EXPERIMENTAL DESIGNS AND DATA ANALYSISEXPERIMENTAL DESIGNS AND
DATA ANALYSIS
The focus is on experimental design, the methods used for data
collection and analysis
The goal in the chemistry laboratory is to obtain reliable results while realizing
there are errors inherent in any laboratory technique. Some laboratory errors
are more obvious than others. Replication of a particular experiment allows an
analysis of the reproducibility (precision) of a measurement, while using
different methods to perform the same measurement allows a gauge of the truth
of the data (accuracy).
There are two types of experimental error: systematic and random error.
Systematic error results in a flaw in experimental design or equipment and can
be detected and corrected. This type of error leads to inaccurate
measurements of the true value.
On the other hand, random error is always present and cannot be corrected. An
example of random error is that of reading a burette, which is somewhat
subjective and therefore varies with the person making the reading. Another
random error may be when the sample or subjects vary slightly. These types of
error impact the precision or reproducibility of the measurement. The goal in a
chemistry experiment is to eliminate systematic error and minimize random
error to obtain a high degree of both accuracy and precision.
Expression of experimental results is best done after replicate trials that report
the average of the measurements (the mean) and the size of the uncertainty,
the standard deviation. Both are easily calculated in such programs as Excel.
The standard deviation of the trial reflects the precision of the measurements.
Whenever possible you should provide a quantitative estimate of the precision
of your measurements. The accuracy is often reported by calculating the
experimental error.
You should then reflect upon and discuss possible sources for random error in
your measurements that contribute to the observed random error. Sources of
random error will vary depending on the specific experimental techniques used.
Some examples might include reading a burette, the error tolerances for an
electronic balance etc. Sources of random error do NOT include calculation
error (a systematic error that can be corrected), mistakes in making solutions
(also a systematic error), or your lab partner (who might be saying the same
thing about you!)
Basic principles of Error
Three basic principles necessary to provide valid and efficient control of
experimental error should be followed in the design and layout of experiments.
These are:
Replication - Replication provides an estimate of experimental error; improves
the precision of the experiment by reducing standard error of the mean, and
increases the scope of inference of the experimental results. If you replicate
the results, one time they may come out high and the next time low, and then
find the mean, error may be minimized.
Randomization. This is practiced to avoid bias in the estimate of experimental
error and to ensure the validity of the statistical tests. This means that your
sample should be as random as possible. You shouldn’t select your sample.
Sample size - The recommended sample size for each experiment should be as
large as possible while still manageable. The larger the sample size the less the
impact of individual differences. A lower sample size could be used for multiple
trials, but maintaining the same number in each trial is advantageous. Using an
unequal sampling, then averaging would lead to a weighted error in the direction
of the smaller sample.



The design of an experiment is the most critical part of any research.
Even the very best data analysis is rendered useless by a flawed
design.
There are three areas to consider during your design.
Percent Error (Percent Deviation, Relative Error) and Accuracy
When scientists need to compare the results of two different
measurements, the absolute difference between the values is of very little
use. The magnitude of error of being off by 10 cm depends on whether you
are measuring the length of a piece of paper or the distance from New
Orleans to Houston. To express the magnitude of the error (or deviation)
between two measurements scientists invariably use percent error.
If you are comparing your value to an accepted value, you first subtract the
two values so that the difference you get is a positive number. This is
called taking the absolute value of the difference. Then you divide this
result (the difference) by the accepted value to get a fraction, and then
multiply by 100% to get the percent error.
So,
% error = | your result - accepted value |
accepted value
x 100%
Several points should be noted when using this equation to obtain a percent
error.
1) When you subtract note how many significant figures remain after the
subtraction, and
express your final answer to no more than that number of digits.
2) Treat the % symbol as a unit. The fraction is dimensionless because
units
in the values will cancel.
3) Notice that the error is a positive number if the experimental value is
high, and is a negative number if the experimental value is low.
4) Usually you can minimize the margin of error by using a larger sample
size.
5) The margin (size) of error depends on what you are measuring. If you
are measuring the length of a room and have an error of 1cm, the error is
minimal compared to measuring a worm and having a 1cm error. The error is
larger even though it is 1cm in both cases.
Example of % error: A student measures the volume of a 2.50 liter
container to be 2.38 liters.
What is the percent error in the student's measurement?
%error = = (accepted value – experimental value) x 100%
accepted value
Ans.
% error = (2.50 liters - 2.38 liters) x 100%
2.50 liters
= (.12 liters)
2.50 liters
= .048
x
100%
x 100%
= 4.8% error
Error Reduction
Consider the following questions.




