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Hands-OnLabs SM-1 Lab Manual
43
EXPERIMENT 3:
Experimental Errors and Uncertainty
Read the entire experiment and organize time, materials, and work space before beginning.
Remember to review the safety sections and wear goggles when appropriate.
Objective: To gain an understanding of experimental errors and uncertainty.
Materials: Student Provides: Pen and pencils
Paper, plain and graph
Computer and spreadsheet program
From LabPaq: No supplies are required for this experiment.
Discussion and Review: No physical quantity can be measured with perfect certainty;
there are always errors in any measurement. This means that if we measure some
quantity and then repeat the measurement we will almost certainly measure a different
value the second time. How then can we know the “true” value of a physical quantity?
The short answer is that we cannot. However, as we take greater care in our
measurements and apply ever more refined experimental methods we can reduce the
errors and thereby gain greater confidence that our measurements approximate ever
more closely the true value.
“Error analysis” is the study of uncertainties in physical measurements. A complete
description of error analysis would require much more time and space than we have in
this course. However, by taking the time to learn some basic principles of error analysis
we can:
Understand how to measure experimental error;
Understand the types and sources of experimental errors;
Clearly and correctly report measurements and the uncertainties in
measurements; and
Design experimental methods and techniques plus improve our measurement
skills to reduce experimental errors.
Two excellent references on error analysis are:
John R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in
Physical Measurements, 2d Edition, University Science Books, 1997; and
Philip R. Bevington and D. Keith Robinson, Data Reduction and Error Analysis
for the Physical Sciences, 2d Edition, WCB/McGraw-Hill, 1992.
Hands-On Labs SM-1 Lab Manual
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Accuracy and Precision
Experimental error is the difference between a measurement and the true value or
between two measured values. Experimental error itself is measured by its accuracy
and precision.
Accuracy measures how close a measured value is to the true value or accepted
value. Since a true or accepted value for a physical quantity may be unknown, it is
sometimes not possible to determine the accuracy of a measurement.
Precision measures how closely two or more measurements agree with each other.
Precision is sometimes referred to as “repeatability” or “reproducibility”. A measurement
that is highly reproducible tends to give values which are very close to each other.
Figure 1 defines accuracy and precision with an analogy of the grouping of arrows in a
target.
Figure 1: Accuracy vs. Precision
Types and Sources of Experimental Errors
When scientists refer to experimental errors they are not referring to what are commonly
called mistakes, blunders, or miscalculations or sometimes illegitimate, human or
personal errors. Personal errors can result from measuring a width when the length
should have been measured, or measuring the voltage across the wrong portion of an
electrical circuit, or misreading the scale on an instrument, or forgetting to divide the
diameter by 2 before calculating the area of a circle with the formula A = π r2. Such
errors are certainly significant but they can be eliminated by performing the experiment
again correctly the next time.
On the other hand, experimental errors are inherent in the measurement process. They
cannot be eliminated simply by repeating the experiment, no matter how carefully.
There are two types of experimental errors: systematic errors and random errors.
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Systematic Errors: Systematic errors are errors that affect the accuracy of a
measurement. Systematic errors are “one-sided” errors because, in the absence of
other types of errors, repeated measurements yield results that differ from the true or
accepted value by the same amount. The accuracy of measurements subject to
systematic errors cannot be improved by repeating those measurements. Systematic
errors cannot easily be analyzed by statistical analysis. Systematic errors can be
difficult to detect, and once detected they can only be reduced by refining the
measurement method or technique.
Common sources of systematic errors are faulty calibration of measuring instruments,
poorly maintained instruments, or faulty reading of instruments by the user. A common
form of this last source of systematic error is called “parallax error”, which results from
the user reading an instrument at an angle resulting in a reading which is consistently
high or consistently low.
Random Errors: Random errors are errors that affect the precision of a measurement.
