Purpose:

The purpose of this experiment is to determine whether pennies gain or loose mass as they remain in circulation.  It is fairly easy to think of several scenarios that would result in a penny either gaining or losing mass over time.  To answer this question absolutely, we would have to weigh every penny in circulation.  Since this is clearly impossible, we will have to use a subset to represent the whole population.  This is the purview of statistics.  We will use several statistical and graphing techniques to help uncover any trends in the mass of pennies over time.

Introduction:

As a student beginning your journey into the world of science, it may seem as though the textbooks and lab manuals are filled with wonderful magic numbers to learn and use.  For example, you have probably seen the relationship:

You may have used them in your dimensional analysis of problems on homework and tests or perhaps in laboratory calculations.  In these and other cases you have accepted them on faith that they are true values.

In this lab we will see where values like these come from and how we can evaluate the results we (and our fellow scientists) come up with in our experiments for accuracy and precision.  It is hoped that after studying this topic, you will begin to appreciate the work of others who refined these values we use everyday as well as learn to view all numbers presented to you with new and healthy skepticism.  It is also intended that you will acquire some basic skills in working with data.  Values in science are wonderful representations of our understanding that are used everyday to enhance our lives, but we must also understand the inherent limitations of the methods behind all these values in science.

Definitions:

True Values:

When a scientist observes a measurement in their work, s/he will record a value in a laboratory or field notebook.  This value may be, for example, a length.  The object that the scientist has measured to get this recorded value has a true, precise length regardless of the value that the scientist recorded.  The actual value of the length of the object is the true value.  We can argue all day about what the number is, but it exists and has a definite value regardless of what we think it may be.  Fortunately, there are organizations that exist that specify several standards we can use a true value so that we won't spend all our time arguing.

Accuracy and Precision:

Now that we know what a true value is we can talk about how accurate and/or how precise our work is compared to that the true values should be.  Accuracy asks the question, “How close to the true value are our results? ”, whereas precision asks, “How close together are all my results? ”  For example, suppose you are determining the weight of an object:
 

The values in the above table are all numerically close together which implies that you are very precise in your work.  If the last two values had been 0.1149 and 0.1119 then we might say that the work was not as precise.  If the actual true value for the weight of the samples was 0.1134 g, then we would say that your results are accurate as well because the average of your results is 0.1134 g.  If, however, the actual true value for the weight was 0.1194 g, then your work would not be very accurate and we might begin to look for some reasons for the discrepancy.

Errors:

When the value we obtain in our experimental work differs from the true value, we say that an error has occurred.  Our first reaction to this statement might be that we have in some way caused the error.  This may or may not be the case.  Not every error we will encounter is the blame of the person controlling the experiment.  There are other types of error as well.

There are, for our purposes, two general types of errors:. Determinate (Systematic) and Indeterminate (Random).  Determinate errors are usually errors in the method, equipment, and/or the scientist.  Indeterminate errors are usually errors in the limitations of the measuring device.

For example, if a scientist does not know how to perform a technique properly or misuses some of the laboratory apparatus, or the apparatus is simply inoperative, then the results obtained are invalid and we say that a systematic or determinate error has occurred.  However, it the same scientist is reading a buret and decides that the meniscus is sitting at 24.35 when it could just a easily be argued that it is 24.36 or 24.34, then the results s/he obtains would have an indeterminate or random error.  This second group of errors is out of our control.  They are inherent in the quality and resolution of the instruments we are using.  These types of errors are taken into account when the results are reported (this is where significant figures come into play).  The first group is a matter of proper technique and attention on the part of the scientist, as well as maintaining proper working equipment. This may be as simple as asking, “Is this glassware clean?”

Data Analysis:

We have looked at some data from an imaginary experiment and have made some observations as to its precision and comparisons to the real values.  This type of work is what we call data analysis.  We need to look at some of the specific things we can do with data to understand what the numbers mean in a more scientific way.  These analysis tasks come from a great deal of work in the area of mathematics known as statistics.  We do not need to understand all the theory of modern statistics but we will borrow the techniques that have developed from that field of mathematics.  For our purposes, we will learn several techniques to analyze our data:  Median, Mean or average, Range, Standard deviation, and the Q-test.

