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Introduction The experiments that are done day in and day out by scientists can involve both qualitative and quantitative data. Qualitative data deals with the observations of what is seen by the eye and can be explained using simple descriptions i.e. words. On the other hand, quantitative data involves numbers. A majority of the time this type of data also involves calculations. Both qualitative and quantitative data are equally important to the progress of scientific research. But neither type of data has value if not expressed correctly. The data must be processed and presented in a format that others can understand and believe. Since the need to communicate observations to others is so important, a set of agreed upon standards has evolved. The current system, established in 1960, is call the International System of Units (abbreviated SI) and has 7 basic units from which all others can be derived. They are as follows: meter (m), kilogram (kg), second (s), Kelvin (K), mole (mol), ampere (amp), and candela (cd). Some of these are quite familiar, others less so. The advantage of this system, as of the metric system which preceded it, is that through the use of various prefixes, powers of 10, and exponential notation, the units can be adjusted to fit almost any sensitivity of measurement, from the mass of the earth (~5.98 x 1024 kg) to the mass of a helium atom (6.644 x 10–27 kg). For work in the laboratory we will continue to use some of the older and more familiar units such as liters for volume, atmospheres, torr or mm Hg for pressure, and Celsius for temperature. By and large, these are easily converted to SI units when necessary. We must also be familiar with other units which have been used in the past, including the English system of measures. Although SI and metric units are easily convertible using powers of 10, conversions between English and SI or metric units are often inexact, depending upon the accuracy of the conversion factor between the two systems. Several useful tables are found on the inside back cover of the laboratory notebook, including tables of conversion factors, physical constants and metric prefixes. Measurement When measurements are taken, there is always some degree of uncertainty. The size and type of uncertainty (and therefore possible errors) depend upon the care with which a measurement is made. These errors typically take two forms: errors arising from the imperfections of the instrument used to make the measurement (mis-calibration) and errors arising from the skill or technique of the experimenter. In the first case, the errors are systematic (all in the same direction). In the second case, the errors will tend to be random (sometimes in one direction, sometimes in another). The two types of errors affect measurements differently. Systematic errors affect accuracy (closeness to the “true” value). Random errors affect precision (how well a set of data agree). In taking measurements in the laboratory, we are concerned with both accuracy and precision of measurements. One way to minimize the impact of any error is to take multiple measurements of the same value and use a statistical treatment of the set of values obtained, such as an average and/or a standard deviation to determine the value. While it is seldom possible to determine accuracy (the “true” value is not often known), the standard deviation does give some indication of the precision of a series of measurements. Significant Figures In the chemistry laboratory, a number of different tools and instruments are used to make measurements. Most of these instruments are very finely machined or calibrated to produce accurate and precise values. For instance, which do you think would produce a better value for the length of a room, a measurement in notebook lengths, or a measurement produced with use of one of the small rulers on the fold over cover of the notebook? No matter how well calibrated a given instrument may be, there is still some degree of uncertainty in the measurement made. Look at the smallest delineating mark on the metric ruler. The numbered marks are centimeters and the smallest mark is a millimeter (0.001 m). If an object is measured, it may fall between the 14.1 cm mark and the 14.2 cm mark. An estimate of the length is then made in the next decimal place (one place past the marks), and the object’s length is determined to be 14.13 cm. In this example, the uncertainty is ± 0.01 cm, because we have estimated the final decimal place. Although the last digit is not certain, it is still reasonably reliable and should be reported. Any digits that are considered reliable are important for calculations involving a measurement, and therefore, these digits are known as significant digits or significant figures. In using any measuring device, an estimate of one decimal place past the marks should be made, thus the uncertainty in any measurement is in the last digit. On some instruments such as the barometer, there is a built-in estimating device called a Vernier scale, which allows the final decimal to be estimated more quickly. In reading various digital displays, the last digit is also assumed to be an estimate, unless the instrument states a different uncertainty. Often, very sensitive devices will state explicitly what the error in the measurement is calibrated to be. It stands to reason that any time a series of measured values is combined mathematically, the value which is least precisely known determines the precision of the whole set. The following set of rules will help to clarify how to determine the number of significant digits in a measurement and how any combination of measured values should be rounded to show an appropriate number of significant figures in the end result. Rules for determining significant figures
Examples: If the decimal point is to the right of this type of zero, then it is necessary to know what the number represents in order to determine the amount of significant figures. Example: The number 100 could have one or three significant figures depending on its context. If it represented an approximation of the volume of water in a glass (about 100 mL), then it would only be considered to have one significant figure. However, if the water had been measured in a graduated cylinder, then all three digits would be significant. Often, this kind of guessing can be avoided if the measurement is recorded in scientific notation. For instance, in the above example, the number could be written as 1.00 x 102 mL, indicating that there are 3 significant digits rather than one in the number. Rounding Numbers The number of significant figures in a measurement becomes very important when doing calculations with collected data. For example, the density of an object should not be reported to four significant figures if its weight or volume can only be recorded to three significant figures. Similarly, the density should not be reported to only two significant figures, because useful information would be discarded. Generally, the calculated result is only as reliable as the least precisely measured value used in the calculation. (In the examples: below, all numbers are rounded to the hundredth place for simplicity. The least number of significant figures present in the calculation would determine the actual number to be reported.) Rules for Rounding
Rounding During Addition or Subtraction Rounding in addition and subtraction is determined by the least precisely measured value, not just the count of significant figures. It is the placement of the decimal relative to the significant figures which determines how many significant figures the answer should have.