Random errors are “two-sided” errors because, in the absence of other types of errors,
repeated measurements yield results that fluctuate above and below the true or
accepted value. Measurements subject to random errors differ from each other due to
random, unpredictable variations in the measurement process. The precision of
measurements subject to random errors can be improved by repeating those
measurements. Random errors are easily analyzed by statistical analysis. Random
errors can be detected and reduced by repeating the measurement or by refining the
measurement method or technique.
Common sources of random errors are problems estimating a quantity that lies between
the graduations (the measurement lines) on an instrument and the inability to read an
instrument because the reading fluctuates during the measurement.
Calculating Experimental Error
When a scientist reports the results of an experiment the report must describe the
accuracy and precision of the experimental measurements. Some common ways to
describe accuracy and precision are described below.
Significant Figures: The least significant digit in a measurement depends on the
smallest unit that can be measured using the measuring instrument. The precision of a
measurement can then be estimated by the number of significant digits with which the
measurement is reported. In general, any measurement is reported to a precision equal
to 1/10 of the smallest graduation on the measuring instrument, and the precision of the
measurement is said to be 1/10 of the smallest graduation.
For example, a measurement of length using a meter tape with 1-mm graduations will
be reported with a precision of ±0.1 mm. A measurement of volume using a graduated
cylinder with 1 mL graduations will be reported with a precision of ±0.1 mL.
Hands-On Labs SM-1 Lab Manual
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Digital instruments are treated differently. Unless the instrument manufacturer indicates
otherwise, the precision of measurement made with digital instruments are reported with
a precision of ±½ of the smallest unit of the instrument. For example, a digital voltmeter
reads 1.493 volts; the precision of the voltage measurement is ±½ of 0.001 volts or
±0.0005 volt.
Percent Error: Percent error measures the accuracy of a measurement by the
difference between a measured or experimental value E and a true or accepted value A.
The percent error is calculated from the following equation:
Equation 1 % Error = | E – A| x 100%
A
Percent Difference: Percent difference measures precision of two measurements by
the difference between the measured or experimental values E1 and E2 expressed as a
fraction of the average of the two values. The equation used to calculate the percent
difference is:
Equation 2
Mean and Standard Deviation: When a measurement is repeated several times we
see the measured values are grouped around some central value. This grouping or
distribution can be described with two numbers: the mean, which measures the central
value and the standard deviation, which describes the spread or deviation of the
measured values about the mean. For a set of N measured values for some quantity x,
the mean of x is represented by the symbol
formula:
Equation 3
Where xi is the i-th measured value of x. The mean is simply the sum of the measured
values divided by the number of measured values. The standard deviation of the
measured values is represented by the symbol σx and is given by the formula:
Hands-On Labs SM-1 Lab Manual
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Equation 4
The standard deviation is sometimes referred to as the “mean square deviation.” It
measures how widely spread the measured values are on either side of the mean. The
meaning of the standard
deviation can be seen
from the figure on the
right. This is a plot of
data with a mean of 0.5.
As shown in this graph,
the larger the standard
deviation, the more
widely spread the data is
about the mean. For
measurements that have
only random errors the
standard deviation
shows that 68% of the
measured values are within σx from the mean, 95% are within 2σx from the mean, and
99% are within 3σx from the mean.
Reporting the Results of an Experimental
Measurement
When a scientist reports the result of an experimental measurement of a quantity x, that
result is reported with two parts. First, the best estimate of the measurement is reported.
The best estimate of a set of measurement is usually reported as the mean
measurements. Second, the variation of the measurements is reported. The variation in
the measurements is usually reported by the standard deviation σx of the
measurements.
The measured quantity is then known to have a best estimate equal to the average, but
it may also vary from
be reported in the following form:
x =
Example: Consider Table 1 below that lists 30 measurements of the mass m of a
sample of some unknown material.
