An illustration of significant
figures, accuracy,
precision and data analysis

Objective:

To introduce some
fundamentals techniques for measuring mass and volume and to learn
apply the rules of significant figures.

Discussion:

In
science it is extremely important to be able to make measurements
correctly. Reliable measurements allow for factual knowledge about
measurement. As you continue your studies in chemistry and science you
will learn by performing experiments that important properties of
matter such as density depend on specific measurements.

Numbers
and measurement

There are two
general categories of numbers: Exact
numbers are of absolute accuracy, having infinite
precision,whereas measured
numbers are of finite
accuracy with limited precision

Exact numbers are absolute
values because there is no uncertainty in the value. For example
counted numbers are exact numbers the number of people in your lab
section or the number of fingers on your hand defined numbers such as
12 inches per foot. 16 ounces per pound are also exact numbers. To say
there are 28.5 people in your lab section is meaningless. Anything
other than 16 ounces is not a pound. These are exact numbers. The
accuracy and precision of any calculation does not depend on exact
numbers. Measured numbers are the numbers can be the length of a
material measured by a linear dale or the mass of a substance measured
by a balance. Several measurements should be made to verify measured
numbers. The accuracy of a measured numbers is calculated from the
average of the measurements, how close the average is to the “true
value” In the absence of a determinate error or systematic error the
accuracy of a group of measurements should be very high the precision
of a measurement depends upon the uncertainty of the individual
measurements. The precision of a measured numbers maybe be large if the
measured values vary, but the accuracy could be very high if the
average comes close to the true value. On the other hand the precision
of a measured number may be small if all the measurements have similar
values but the accuracy could be very low if the average of those
similar numbers is far from the true value

Every measurement
has two parts: the numerical value and the unit. Both must appear when
quoting or recording a measurement. The numerical value provides the
accuracy of the measurement while the unit tells us the dimension or
property which has been measured. Without both parts the unit and the
value the measurement can be confusing or meaningless.

Measurement
and the 10% rule.

The number of significant
figures in a measurement
always includes one estimated digit when reading the measure value on a
calibrated scale. We include one estimated digit because it is standard
practice in making a measurement to complete the measurement by reading
or estimating 0.1 times (or 10%) of the calibrated separation. Between
the nearest adjacent calibrations of the measuring device. The
estimated digit represents the uncertainty of the measurement. When
reading the measurement on a digital output, the instrument will
usually list the uncertainty of +/-1 for the last digit.

Volume
measurements

The
volume of a sample is the total
amount of
space occupied by the sample. When cooking, liquid volumes are measures
in units of teaspoons, and tablespoons and cups. In the lab liquid
volume are typically measured by using graduated cylinders or
volumetric glass wear. Such glass wear is read by observing the bottom
of the meniscus level of liquid and reading to 0.1 times (the 10% rule)
of the smallest calibrated mark.

Mass measurement and
weight

Mass is
measured in the laboratory by using a
balance. Electronic balances can be tared when performing mass
measurements to tare a balance means to set the display equal to zero
while the container is on the balance then the mass of the matter being
weighted can be directly reads from the balance without having to
subtract the mass of the container.

Average Results and
Standard Deviation

Consider a set of valid experimental
results. Set means a group of results which are related in
that each result refers to the same quantity. Valid means
that each result is acceptable and has been obtained by performing
measurements correctly within the limits of the experimental
accuracy. Rather than report the entire set of values, we
prefer to report the average result for each set. The average
result is the sum of all individual results in the set divided by the
number of results.

Before lab, you must be familiar
with how
to calculate averages and standard deviations (SD). Standard
deviation is defined as

Where
Xi
= the value of each data point

= the average of all the data points

∑
= the Greek letter sigma, meaning “sum of”

n
= the total number of data points

If you do not know how to 'use' this equation, you might want to do some research. Internet?

The
Penny

From 1959
through to 1981, the U.S. Mint produced more
than 130 billion pennies all of pure copper with a density of 8.96 g/mL. Then, in 1982 the Mint introduced a
penny that is 97.6 percent zinc and only 2.4 percent copper with a
density of 7.07 g/mL. The copper-plated zinc penny is
identical in size and appearance to its copper-rich counterpart, but
the current cent is 19% lighter.

The Experiment

In
this experiment, you will determine the density of individual coins as
well as a group of coins. The masses will be measured with
analytical balances. The volume of individual coins and the
group must be done by two different methods. This will give a
total of 8 different groups of densities for you and your lab partner.

(5x
individual old pennies) x 2 Volume methods

one group of 10 old
pennies x 2 Volume methods

(5x individual new pennies) x 2
Volume methods

one group of 10 new pennies x 2 Volume methods

You
will then calculate the average density of a single penny from the 8
methods. You will also calculate and report the standard
deviation of the two sets of individual 5 coin measurements.

A picture:

Questions
to be answered in lab report

No
need to
re-write the question, simply label the answer with question #’s.

1)
Compare the volume and average density that were measured for the stack
of ten pennies by linear measurement and displacement methods for both
old and new pennies. Which method gave a larger value?

2)
By comparing to the actual density, which method, dimensions or
displacement, gives a more accurate measurement of the density of the
penny? Why?

3) Check
all of your results with at least two other groups. Were your
averages comparable? Explain

4)
Which method is more accurate, the stack of ten pennies or individual
pennies? Why?

5) Which
method is more precise, the dimensions or the displacement?
Why?

6) Which method is more
precise, the stack of ten pennies or individual pennies? Why?

7)
Comment on the measuring devices used in this experiment and how they
effected the accuracy or precision.

8)
What did you learn about accuracy and precision from this experiment?

Lab
Report

Your computer generated lab
report (no hand written)
will consist of the answers to the above questions as well as a well
labeled and understandable computer generated (Excel?) table with all
your individual data points as well as densities and SD
calculations. For each average density, also give the % error. You must include only ONE sample calculation of
each calculation in your report. So, show me (in gory
detail) one of your S.D. calculations. For the rest, just
give me what the S.D. actually is. Your sample calculations
CAN BE hand written, but only calculations are allowed to be hand
written. This rule goes for ALL lab reports.

Pre-lab
questions (written on recycled paper, due at start of lab)

1.
Is the assumption that a penny has a regular cylinder valid?

2.
What is the average and standard deviation of the following data set:
3.40 cm, 3.71 cm, 3.20 cm, 3.89 cm, 3.11 cm, 3.39 cm? Show your
work and be detailed in showing your method. Do NOT simply
put the data in your calculator and give me the final result.

3.
As a coin gets “older” what do you expect to happen to the mass of the
coin? There is no right/wrong answer.
Defend/explain your answer.

Last Updated by MEO :28 Dec 10