Download Uncertainties

Scientific Notation
When expressing a very large number or a very small number, scientists use scientific
notation. The basic format of scientific notation is # x 10n, where # and n represent any
number.
Examples:
1 x 100 = 1
100 =
0.1 =
0.000012 =
12, 000, 000 =
Convert from scientific notation to a number
8.343 x 102 =
1.02 x 10-4 =
4 x 102 + 3.2 x 103 =
1 x 10-1 + 4.2 x 10-3 =
Significant Digits
The significant digits represent the valid digits of a number. The following are rules to
remember:
1. Nonzero digits are always significant.
2. All final zeros after the decimal points are significant.
3. Zeros between two other significant digits are always significant.
4. Zeros used solely for spacing the decimal point are not significant.
Example: 1000C
Practice:
Values
0.0030
1.205
0.59060
503
1.000 x 104
7.80
12
3.21
9.03
10.0030
Significant digits
Values
0.0012003
505.0
950.010
2.0 x 10-3
2.4
100001
54.005
20.50 x 105
0.000345
905.02000
Significant digits
Addition and Subtraction
When adding or subtracting numbers, always round up your answer to the least precise
measurement.
Procedure: Add the numbers up as usual and then examine each set of numbers and
determine which one has the least number of significant figures (this number is the least
precise one) and will determine the overall number of significant figures. In some cases, you
may to have to express the number in scientific notation in order to express the number to the
least precise measurement.
For example: 24.696 + 1.17 =
Practice:
15 + 100 + 3.25 =
1.125 + 5.98 + 63.257 =
Multiplication and Division
When multiplying or dividing a number, always round your answer up to the least number of
significant digits.
0.30 x 0.1 =
5.2 x 1.200 =
Rounding Rules
When rounding always round up if the value is >5
For example: 19.67 can be rounded to 19.7
But what if the following number had to be expressed to three significant figures:
87.35 becomes _____________________
Why? because there are no numbers after 5 that tell us definitely that the 5 is the actual
number.
So, how about: 87.351 to three significant figures would be_______________________.
Why? this will change the answer because now we know that 5 is definitely a number and not
due to rounding up by the instrument.
Examples: Round the following values to 3 significant figures
1. 77.45 __________________
2. 77.55 __________________
Note: depending on whether the preceding digit is an EVEN number followed by a 5, the digit
is NOT rounded up, while if the preceding digit is an ODD number followed by a 5, the digit IS
rounded up. This is done to even out rounding errors. So that we are not always rounding
up.
Graphing Rules:
All graphs must contain an appropriate title, labelled x-axis and y-axis with the appropriate
units. Make sure you choose a suitable scale for your graph. You don't have to start all
graphs at 0,0 (the origin).
Note: all computer-generated graphs start all axis at the zero unless you specify.
According to IB: a graph can only be plotted when a minimum of 5 points.
Once all points have been plotted on the graph, you must decided if you should:
1. Join the points directly
2. Draw the line of best fit – this is usually the case
Ideally, you want a graph in which a mathematical relationship can be generated (a straight
line). This means that you may have to manipulate data to get the straight line.
Equation of a straight line:
y = mx + b
Slope of a line:
m =
y2 – y1
x2 – x1
Measuring the slope of a curve: Draw a tangent to the curve and determine slope.
A tangent to the curve is drawing a line that only touches the curve at one point and taking its
slope.
Measuring the average of two points along a curve of a graph: Draw a secant (cord) to the
curve and determine the slope.
A secant is a line that touches a curve at two points (the average) and this is used to
determine the average of the two points.
ACCURACY, PRECISION AND UNCERTAINTY
Accuracy - how close a value is to an accepted value.
Precision
- measured over a data set
- average or mean of data set
- can you reproduce the results.
Systematic error: error due to apparatus itself or the way in which readings are taken. For
example, not measuring volume by looking at the bottom of the meniscus.
Random error: usually due to the equipment or judgement. For example, each person may
perceive a colour change different or the parallax of the eye. Repeating the measurement
can reduce this.
Uncertainties - can be determined in two days
1. Written on the measurement devices or degree of tolerance:
a. Digital balance always ± 1 of last digit
i.
Centigram balance ±0.01g
b. Graduated cylinder ±1 ml
2. Estimate uncertainty from the measurement devices:
a. It is always half of the last place of significant figures (± half of smallest increment)
i.
Example: smallest division is 0.1 mL, therefore the estimated uncertainty is 0.1
/ 2 = ± 0.05 mL.
Reading Graduated Devices: Estimate the value to one decimal place more than the level
of graduation.
Note: using a beaker to measure a volume of a substance will produce low accuracy and high
error due to the high level of uncertainty involved.
Mathematical Manipulation of Uncertainties
Addition: adding numbers together means adding the uncertainty values.
Example: 5.0 cm ± 0.05cm + 6.0 cm ± 0.05cm
Subtraction: subtracting numbers means adding the uncertainty values.
Example:
mass of beaker and water : 25.0g ± 0.1g
mass of beaker
: 5.0g ± 0.1g
mass of water:
Multiplication and Division and Uncertainties
Problems involving multiplication and division require a different approach. Three steps are
involved:
1. Calculate relative uncertainties of each value (percentage uncertainty) can be determined
using the following formula:
relative uncertainty = uncertainty value/ measurement x 100%
2. Add values of relative uncertainties together (total uncertainty)
3. convert the relative uncertainties (percentage) into absolute uncertainty (a number), using
the following formula:
Absolute
=
Uncertainty
calculated answer x % total uncertainty
100%
Example: Calculate the density of 10.00 ± 0.01 mL of water with a mass of 9.9643
±0.0002g.
Step 1
Step 2
Step 3
(answer: 0.99643g/ml ± 0.0010g/ml)
Example 2: Calculate the density of a block with a mass of 9.240 ± 0.005g and volume of
14.1 ±0.05 cm3.
(answer: 0.655g/cm3 ± 0.003g/cm3)
Uncertainties and Significant digits
the number of significant digits in your question should be reflected in your uncertainty value
as well. In other words, the number of decimal places in your uncertainty value will have the
same number of decimal places as the your calculated answer.
Percent Error Versus Percent Uncertainty
Calculating percent error = l experimental value-theoretical value/ theoretical value l x 100%
During lab discussion, whenever possible, you should always try to compare the percent error
to the relative uncertainty.
For example: the calculated mass of a substance is 129 +- 1.3% g/mol but the percent error
has been calculated to be 2.4%.
The 2.4 % lies just outside the uncertainty value, therefore other factors would have affected
the results and should be discussed in the lab.
However if the same example has a percent error to be 1.4%, then one can conclude that the
inaccuracies are due to systematic errors within the lab.
Note: Some uncertainties in a lab will have a much greater value than others. In those
cases, it is possible to ignore the other uncertainties because of this will have a major effect
on the final result. For example, a thermometer.