Median:

The median of a set of data is the middle value in the set.  If the set has an odd number of values then it is the symmetrical middle value of the sorted data set.  If the set has an even number of values then the two middle values are averaged to arrive at the median.  Again, using our table of data from above, we can calculate the median:

The median gives us a value that we may use to compare to the true value or that we may report as the result.  It is probably not as accurate a value to report or compare as the average or mean of the set of data.

Mean or average:

The mean or average of a set of data is defined by the following mathematical formula:

For our data set we calculate the mean (X) as:

In general, this is the best method to obtain a central tendency of the data that you may have generated during an experiment.  Now lets look at how we can determine if we are being precise in our work.

Range:

The range of a set of data is the absolute value of the difference between the lowest value in the data set and highest value in the data set.  In our data from the table above our lowest value was 0.1132 M and our highest value was 0.1135 M.  The range our our data is:

This value can roughly indicate how precise our values are; the smaller the range, the more precise our work has been.  A better indication of the precision of our work is the standard deviation.

Standard deviation:

The standard deviation for a set of data is defined by the following mathematical formula:

This is the best method to determine the precision of the values you have generated during an experiment.

Percent Standard deviation (%SD):

As discussed above, the standard deviation is an important tool in determining the precision of your data.  However, without some other information, it is impossible to interpret a standard deviation.  For example, if someone came up to you and asked if a standard deviation of 0.01 is a good result or not, what would you tell them?  Without knowing something more about their data, you can would be hard pressed to answer their question.  You need to compare their standard deviation to their data.  The easiest way to do this is with the % Standard Deviation (%SD):

Lets look at the results from two experiments:  

Although both experiments have the same standard deviation (0.096), the average for the data is much smaller in Experiment #1, than it is in Experiment #2.  This means that there is much more scatter (poorer reproducibility, poorer precision) in Experiment #1, than in Experiment #2.  By dividing the standard deviation by the average, we see that the %SD for Experiment #1 is a whooping 55%!!, while the %SD for Experiment #2 is a very respectable 0.94%.  To give you a frame of reference, it is expected that a well trained chemist or technician should be able to run an analysis and obtain %SDs of less than 2%.

Q-test:

The Q-test is a numerical analysis technique that reveals whether or not a value is far enough away from its closest neighboring number to warrant "throwing it out".  Sometimes, at first glance, a value obtained during an experiment can seem to be wrong and we may be tempted to question its validity.  We may become concerned that if we use that number in our averaging and other calculations that our answer may be skewed as a result.  The Q-test will provide a mathematical tool we can employ to check these values.  The Q-test is defined as follows:

where

The value of Q_exp is then compared to a table of Critical values, and if the Q_exp > Q_crit then the number in question is thrown out.  The following is a table of critical values:

Using our data from the weighing experiment above, it looks like the value of 0.1132 is a little lower than the other three values and could be in error.  We can check this with the Q-test:

Searching the table, we find that the value of Q_crit (4 observations, 95% confidence) is 0.829.  Since Q_exp (0.667) is less than Q_crit (0.829), we would NOT throw out the 0.1132 result.  You will use the Q-test frequently in your chemistry labs to check the reproducibility of your results.

Procedure:

Analysis of 10 pennies:

1. Obtain a collection of 10 pennies from your instructor.
2. Carefully weigh each penny individually and record the weight, the year minted, where minted, and a brief description of its condition.
3. Obtain the similar data data from your classmates from their pennies.

1. Calculate the average and standard deviation of your ten pennies.
2. Calculate the average and standard deviation for the entire set of pennies used in the class.
3. Separate your ten pennies by year and calculate the average and standard deviation for each year (if you have enough pennies for a given year).
4. Separate the penny data from the whole class by year and calculate the average and standard deviation for each year (if you have enough pennies for a given year).
5. Use 'Mr. Plot' or a spreadsheet to create a plot of 'Year Minted' (x-axis) vs 'Weight of Penny' (y-axis) for your 10 pennies.  How do you interpret this graph?
6. Use 'Mr. Plot' or a spreadsheet to create a plot of 'Year Minted' (x-axis) vs 'Weight of Penny' (y-axis) for all of the pennies used in the class.  How do you interpret this graph?
7. Finally, after sifting through all of your data with this sophisticated analysis, do pennies gain or loose weight over time?  
   

(Updated 6/7/07 by C.R. Snelling)