Dealing with Numbers
A guide to Numerical &
Graphical Methods
1.0 The Importance of
Experiments
Scientists and engineers spend a lot of time
performing experiments. Why ?
1. They form the basis for scientific and technical
advances.
2. They allow theory to be “put to the test”.
3. They may reveal new, unexpected effects leading
to new or modified theoretical models or
explanations.
In the case of students in a VCE class, experiments
are unlikely to break new ground, but they do
provide you with the opportunity to acquire:
Knowledge
Skills
Understanding
through investigating the “real world”
1.1 Experimental Results –
The Data
OK you’ve done an experiment and collected some
results.
What are the important features of the data you have
collected ?
Measurements made or taken during an experiment
generate “raw” data.
This data must be recorded then presented and
analysed.
All data will have some uncertainty attached.
It doesn’t matter how good the experimenter, how well
designed the experiment or how sophisticated the
measuring device, ALL collected data has some
uncertainty.
(27.5 ± 0.5)0C
This statement of temperature indicates both its
measured value and the uncertainty.
The temperature could be anywhere between
27.5 – 0.5 = 27.00 and 27.5 + 0.5 = 28.00
1.2 Uncertainty
The uncertainty of the measurement
is determined by the scale of the
measuring device.
The uncertainty quantifies (gives a number to)
the amount of variation that has been found in a
measured value.
An alternative term to that of uncertainty is to use
the term EXPERIMENTAL ERROR.
This does NOT imply a mistake in your results,
but simply the natural spread in the values of a
repeatedly measured quantity.
Uncertainty generally comes in three forms:
Resolution Uncertainty – how fine is the scale on the measuring
device ?
Calibration Uncertainty – how well does the measuring device conform
to the standard ?
Reading Uncertainty - how well did the operator use the device ?
1.3 Systematic and Random
Uncertainty
Each form of uncertainty can have
2 categories:
1. Systematic Uncertainty – can
exist without the experimenters
knowledge.
Can skew all readings or values
one way.
Mostly due to instruments
rather than humans.
2. Random Uncertainty –
produces scatter in
measurements.
Environmental factors often
cause this type of error.
Mostly due to humans rather
than instruments.
Elimination of these
“experimental errors” is the
“holy grail” of experimental
scientists and engineers.
Systematic uncertainties can be
reduced or eliminated from the
measuring device by
“calibrating” (comparing to a
known standard) known to a high
degree of both accuracy and
precision.
Random uncertainties can be
controlled (but not eliminated) by
taking multiple readings and
using statistical analysis on the
collected results.
1.4 Precision
Precision is a measure of how closely a group of measurements agree
with one another.
Close agreement translates to a small uncertainty.
However, precision DOES NOT mean that the measurements are close to
the “true value”.
An example here should explain:
The “true value” on a dart board is the bullseye.
A player throws 5 darts
This player is precise - all
darts fall within a small area
(small uncertainty) – but he is
certainly not accurate
1.5 Accuracy
Accuracy is how closely the measurements agree with the true value.
Again using the darts analogy:
This player is BOTH accurate AND precise.
What can you say about the following
measurements ? Each dot represents one
person’s attempt to measure the length of
a piece of string
True
Value
Precise
Imprecise
Inaccurate Inaccurate
Imprecise
Accurate
Precise
Accurate
1.6 Significant Figures
Significant Figures can be regarded as another method of indicating
the uncertainty in a measured quantity.
Significant Figures – THE RULES:
1. All NON ZERO integers are significant.
2. Zeros
(a) Captive Zeros –
they fall between two non zero numbers
they always count as significant figures.
(b) Decimal Point Zeros – Zeros used to place a decimal
point are NOT significant.
(c) Trailing Zeros –
any zeros following a decimal point are
significant.
Number
12.5
0.003002
49,000
0.000234
123.00
Significant
Figures
3
4
2
3
5
1.7 Significant Figure
Manipulation
1. ADDITION & SUBTRACTION:
When adding and subtracting
numbers, round the result of the
calculation to the same number of
decimal places as the number with
the fewest decimal places used in
the calculation.