were the control and/or experimental groups equal before the beginning of
the experiment
How are individual differences controlled
What was done to control for selection bias?
How appropriate is the measuring device employed?
Averaging to reduce error





The effects of unusual values (observations) can be reduced by taking an
average.
This average is referred to as the arithmetic mean.
In general, the larger the sample size the more chance mean is correct.
In experimentation where there are no groups averaging, it is
accomplished by conducting numerous trials. But this would be a problem
if this design were used alone if your instrument is reading high because
all your trials would be high.
Error is generally reduced through averaging averages. It’s true a larger
sample would give a better average, but if there’s only one lab doing the
experiment they can’t feasibly use 2000 items for their sample. Then it
would be better to do the experiment 4 times with samples of 50 each time
and average that data.
Random assignment means that each subject has an equal chance (probability)
to be assigned to a control or experimental group.

Best accomplished by trusting a computer program that picks random
numbers without replacement.
MEASUREMENT
A flawless design will be invalidated by an inappropriate measuring instrument.
DATA COLLECTION, ORGANIZATION AND DISPLAY
INTRODUCTION
The purpose of experimentation or observation is to collect data to test the
hypothesis.




Data is a plural noun and datum is its singular form.
Data refers to things that are known, observed, or assumed.
Based on the facts and figures collected during an experiment,
conclusions about the hypothesis can be inferred.
data include stables, graphs, illustrations, photographs and journals.
Information should include, but is not limited to,












step-by-step procedures,
quantities used,
formulas,
equations,
quantitative and qualitative observations,
material lists,
references,
special instructions,
diagrams,
procedures,
data tables,
flow charts
DATA TABLES
The "best" tables provide the most information in the least confusing manner.






contain qualitative or quantitative observations.
Qualitative data is descriptive but contains no measurements. Quantitative
data is also descriptive but is based on measurements.
titled and preceded with consecutive identification references such as
"Figure 1, Figure 2"
labeled columns.
quantity labels in the column headers and not the columns themselves.
consistent significant digits
DRAWING CONCLUSIONS
INTRODUCTION
The conclusion should contain the following sections:




A restatement of the problem, purpose, or hypothesis.
A rejection or failure to reject the hypothesis.
Rational (support) for the decision on the hypothesis
Discussion on the significance (value) and implications regarding the
experiment.
discuss the "significance" (value) and/or implications of the research.
Questions to be considered might include the following:







Why was the research conducted?
Of what practical value is the research?
How could the results be applied to a "real" situation or problem?
What implications could this research have on solving future problems?
Can the results be used to predict future events and if so how accurately?
What can be inferred about the total population, based on the analysis of
the sample population?
Did the research suggest other avenues for further investigations?
SUPPORTING CONCLUSIONS



Be sure to refer to the data in explaining how the conclusion was drawn.
Make direct references to the appropriate illustrations, tables, and graphs.
Make comparisons among the control and experimental groups.


Each comparison should have a paragraph of its own.
Point out similarities and differences.
More on Making comparisons
When making comparisons...






use terms that are quantitative.
Describe in magnitudes such as more than, or less than.
rank the results
compare your results with that of other authors
Avoid the phrase "significant difference" unless you used a test of
significance
Differences among groups, even if numerically large, may not be
significant.
MAKING PREDICTIONS
One of the strongest supports for a cause and effect relationship is to be able to
predict the effects of the independent variable on the dependent variable.





not everything is the result of a single cause.
the effect is due to the interactions of several variables which were not
controlled.
the effect may be a correlation... such as increasing height with increasing
age.
experimental error or chance can give the appearance of a cause and
effect relationship.
The strongest cases for cause and effect are defined by mathematical
equations
EXPLAINING DISCREPANCIES
Flaws in the experimental design should be pointed out with suggestions for
their elimination or reduction.
Questions to be considered might be...