Hands-On Labs SM-1 Lab Manual
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Table 1: Measured Mass (kg) of Unknown
We can represent this data on a type of bar chart called a histogram (Figure 3), which
shows the number of measured values which lie in a range of mass values with the
given midpoint.
Figure 3: Mass of Unknown Sample
For the 30 mass measurements the mean mass is given by:
We see from the histogram that the data does appear to be centered on a mass value
of 1.10 kg. The standard deviation is given by:
We also see from the histogram that the data does, indeed, appear to be spread about
the mean of 1.10 kg so that approximately 70% (= 20/30×100) of the values are within
σm from the mean.
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The measured mass of the unknown sample is then reported as:
m = 1.10± 0.05 kg
PROCEDURES: The data table that follows shows data taken in a free-fall
experiment. Measurements were made of the distance of fall (Y) at each of the four
precisely measured times. From this data perform the following:
1. Complete the table.
2. Plot a graph
3. Plot a graph
motion for an object in free fall starting from rest is y = ½ gt2, where g is the
acceleration due to gravity. This is the equation of a parabola, which has the general
form y = ax2.
4. Determine the slope of the line and compute an experimental value of g from the
slope value. Remember, the slope of this graph represents ½ g.
5. Compute the percent error of the experimental value of g determined from the graph
in part d. (Accepted value of g = 9.8 m/s2)
6. Use a spreadsheet to perform the calculations and plot the graphs indicated.
Time, t
(s)
Dist.
y1 (m)
Dist.
y2 (m)
Dist.
y3 (m)
Dist.
y4 (m)
Dist.
y5 (m)
0 0 0 0 0 0
0.5 1.0 1.4 1.1 1.4 1.5
0.75 2.6 3.2 2.8 2.5 3.1
1.0 4.8 4.4 5.1 4.7 4.8
1.25 8.2 7.9 7.5 8.1 7.4
- SM-1 Manual COLOR 105 08-17-07
Discussion Questions
1. Consider two concepts you encountered in Experiment 3: 1. Accuracy and Precision, and 2. Means. Based on your readings and experiences in conducting Experiment 3:
2. What problems do you envision students having with these concepts?
3.
Describe how you would present these concepts to students, based on what you now know about how children think at different cognitive stages. (Address the question using the age group, and corresponding cognitive state, that you are most interested in teaching.)
Hands-OnLabs SM-1 Lab Manual
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EXPERIMENT 4:
Separation of a Mixture of Solids
Read the entire experiment and organize time, materials, and work space before beginning.
Remember to review the safety sections and wear goggles when appropriate.
Objective: To become familiar with the separation of mixtures of solids.
Materials: Student Provides: Distilled water
2 Coffee cups 2 Small paper or Styrofoam® cups
2 Sheets of paper Small spoon or stirrer
Small saucer Piece of plastic wrap or a plastic baggie
Crushed ice Paper towels
From LabPaq: Goggles
Magnet 100-mL Beaker
Funnel Burner stand
Graduated cylinder Burner fuel
Digital scale
From Experiment 4 Bag: Mixture of solids
Circular filter paper Plastic weighing dish
Discussion and Review: Many materials are actually mixtures of pure substances.
How to separate mixtures into their component substances is a frequent problem for
chemists. The essential distinction between mixtures and “pure” substances is whether
or not they can be separated by physical means. Physical means of separation are
those techniques that utilize the physical properties of a substance such as melting
point and solubility.
In this experiment you will separate a mixture of four substances: sodium chloride
(NaCl, table salt); benzoic acid (C6H5COOH, a common food preservative); silicon
dioxide (SiO2, sand); and iron (Fe, powder and/or filings) into pure substances. The
separation will be accomplished by utilizing the unique properties of each material and
their differences in water solubility.