16.54
8.269
0.47
+21.1
46.379
Rounding to the least number
of decimal places of those
numbers added (21.1 with 1
decimal place).
Answer = 46.4
2.
MULTIPLICATION AND DIVISION:
Identify the number in the
calculation with the least
number of significant figures.
Give your answer to the same
number of sig figs.
65.64 (32.787 + 98.443)
56.4
= 152.729383
With 56.4 having 3 sig figs,
the answer should have 3 sig
figs
Answer = 153
1.8 Scientific Notation
It is not always clear how many
figures in a number are significant.
By changing the unit in which a
number is expressed it can appear
that the amount of significant
numbers changes.
For example a time measurement
could be 125 sec.
Writing this time in milliseconds
would give 125,000 ms.
Both numbers have 3 significant
figures.
However, say somebody asks for the
time measurement in ms and assumes
(incorrectly) that our measuring
device is accurate to within ±1 ms,
then the time would be seen as a 6
significant number.
To get around this problem, Scientific Notation can be
used.
This has all numbers expressed as a “number between 1
and 10, multiplied by a power of 10”
The time 125,000 ms becomes 1.25 x 105 sec and now
only the numbers to the left of the multiply (x) sign are
significant.
1.9 Orders of Magnitude
When performing experiments,
such as measuring the distance to
the stars, determining the strength
of gravity or measuring the speed
of cars passing the college, you
expect to gets answers within a
certain range.
For gravity you would expect to get an answer in the
range 9.7 to 9.9 Nkg-1
For the speeds of the cars maybe a range between 40 to
80 kmh-1.
If your measurements and
subsequent calculations gave
answers for g of 99 Nkg-1 or
speeds of 400 kmh-1 hopefully
you would suspect your
calculations or
measurements.
The ability to make an estimate of an
expected answer at least to within a factor
of 10 can often save an embarrassing and
career threatening mistake.
This ability is called knowing an answer to
within an “order of magnitude”.
Chapter 2
Mathematical Processes
2.0 Rounding
When the result of a calculation has too many figures,
which normally happens when using a calculator, you may
need to reduce the number of figures that appear in the
answer, so that it is becomes both meaningful and
acceptable.
For example, you are asked to measure the length of a thigh
bone (femur) from a skeleton and put that measurement into
a formula to calculate the height of the person before death.
You do this and the calculator gives you an answer of
2.064655089. Your original measurement for the
femur was 0.33 m
Since the original measurement had 2
The process of
significant figures, the answer you quote
reducing the number
should be no more that 2 sig figs.
of significant figures is
Thus the height of the person was 2.1 m.
called ROUNDING the
number.
When a calculation has a number of steps don’t round
until you get to your final answer, as rounding during the
calculation could lead to large errors in the final answer.
2.1 The Mean
A team of students collected the following data in
an experiment aimed at finding the Speed of
Sound.
Speed of
341.5
Sound (ms-1)
342.4
342.2
345.5
341.1
338.5
340.3
342.7
To determine the average or MEAN (usually labelled as x) of these values:
add them and divide by the number of measurements:
x = 2734.2
8
= 341.775 ms-1
How many Significant Figures should the Mean be quoted to ?
The data has 4, so the mean should also have 4, right ?
So, in this case the Mean or
Average speed for sound on
this day was 341.8 ms-1
Is there an uncertainty in the Mean ? If so, how is it calculated ?
2.2 Uncertainty in the Mean
What is the uncertainty
associated with the calculated
speed of sound of 342.8 ms-1 ?
To calculate the uncertainty in
the mean:
1. Calculate the range of values (largest – smallest)
2. Divide the range by the number of terms
Since uncertainties are about
determining the probable range
of a measured or calculated
Thus uncertainty = 345.5 – 338.5
quantity, there is little use in
8
quoting them to any more than 1
= 0.875 ms-1
Significant Figure.
So the uncertainty here
Thus, the speed of sound, and its
becomes ± 0.9 ms-1
associated uncertainty, as
(NOTE: if the first number of the
determined by the students is
uncertainty is a 1, then quote to
(342.8 ± 0.9) ms-1
two sig figs., so an uncertainty
of ±1.425 becomes ± 1.4)
2.3 Fractional and Percentage
Uncertainty
The function of uncertainties is to quantify the probable
range of the values of the measured quantity.