Rounding During Multiplication and Division As stated earlier, a calculated result can only be as reliable as its least precisely measured value. In multiplication and division, this means that the result of a calculation should have the same number of significant figures as the number with the fewest significant figures used. If this is not done, the precision of the calculated result is overestimated.
The Role of Conversion Factors in Rounding Numbers in conversion factors can be either exact or inexact. Exact numbers are used when conversions are made from one set of units in the metric system to another set also in the metric system. The prefixes in the metric system are similar to definitions. There are exactly 100 centimeters in 1 meter, because of this both 100 and 1 are exact numbers. Exact numbers are considered to have an infinite number of significant figures. Therefore, when doing this type of conversion, the conversion factors do not limit the number of significant figures the answer can have. Conversions from the metric system to the English system (and vice versa) involve inexact numbers. There are approximately 30.5 cm in 1 foot. The 1 foot is the quantity being defined, so it is an exact number. However, 30.5 is a rounded value, which makes it an inexact number. Inexact numbers do affect the number of significant figures that can be present in the answer.
Scientific Notation One way to express very large or very small quantities is to use exponential or “scientific” notation. Using powers of 10 is a reasonable way to specify the precision of a measurement. Very small numbers have a negative exponent, and very large numbers have a positive exponent. The rules on rounding remain the same. Remember in addition and subtraction to adjust the values to have the same exponent before performing the operation.
Significant Figures Using Logarithms Logarithms have two parts: 1) The characteristic is the portion of a logarithms to the left of the decimal point, and reflects the exponent. 2) The mantissa is the portion to the right of the decimal and reflects the value of the measured quantity.
The rule for determining the number of significant figures in a logarithm is that the mantissa should have the same number of significant figures as the measured quantity.
Likewise, the same rule applies when the operation is reversed (antilog).
Graphing Using graphs to illustrate a set of data and make predictions is a typical experimental approach. In many experiments, one parameter is varied systematically while observing what changes there are in a second parameter, such as observing the volume of a gas when pressure is increased or decreased. If a linear relationship can be found between two parameters, then predictions are possible. Thus graphs can be used both to visually inspect a set of data for linear behavior and to predict other information. For construction of graphs required for this lab, we recommend using Excel. Excel is a computer program that can be used at a wide range of levels. Lon-Capa will teach you the very basics in using Excel for this experiment. It is important to remember a few general guidelines:
Statistics Mention has been made previously of the use and value of statistical manipulation of data in relation to the discussion of accuracy and precision. This course does not depend heavily on statistical treatment of data. However, the formulas for several useful data manipulations are found on the back of the laboratory notebook, such as percent yield, average, and standard deviation. Calculators and Computers It is appropriate here to discuss the use of calculators, graphing and otherwise, in this course. In most cases, a typical scientific calculator will be able to perform any manipulations necessary in the completion of this course. In the routine manipulation of numbers, almost nothing beats a calculator. The correct use of the device, one of the many tools and instruments used in the laboratory, depends entirely on the student operator. As with any instrument, confident and correct use comes with practice. Calculators do very well at producing numbers, but have no discrimination whatsoever. If numbers are not entered correctly or operations are ill-defined, then the results will be suspect. Using the rules about significant figures, rounding, and so forth are entirely up to the operator. Thus, it is much more important to know how to operate the calculator than to have the most sophisticated calculator on the market. In particular, make sure you know how to do basic mathematic operations and keep the owners’ manual handy.
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Measurement and Expression of Experimental Data