Solubility is defined as the amount of the solute that will dissolve in a given amount of
solvent. The extent to which a substance dissolves depends mainly upon the physical
properties of the solvent and of the solute and to some extent upon the solvent’s
temperature. Sodium chloride (table salt) is an ionic substance that dissolves readily in
cold water. Benzoic acid is a polar covalent compound that is only slightly soluble in
cold water but is very soluble in hot water. The Handbook of Chemistry and Physics
reflects that the solubility of benzoic acid in water is 6.8 g/100ml at 95°C and only 0.2
g/100ml at 10°C.
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Table of Solubility of NaCl in Water at Various Temperatures:
Temp in degrees C 0 10 20 30 40 50 60 70 80 90 100
Grams/100 mL H20 35.7 35.8 36.0 36.3 36.6 37.0 37.1 37.5 38 38.5 39.2
From the above information and data, it is apparent that sodium chloride and benzoic
acid can be dissolved at different temperatures in water. Further, we know that sand is
not soluble in water. That leaves the iron filings or iron powder, but we also know that
iron filings can be easily removed by a magnet. With this information we can now devise
a plan or flow chart on how to separate the mixture.
PROCEDURES: Some sections of this experiment may require several days for
drying and evaporation. Plan your time accordingly. Remember to completely read all
instructions and assemble all equipment and supplies before beginning your work.
Before you start the actual separation, challenge yourself. Think about and prepare a
flow chart on how the four substances might be separated. Then read the instructions
and compare your proposed procedure or flow chart to the one presented here.
1. Separating out the Iron:
A. Use your digital scale to determine the mass of your weighing dish.
B. Empty the entire mixture of solids from the plastic bag into the weighing dish and
determine the gross mass of the total mixture and weighing dish. Compute the
net mass of the mixture: this is equal to the gross mass of the weighing dish with
the mixture less the mass of just the weighing dish determined in 1-A.
C. Spread the mixture into a very thin layer over a full sized piece of paper.
D. Cut a second piece of paper into a 10-cm square. Weigh and record its mass and
set it aside.
E. Wrap a small square of clear plastic over the magnet. Remove the iron
powder/filings by passing the magnet closely over the surface of the entire
mixture. Repeat several times to make sure you’ve collected all the iron.
F. Holding the magnet over the 10-cm square of paper, carefully remove the plastic
and allow all the iron to fall onto the paper. Weigh and determine the net mass of
the iron powder/filings.
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2. Separating out the Sand:
A. Put the remaining mixture, containing sand, benzoic acid, and table salt into your
beaker and add 50 mL of distilled water.
B. Set up the beaker stand and burner fuel and heat the beaker of solids and water
to near boiling. Stir the mixture to make sure all soluble material dissolves. At this
point, the benzoic acid and the sodium chloride should have dissolved and been
extracted from the insoluble sand.
C. Decant (pour) the liquid while it is hot into a small paper or Styrofoam® cup.
D. Pour another 10 to 15 mL of distilled water into the beaker containing the sand,
bring the mixture to a boil, and decant again into the same cup used in 2-C. This
assures that any remaining salt and benzoic acid is removed from the sand.
E. Make an ice bath by placing a small amount of crushed ice and tap water into a
coffee cup or similar container that is large enough to hold your paper cup of
benzoic acid and salt solution. Make sure the ice bath level is higher than the
solution level but low enough so that no additional water can pour into the
solution cup.
F. Place the cup containing the water solution of benzoic acid and salt into the ice
bath. Observe the benzoic acid crystallizing out of the solution as it cools. Set
this water bath assembly aside until the next section.
G. Heat the sand in the beaker over low heat until the sand is completely dry. Sand
has a tendency to splatter if heated too rapidly. The possibility of sample loss can
be reduced by covering the beaker with a small saucer and heating it very slowly.
You might accomplish this also by placing the beaker in a warm oven.
Alternatively, you may dump the wet sand onto a double layer of paper towels
and let it air-dry.
H. When the sand is completely dry allow the beaker to cool to room temperature.
I. After the sand and any paper towels used are completely dry transfer the sand to
a weighing dish of known mass and determine the net mass of the sand.