Thus it is usual to quote uncertainty to, at the most, 2
significant figures and often only 1 significant figure.
For the speed of sound - (342.8 ± 0.9) ms-1
FRACTIONAL UNCERTAINTY = Uncertainty in Quantity
Value of Quantity
=
0.9
342.8
= 0.0026
NOTE: Fractional Uncertainty has NO units
PERCENTAGE UNCERTAINTY = Fractional Uncertainty x 100
= 0.0026 x 100
= 0.26%
2.4 Combining Uncertainties
In experiments often you will
collect two or more sets of
data which need to be used in
an equation to calculate a final
result.
The uncertainties in each of the
pieces of data will affect the final
result in a process called error
propagation.
Mathematical Operations:
1. Sums and Differences
If V = a + b
or V = a - b
then ∂V = ∂a + ∂b
In words, the uncertainty
in V is the sum of the
uncertainties of a and b
Uncertainties in measured or
calculated quantities are quoted
in a number of ways:
If the quantity measured is a
Volume (V), its uncertainty could
be quoted as ∂V or ∆V or σV
Mathematical Operations:
Products and Quotients
If V = a x b
or V = a / b
then ∂V = ∂a + ∂b
V a
b
In words, the fractional
uncertainty in V equals the
sum of the fractional
uncertainties in a and b
2.5 Data Selection
A vital question for all experimental
scientists and engineers is:
Are ALL my data equal ?
For many investigators ALL data
is valid and NONE can ever be
rejected.
While others can simply look at a
set of data and label it as
spurious and reject the lot.
And there are yet others who can
look at individual data points and
reject them whilst keeping the
rest.
Confidence in the “correctness” of
experimental data really comes when you
are satisfied that the experiment is
repeatable.
If you do have a suspect data point the
best thing to do is to repeat the
experiment.
Of course this is not always possible,
especially when testing to destruction,
as in breaking a wire or bursting a
balloon.
There are situations where a data point
may be neglected or rejected. For
example, during a series of events being
hand timed, the operator lost
concentration during one of the events.
Statistical tests which help eliminate “spurious” data do exist, but
their rigid and unquestioning application to all data may mask a trend
that you should know about.
Chapter 3
Graphical Methods
3.0 Why Graphs ?
A picture is worth a thousand words.
Humans generally find it easier to
understand information when presented as
a picture rather than as a table of figures.
A graph will indicate:
(a) The range of the measurements taken
(b) The uncertainty in each measurement
(c) The existence or otherwise of trends
(d) The existence of “outlying data”
Temperature
0C
80
60
Data point showing
error bars for both
Temperature (vertical)
and Time (horizontal)
40
20
0
0
outlying data point
20
40
60
80
100
120
Time (sec)
3.1 Graphs – The Basics
The most used graph in
science is the Cartesian
Coordinate Graph, better
known as the x – y graph.
The y axis is known as the
ordinate and the x axis as the
abscissa.
Temperature versus Time
Y
axis (0C)
Temp
Ordinate
The quantity that is controlled or
deliberately varied throughout the
experiment is the INDEPENDENT
Variable and is plotted on the x
axis
The quantity that varies in
response to changes in the
independent variable is called the
DEPENDENT Variable and is
plotted on the y axis
Dependent Variable
ALL graphs require a TITLE,
and AXIS labels and UNITS
X axis
Time (Sec)
Abscissa
Independent Variable
3.2 Graphs – Origins, Scales &
Symbols
On most graphs the numbering
of both the axes begins at zero,
so the bottom left hand corner of
the graph is the point (0,0) and is
called the ORIGIN.
Bad data point
Temperature
0C
Good data points
However there is no law that states
that an origin must be included in a
graph.
Sometimes including an origin will
produce too coarse a scale which
may hide important information.
80
.
ORIGIN
60
40
20
0
0
20
40
60
80
100
120
Time (sec)
The scale should be
chosen so as to allow the
graph to fill the whole
page, while leaving enough
space for labels units and
a title
Data points (with or
without error bars) should
be too big rather than too
small so as they cannot be
mistaken for a smudge on
the page
3.3 Error Bars & Line Drawing
Uncertainties in the quantities
being graphed are indicated
by attaching “error bars” to
each of the data points. They
can be vertical, horizontal or
both.
