__EXPERIMENTAL
DESIGNS AND DATA ANALYSIS__

The focus of this site 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

** Systematic error** results in a flaw in experimental
design or equipment and can be detected and corrected. For example if your equipment is not
calibrated to zero; this can be fixed.
You should control this type of error by periodically checking the scale
throughout the experiment to ensure that calibration doesn’t degrade over
time. 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. Random error is inherent in measurement. An
example of random error is that of reading a burette, which is somewhat subjective and
therefore varies with the person making the reading. Sometimes even the same sample could yield
different measurements on the same instrument!

This type
of error impacts 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**

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, then find
the mean, and error may be minimized.

Although if
for example you are massing individual pennies 20 times, and each measurement
is above the true value, and then you are using that data in a multiplication
calculation, the error is propagated.
Meaning, you are multiplying the error too (not good.)

*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.

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

- 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
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.

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

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 generalizing 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.

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.

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 cl
(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:
- M x 10
^{n}... 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 10
^{n }format. - division ... divide the values of M and
subtract the values of n... Express answer in M x 10
^{n }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 an 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 ...
- mean ... arithmetic, harmonic,
and geometric
- median
- 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 mid-point 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"