If the experiment were to be repeated, what would be changed and why?



Is there reason to suspect error as a result of the measuring instrument?
How were individual variations controlled?
Is the sample being studied representative of the entire population and
how was it selected.
Discuss any discrepancies in the data or its analysis.


Attempt to explain any unusual observations or discrepancies in the data.
Refer to the data to build support.
EXPERIMENTAL ERROR AND ITS CONTROL







There will be "real differences" among the control and variable groups.
All error can never be entirely eliminated...Random Error occurs in all
experimentation.
Good experimental design strives to reduce error to its minimum.
Systematic error is inherent in all measuring devices
It can be reduced by using an appropriate measuring instrument
and/or careful calibration.
Conclusions can be challenged on the basis of the accuracy and precision
of the measuring devices.
Sampling errors are generally the result of individual differences and/or
the method of selection.
LIMITATIONS OF A STUDY
Do not generalize the results to an entire population... Limit the conclusions to
what was tested.
 "In vitro" (in the laboratory) may not produce the same results as those
done "in situ" (under natural conditions). It is not always possible to control
the interactions of variables under natural conditions.
 Synergistic effects occur when variables interact.
o These may modify or create entirely new effects that neither variable
would produce if tested alone.
Do not propose assumptions that can not be supported.
Avoid editorializing. Make only objective observations, not subjective.
EXPRESSING MEASUREMENT





Measurements are to be recorded using the primary or alternative metric
units in the SI
All measured or calculated values using measurement must have unit
labels.
Decimals are to be used in place of fractions.
A counted number is not a measurement.
When expressing a measured value less than one, place a zero in front of
the decimal point.
Rules for the written expression of SI units are as follow:
Capitalization:



Symbols for SI units are NOT capitalized unless the unit was derived from
a proper name.
Unabbreviated units are NEVER capitalized.
Numerical prefixes and their symbols are NOT capitalized except for the
symbols T, G, M.
Plurals:


Unabbreviated symbols form their plurals in the usual way by adding an
"s" as in newtons.
SI symbols are ALWAYS WRITTEN in their singular form.
Punctuation:

Periods SHOULD NOT be used after a SI unit unless it is the end of a
sentence.
SIGNIFICANT DIGITS
Rules to determine significant digits.
1. Digits other than zero are always significant.... 23.45 ml (4) 0.43 g (2)
69991 km (5)
2. One or more final zeros used after the decimal point....8.600 mg (4) 29.0
cm (3) 0.1390 g (4)
3. Zeros between two other significant digits....10025 mm (5) 3.09 cm (3)
0.704 dc (3)
4. Zeros used for spacing the decimal point or place holding are not
significant....5000 m (1) 0.0001 ml (1) 0.01020 (4)
Rules for determining significant digits when calculations are involved.
An answer can not be any more accurate than the value with the least number
of significant digits.
1. Addition and Subtraction - after making the computation the answer is
rounded off to the decimal place of the least accurate digit in the problem.
2. Multiplication and Division - after making the computation the answer
should have the same number of significant digits as the term with the
least number of significant digits in the problem.
When using calculators…


Significant digits should be determined before placing values in the
calculator.
Answers must then be rounded to the proper number of significant digits.
SCIENTIFIC NOTATION



Scientific or exponential notation is a method for expressing very large or
very small numbers.
All numbers are expressed as a product between the integer (M) to a
power of ten (n).
The format is expressed as:
n
o M x 10 ... where M = an integer from 1 to 9 and n = any integer
Examples of scientific notation.
1. Avogadro's Number = 602 217 000 000 000 000 000 000 molecules =
6.02217 X 1023 molecules
2. Angstrom = 0.000 000 000 1 m = 1 x 10-9 m
Calculations with Scientific Notation



there can only be one number to the left of the decimal point.
multiplication... multiply the values of M and add the values of n ...
Express answer in M x 10n format.
division ... divide the values of M and subtract the values of n... Express
answer in M x 10n format.
Calculator rounding