3. Separating out the Benzoic Acid:
A. The benzoic acid crystals from Step 2-F above can be separated out by filtration.
Use the following instructions to set up a filtration assembly:
1) Weigh a paper cup and record the weight (mass).
Hands-On Labs SM-1 Lab Manual
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2) Set the paper cup inside a slightly larger coffee cup or
similar container to give the paper cup support and
prevent it from tipping over when you add a funnel.
3) Fold a sheet of filter paper in half and then in half again
as illustrated. Weigh it.
4) Open one section of the folded filter paper as shown in
the bottom illustration.
5) Place the opened filter paper into the funnel and the
funnel into the paper cup supported by the coffee cup.
B. Remove the paper cup of salt and benzoic acid crystals
from Step2-F from its ice bath. Fill a graduated cylinder with about 5 mL of
distilled water and place the cylinder in the ice bath to chill the distilled water.
C. Swirl the cup containing the salt and benzoic acid crystals to dislodge any
crystals from the sides. Then, while holding the filter paper in place and open,
pour the contents of this cup into the filter paper-lined funnel.
D. After the sodium chloride solution has fully drained through the filter paper, slowly
pour 2 to 5 mL of chilled distilled water around the inside surfaces of the filter
paper-lined funnel to make sure all the sodium chloride has been removed from
the benzoic acid crystals.
E. After all the liquid has drained from the funnel lay the filter paper containing the
benzoic acid crystals on folded layers of paper towels and put this someplace
where it will not be disturbed while the filter paper and its contents air dry.
Depending upon the humidity in your area this can take several hours or days.
F. When the filter paper containing the benzoic acid crystals is completely dry,
weigh it and subtract the weight of the filter paper to obtain the net weight of the
benzoic acid crystals.
NOTE: A very small amount of benzoic acid may not have precipitated but rather
may have remained in the salt solution and have passed through the filter paper.
This will cause a very small experimental error in your final results.
4. Separating out the Salt:
A. Remove the funnel from the above filtration assembly and set the paper cup of
sodium chloride solution someplace where it will not be disturbed while the water
evaporates. Depending upon the humidity in your area this might take several
days. When all the water has completely evaporated only sodium chloride will be
left in the paper cup.
Hands-On Labs SM-1 Lab Manual
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B. Weigh the paper cup with the dried salt crystals inside and then subtract the
weight of the cup to get the net weight of the table salt.
C. Prepare a data table listing the various components of the mixture and record
both their masses in grams to at least 1 decimal place, (i.e., .1 or 1/10th of a
gram) and their percentage of the total mixture.
Sample data table:
Grams Percent of mixture
Iron filings .8 .8/4 * 100 = 20 %
Sand 1.4 1.4/4 * 100 = 35 %
Table salt 1.2 1.2/4 * 100 = 30 %
Benzoic acid .6 .6/4 * 100 = 15 %
Total 4.0 100 %
Questions:
A. How did your proposed procedures or flow charts at the beginning of this experiment
compare to the actual procedures of this lab exercise?
B. Discuss potential advantages or disadvantages of your proposed procedure
compared to the one actually used.
C. What were potential sources of error in this experiment?
Cleanup: Thoroughly clean, rinse, and dry all equipment and return it to the LabPaq.
Throw all used paper cups and paper towels in the trash.
- SM-1 Manual COLOR 105 08-17-07
Discussion Questions
1. Consider two concepts you encountered in Experiment 3: 1. Accuracy and Precision, and 2. Means. Based on your readings and experiences in conducting Experiment 3, what problems do you envision students having with these concepts? Describe how you would present these concepts to students, based on what you now know about how children think at different cognitive stages. (Address the question using the age group, and corresponding cognitive state, that you are most interested in teaching.)
2.
Consider Experiment 4. What changes would you make when conducting this lab with elementary students? What types of scaffolding would you provide, and at what points? Why?
