1Introduction to Data Types and Curve Fitting I. Data types Physical measurement data and analytical data generally fall into one of two categories. For the lack of better terms, we will refer to these categories as replicate data and as derivative data. Replicate data results from multiple analysis of the same sample. This type of data is what you generated most often in the quantitative analysis course and is characterized by having that all members of a particular data set with, ideally, the same value. Table 1 Examples of Replicate Data Molecular mass of an unknown gas Trial Molecular mass (g/mol) 1 234.15 2 233.89 3 233.17 4 234.03 Concentration of an unknown solution Trial Concentration (ppm) 1 15.234 2 15.916 3 15.197 4 16.008 Questions a) Theoretically, all members of a replicate data set should have the same value. Why are all values in a replicate data set not the same? b) How is the quality of this type of data evaluated? c) How can any single piece of data be rejected legitimately from a particular replicate data set? For example, how would one decide if the value of 233.17 g/mol in Table 1 should be kept in the data set or if it should be discarded?2The second type of data results from measurements/experiments that are intended to determine how a change in one of the system variables will affect the value of another variable. For example, the volume of a gas depends on the pressure, temperature and number of moles of the gas. If we wanted to understand how pressure changes affect the volume, we would measure the volume of a gas sample at various pressures while keeping the temperature and amount of gas constant. This type of data does not have a common name but we shall refer to it as derivative data. Just as the derivative in calculus connects the change in x to the change in y, derivative data connects the change in one variable to the change in another variable. Derivative data is often presented in tabular form (Tables 2 and 3) but it is most informative when presented in graphical form (Graphs 1 and 2). Table 2 Examples of Derivative Data Pressure-volume data for an unknown gas Pressure (torr) Volume (mL) 100 1602 186 861 218 735 350 653 372 431 451 355 622 258 Temperature = 25.0 °C Mass of sample = 0.6549 g Questions a) This example illustrates volume as a function of pressure at constant temperature and amount of gas. What other functional relationships can be studied for gases? b) How is the quality of derivative data evaluated? c) Should any of the data points in Table 2 be eliminated? d) What are error bars? Why are they used? How are they calculated?3II. Curve Fitting/Data Smoothing/Regression Analysis Curve fitting/data smoothing/regression analysis are three names for the same process. This tool only has meaning with derivative type data. Regression analysis is the process of finding a mathematical equation which reproduces the experimental data. Consider the following experimentally obtained values for the system variables X and Y. For the purpose of this example, we will assume that X and Y are the only variables for this system and that X is the independent variable and Y is the dependent variable. That is, we are assuming that Y=f(X). Table 3 Experimental Data Y X 0.767 1.00 3.46 2.00 12.97 3.00 20.25 4.00 38.18 5.00 50.13 6.00 75.81 7.00 94.06 8.00 The data from Table 3 is plotted in Graph 1. To fit the data, we need to find a mathematical equation, in the form Ycalc = f(Xexp), that would allow the calculation of an Ycalc for any Xexp value between 1.00 and 8.00.4Question -- Why would the Xexp values be restricted to the range of 1 - 8 for calculating Ycalc? If mathematical equation can be found to represent the experimental data, then plotting both Yexp and Ycalc on the same graph should have an agreement similar to the agreement illustrated in Graph 2. Graph 2 illustrates why this process is also referred to as data smoothing or curve fitting. Regression analysis is just a sophisticated way of drawing the best line through a data plot. The Mechanics of Curve Regression Analysis Experimental Data 5With the development of computers and commercially available software, the process of regression analysis has become much easier. Nevertheless, most curve fitting methods require that the user provide the equation that will be used for the fit. Most of the time, after plotting the experimental data and examining data visually, choosing an appropriate equation is relatively easy. Derivative data will often be based on either a theory/model. If not a model is not available, an empirical relationship, without an underlying theory, is found. Almost all theories in physical science include mathematical equations that the data is expected to follow. For example, the P-V data given in Table 1 may be expected to follow ideal gas behavior. Based on the ideal gas model, a plot of P versus 1/V should be linear if temperature and amount of gas are held constant. Question: Before doing any curve fitting, it is very important that to graph the data. At least two important pieces of information can come from this graph, what information do you think can be gained from this initial plot of the data? Empirical data is data not connected by any theory or model. The fitting equation is either suggested by a plot of the data or it is found by trial and error. A popular general equation for this type of data is the power formula which has the general form given below. Ycalc = a + b Xexp + c X2exp+ d X3exp + e X4exp + .................. Fitting equations can come in many forms. A few other examples are given below. Ycalc = a + b Xexp (linear equation, a special form of the power formula) Ycalc = aebX (exponential equation) ln Ycalc = a + b/Xexp (another form of the exponential equation) 1/Ycalc = a + b ln Xexp + c (ln Xexp)3 (an equation that works for certain systems) The process of curve fitting is to use your experimental data to find the best values of a, b, c etc to give an equation that yields Ycalc that are in good agreement with Yexp. The a's, b's, c's etc. used in the above equations are often called the fitting coefficients or the fitting parameters. The most usual mathematical method used to determine these coefficients is called the method of least squares.