Significant Figures
Accuracy vs. Precision
Percentage Error
• A student measures the mass and volume of a substance and
caluclates its density at 1.40g/mL. The correct, or accepted,
value of the density if 1.30 g/mL. What is the percentage
error?
Percentage error =
𝑉𝑎𝑙𝑢𝑒𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 −𝑉𝑎𝑙𝑢𝑒𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑
𝑉𝑎𝑙𝑢𝑒𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑
X 100
Percentage error
Percentage error=
1.40 𝑔/𝑚𝐿−1.30𝑔/𝑚𝐿
1.30 𝑔/𝑚𝐿
X 100
= 7.7%
Practice:
1.) What is the percentage error for a mass measurement of 17.7
g, given that the correct value is 21.2g?
2.) A volume is measured experimentally at 4.26mL. What is the
percentage error, given that the correct value is 4.15mL?
Percentage Error
17.7 𝑔 −21.2 𝑔
21.2 𝑔
• 1.) Percentage error=
X 100
=-16.5%
4.26 𝑚𝐿 – 4.15 𝑚𝐿
X
4.15 𝑚𝐿
• 2.) Percentage error=
=2.65%
100
Error in Measurement
• Some error or uncertainty always exists in any measurement.
Therefore, the last digit is always the estimation of the value
of a questionable digit.
You know that it is above 39. So
39 is a certain number. You also know
It is above 39.2. What you don’t know
Is where exactly the meniscus is
Between 39.2 and 39.3. It looks like it
Is just below the half way mark, so
39.24 would be an acceptable guess,
With the 4 being the questionable digit.
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.
• The term significant doesn’t mean certain-like the example
with the meniscus, the last digit was significant, but not
certain. There are rules for determining if numbers are
significant or insignficant
Rules for determining Significant
Zeros
Rule
Examples
1. Zeros appearing between nonzero
digits are significant
a. 40.7 L has three significant figures
b. 87,009 km has five significant
figures
2. Zeros appearing in front of all
nonzero digits are not significant
a. 0.095897 has five significant
figures
b. 0.0009 kg has one significant figure
3. Zeros at the end of a number and to
the right of a decimal point are
significant
a. 85.00 g has four significant figures
b. 9.000000000 mm has 10 significant
figures
4. Zeros at the end of a number but to a. 2000m may contain from one to
the left of a decimal point may or may
four significant figures, depending
not be significant. If a zero has not
on how many zeros are place
been measured or estimated but is just
holders. In this class we will
a place holder, it is not signficant. A
assume it only has 1 significant
decimal point placed after zeros
figure
indicates that they are significant.
b. 2000. m contains four significant
Practice with significant figures
• How many significant figures are in the following
measurements?
1.
2.
3.
4.
5.
28.6 g
3440. cm
910 m
0.04604 L
0.0067000 kg
Practice with signficiant figures
• How many significant figures are in the following
measurements?
1.
2.
3.
4.
5.
28.6 g 3 sig figs
3440. cm
4 sig figs
910 m 2 sig figs
0.04604 L
4 sig figs
0.0067000 kg
5 sig figs
Rounding
If the digit following the
last digit retained is:
Then the last digit
should be:
Example (rounded to
three significant figures)
Greater than 5
Be increased by 1
42.68 g
42.7 g
Less than 5
Stay the same
17.32 m
17.3 m
5, followed by nonzero
digit(s)
Be increased by 1
2.7851 cm
5, not followed by
nonzero digit(s), and
preceded by an odd digit
Be increased by 1
4.635 kg
4.64 kg
(because 3 is odd)
5, not followed by
nonzero digit(s), and the
preceding significant
figure is even
Stay the same
78.65 mL
78.6 mL
(because 6 is even)
2.79cm
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: What is the sum of 2.099 g and 0.05681 g?
Addition and subtraction
• 2.099 g + 0.5681 g = 2.156 g
3 sig figs to the
Right of the decimal
• Practice:
1.
2.
87.3 cm – 1.655 cm
5.44 m – 2.6103 m
3 sig figs to the
Right of the decimal
Multiplication and Division with
Significant Figures
• For multiplication and division, the answer can have no more
significant figures than are in the measurement with the
fewest number of significant figures.
• Practice: Calculate the area of a rectangular crystal surface
that measures 1.34 meters and 0.7488 meters (area=length x
width)
Multiplication and Division
• Area= 1.34 m x 0.7488 m = 1.00 m2
3 sig figs
3 sig figs
• Practice:
1.
2.
What is the volume, in cubic meters, of a rectangular solid
that is 0.25m long, 6.1 m wide, and 4.9 m high? (volume=
length x width x height)
12 m x 6.41 m
Scientific Notation
• Numbers are written in the form of M x 10n where M is a
number between 1 and 9 and n is whole number
• Example:
0.00012 mm = 1.2 x 10-4
65,000 km = 6.5 x 104 km (2 sig figs)
1.
2.
•
If you wanted to have 3 sig figs you would write
6.50 x 104
Practice
• Calculate the volume of a sample of aluminum that has a mass
of 3.057 kg. The density of aluminum is 2.70 g/cm3. Pay
attention to your units!!
Direct Proportions
• Two quantaties are directly proportional to each other if
dividing one by the other gives a constant value
• Example: If you make $20.00 an hour, the more hours you
work, the more money you make.
hours
money
Inverse Proportions
• Two quantities are inversely proportional to each other if their
product is constant.
• Example: As the distance from the Earth increases, the
gravitational pull decreases.
distance
gravity