Be aware that rounding inside a calculator or a computer depends on how
it was programmed.
Some programs truncate. ... This means they simply "cut-off" the end
digits without rounding.
Start with the last number the calculator display and round forward to the
proper number of significant digits.
ACCURACY IN MEASUREMENT
Accuracy refers to how close a measurement is to the accepted value. It may be
expressed in terms of absolute or relative error.
Absolute error is the difference between an observed (measured) value and the
accepted value of a physical quantity. In the laboratory it is often referred to as
experimental error.
Absolute Error = Observed-Accepted
Relative error is a ratio of the absolute error compared to the accepted value
and expressed as a percent. It is generally called the percent of error.
The accuracy of a measurement can only be determined if the accepted value
of the measurement is known.
PRECISION IN MEASUREMENT
Precision relates to the uncertainty (+ or -) in a set of measurements. It is an
agreement between the numerical values of a set of measurements taken in the
same way. It is expressed as absolute or relative deviation.
Uncertainty example 1
EXAMPLE 1:
The value read on the triple beam balance is 2.35 grams. This triple beam
balance has accuracy (absolute uncertainty) of plus or minus 0.1 gram.
Because this balance can not measure values less than 0.1 gram, the last digit
in 2.35 grams is referred to as being doubtful. That is to say we are uncertain as
to the accuracy. It could be 2.36 grams, or 2.34 grams, or even 2.33 grams.
While the number of significant digits is three the last number is doubtful. We
are uncertain of its exact value because our measuring device only has an
accuracy of 0.1 gram.
READING INSTRUMENTS





Some metric rulers have rounded ends. Be certain to read beginning at
the zero mark.
When reading the meniscus, read the lowest point for concaved fluids and
the highest point for convex fluids
When reading an analog instrument (one with a dial or meter as opposed
to digital readouts), look "head-on" at the pointer.
Looking from the side will give an improper reading know as a parallax.
High quality instruments will have a mirror on the dial face to reduce
parallax error.
Calibrating Instruments



Be consistent when calibrating instruments.
Allow for the proper warm-up period for electrical instruments
use the same techniques and standards each time you calibrate.
MEASURES OF CENTRAL TENDENCY


Measures of central tendencies summarize data.
Most common ...
o mean ... arithmetic, harmonic, and geometric
o median
o mode
Arithmetic Mean



The most common measure is the arithmetic mean or average.
The mean is calculated by summing the values of each observation and
dividing by the total number of observations.
Arithmetic Used when only a single variable is involved.

Examples are weight, temperature, length, and test scores.

The median is the central observation (measurement) when all
observations are arranged in increasing sequence.
It is the value above and below which lie an equal number of
observations.
If there are an equal number of observations, then the median is the midpoint value between the two central observations.
In grouped data the median is calculated as the mid-point of the central
interval for an odd number of groups.
If there are an equal number of group intervals it lies between the two
central group intervals.
The median provides a better measure of central tendency than the mean
when the data contains extremely large or small observations.





Mode




The mode is the most frequently occurring value(s) in a set of
observations (measurements).
For grouped data the mode is represented by the mid-point of the
interval(s) having the greatest frequency.
It is possible for a population (group of measurements) to have more than
one mode...bimodal
more than one mode may indicate of a potential problem with the
experimental design.
SYMMETRY




A frequency curve is said to be symmetrical when the mean, mode, and
median are equal
It is asymmetrical when the mean, mode, and median are not equal.
Most frequency curves are asymmetrical.
Curves either lean to the right or to the left and are said to be skewed.
Skewed Frequency Curves


If the tail points to the right, it is said to be "positively skewed".
If the tail points to the left, it is "negatively skewed"