
written in scientific notation.
Example: we want to express 300
with two significant figures. If we
write 300, this has 1 significant
figure, but 300. has 3.
Solution: write the number as 3.0 x
102. This has 2 significant
figures.
How many significant figures in
these?
6 x 1010
4.02 x 105
1.00 x 10-3
Solution: write the number as 3.0 x
102. This has two significant
figures.
How many significant figures in
these?
6 x 1010
1
4.02 x 105
3
1.00 x 10-3
3
Rounding
We use rounding to express
calculated numbers to the correct
number of significant figures, and
not suggest greater precision
than actually occurred.
Rules for Rounding
(see Tab. 6, p. 48)
If the number following the last
digit that you want to keep is 6, 7,
8, or 9, round up. Examples:
16 → 20
388 → 390
7784 to 2 sig. fig. → 7800
0.097 → 0.10
If the number following the last digit that
you want to keep is 0, 1, 2, 3, or 4, keep
the digit the same. Examples:
183 → 180
0.009413 to 2 s.f. → 0.0094
If the number following the last digit that
you want to keep is 5 followed by a
nonzero digit, round up. Examples:
1851 to 2 s.f. → 1900
0.0378583 to 3 s.f. → 0.0379
If the number following the last digit that you
want to keep is 5 exactly: some people say
always round up; others say round up or
down to give an even number (to reduce
slight errors that can occur if you have lots
of numbers that are rounded up).
Examples:
2.5 rounds to 3 or to 2
550 rounds to 600
0.00150 to 1 significant figures rounds
to 0.002
865 rounds to 870 or to 860
Addition or Subtraction with Significant
Figures
When adding or subtracting
decimals, the answer must have
the same number of digits to the
right of the decimal point as there
are in the measurement having
the fewest digits to the right of
the decimal point.
Example: 56.27 g + 1.3 g =
57.57 g
Since the second number has
only 1 digit to the right of the
decimal point, the sum must be
rounded to 1 decimal place =
Example: 56.27 g + 1.3 g =
57.57 g
Since the second number has
only1 digit to the right of the
decimal point, the sum must be
rounded to 1 decimal place =
57.6 g
For numbers with no decimal points,
round so that the last significant digit is
in the same place as the leftmost
uncertain digit.
Example: 630 000 – 58 000 =
572 000
Since the leftmost uncertain digit is in
the 10 thousands place (the “3” of
630 000), the difference must be
rounded to the same place =
For numbers with no decimal points,
round so that the last significant digit is
in the same place as the leftmost
uncertain digit.
Example: 630 000 – 58 000 =
572 000
Since the leftmost uncertain digit is in
the 10 thousands place (the “3” of
630 000), the difference must be
rounded to the same place = 570 000
Multiplication or Division with
Significant Figures
For multiplication or division, the
answer can have no more
significant figures than are in
the measurement with the
fewest number of significant
figures.
Examples:
6.50 m x 3 m = 19.50 m2
The first number has 3 s.f., the
second only 1, so the answer
rounds to 1 s.f. =
Examples:
6.50 m x 3 m = 19.50 m2
The first number has 3 s.f., the
second only 1, so the answer
rounds to 1 s.f. = 20 m2
35.03 g
3
25 cm
g
 1.4012
3
cm
Of the two numbers being
divided, one has 4 s.f. and
the other 2, so the answer
must be rounded to 2 s.f.
=
35.03 g
3
25 cm
g
 1.4012
3
cm
Of the two numbers being
divided, one has 4 s.f. and
the other 2, so the answer
must be rounded to 2 s.f.
= 1.4 g/cm3
Exact Numbers and Significant
Figures
Exact numbers are not considered
when determining significant
figures.
1. Conversion factors: example:
1 kg
523 g 
 0.523 kg
1000 g
You don’t need to round this
because 1000 is an exact number.
2. Integers.
Example: you determine the mass of 3
pennies as 2.528 g, 2.517 g, and
2.534 g. What is the average mass?
2.528 g  2.517 g  2.534 g 7.579 g

 2.52633... g
3
3
The “3” in the denominator is an integer.
Round to 4 significant figures, not 1.
Answer =
2. Integers.
Example: you determine the mass of 3
pennies as 2.528 g, 2.517 g, and
2.534 g. What is the average mass?
2.528 g  2.517 g  2.534 g 7.579 g

 2.52633... g
3
3
The “3” in the denominator is an integer.
Round to 4 significant figures, not 1.
Answer = 2.526
Scientific Notation
In scientific notation, numbers are written
in the form M × 10n, where the factor M is a
number greater than or equal to 1 but less
than 10 and n is a whole number (positive
or negative).
1. Determine M by moving the decimal point
in the original number to the left or the right
so that only one nonzero digit remains to
the left of the decimal point.
2. Determine n by counting the
number of places that you
moved the decimal point.
If you moved it to the left, n is
positive (the number is greater
than 1).
If you moved it to the right, n is
negative (the number is less
than 1).
example: 0.000 12 mm =
1.2 × 10−4 mm
Move the decimal point 4
places to the right and
multiply the number by 10−4.
Examples
149 600 000 = 1.496 x 108
move decimal pt. 8 places
2.36 x 104 = 23 600
0.001293 = 1.293 x 10-3
9.42 x 10-3 = 0.00942
Mathematical Operations Using
Scientific Notation
1. Addition and subtraction —These
operations can be performed only if
the values have the same exponent (n
factor).
ex.: 4.2 × 104 kg + 7.9 × 103 kg
4.2 × 10 4 kg
+0.79 × 10 4 kg
4
4.99 × 10 kg
4
rounded to 5.0 × 10 kg
(Why?)
or:
42 x 103 kg
7.9 x 103 kg
49.9 x 103 kg
rounded to 50. x 103 kg
Note that 50. x 103 is not in scientific
notation, so it must be changed to:
5.0 x 104 kg
2. Multiplication —The M factors
(number parts) are multiplied, and
the exponents (powers of 10) are
added algebraically.
ex.: (5.23 × 106 µm) (7.1 × 10−2 µm)
= (5.23 × 7.1) (106 × 10−2)
= 37.133 × 104 µm2
= 3.7 × 105 µm2
(convert to scientific notation and
round to 2 significant figures)
3. Division — The M factors are
divided, and the exponent of the
denominator is subtracted from that
of the numerator.
7
example:
5.44  10 g
8.1  10 mol
4
5.44
7-4
=
 10 g / mol
8.1
= 0.6716049383 × 103
= 6.7  102 g/mol
Direct Proportions
Two quantities are directly proportional
to each other if dividing one by the
other gives a constant value.
y
 constant (k )
x
y  kx
We can also write:
yx
read as “y is proportional to x.”
Graph is a
straight line
passing through
(0,0)
Slope = k
Inverse Proportions
Two quantities are inversely
proportional to each other if their
product is constant:
x•y=k
If one increases, the other decreases.
1
y
x
read as “y is proportional to 1 divided
by x.”
The graph is
a hyperbola.
P•V=k