Their length indicates the size
of the uncertainty associated
with that data point.
Temperature
0C
60
40
20
0
20
Where error bars are very small,
due to the scales used, it is
advisable to omit them from the
graph.
When connecting data points it is
difficult to draw freehand “smooth
curve”
A rubberised flexible ruler called a
“flexi - curve” is probably the best
way to draw curves through data.
As long as the curve fits within the
error bars, the data has been joined
together in a valid way.
80
0
Error bars may:
•remain the same size
for all data points or,
•vary in size from data
point to data point.
40
60
80
100
120
Time (sec)
3.4
Linear
Graphs
It is hard to determine exact
Recognising that the
temperature – time graph
shown previously indicates an
inverse relationship (Temp α
1/Time) and manipulating the
data will give:
mathematical relationships
from curved graphs.
Converting the graph to a linear
or “straight line” graph allows
quantitative relationships to be
determined.
Linear graphs are important in
the analysis of experimental
data because:
(a) The slope or gradient and y
intercept can be calculated
(b) Departure from linearity can
be easily seen
(c) Outliers are easily identified
(d) A mathematical relationship
between “y” and “x” is easily
determined
Temperature
0C
80
60
40
20
0
0
2.5
5.0
1/Time x 10-2 (sec)
Temperature
3.5 Line of Best Fit
0C
Is the red line the only line
that can be drawn to join the
data points ?
Obviously not, other lines
can be drawn.
80
60
40
Is the red line the “best” line
to join the data ?
20
0
0
2.5
5.0
1/Time x 10-2 (sec)
Rules for drawing a Line of Best Fit:
1. Place a clear plastic ruler over the data
points.
2. Move the ruler until the data points are
equally placed above and below the straight
edge.
3. Generally the origin is not a special point,
don’t force the line through it.
4. Use a pencil to draw a fine line along the
straight edge.
Yes, because it meets the
criteria for a “line of best fit”.
It passes through all the error
bars.
It has as many data points
above the line as below and
the distances above and below
total about the same.
3.6 Determining Relationships
Linearising the relationship between variables allows you to use the
general equation for a straight line (y = mx + c) to determine the
mathematical law which relates the variables.
Temperature
In this case:
y = Temperature (oC)
m = Slope of Graph
x = 1/Time (sec)
c = Temperature axis intercept
0C
80
60
40
Rise
20
0
Slope = Rise/Run
= (75 – 5)/(5 x 10-2 - 0)
= 1400
= 1.4 x 103
Run
0
2.5
5.0
1/Time x 10-2 (sec)
c = +5
Thus: Temp = 1.4 x 103 (1/Time) + 5
3.7 Interpolation & Extrapolation
Once a line of best fit has been drawn for the available data,
it becomes quite easy to determine a “y” value from a given “x” value
or visa versa.
When the “y” or “x” value falls
within the range of known data
points INTERPOLATION is
occurring.
Temperature
0C
Determining a value of a
variable (y and/or x) outside the
range of those already known,
EXTRAPOLATION is occurring.
Interpolation Region
80
60
Extrapolation
Regions
40
20
0
0
2.5
5.0
1/Time x 10-2 (sec)
Of the two processes,
interpolation is inherently more
reliable than extrapolation.
4.0 In Summary
It is very easy to enter data incorrectly
into a calculator or computer which will
ultimately lead to ridiculous values for
gradients and intercepts.
This can go unnoticed unless you have
an approximate value obtained from a
hand drawn graph for comparison.
Computers and calculators are excellent
for fast and repetitive calculations.
But they cannot match the eye/brain
combination when it comes to spotting
patterns or anomalies.
Information Sources:
1. Experimental Methods – An Introduction to the Analysis and Presentation of Data
Les Kirkup – Jacaranda Wiley Ltd. ISBN 0 471 33579 7
2. Dr. Fred Omega Garces - Chemistry 100 – Powerpoint - Miramar College