Subject: Re: Eight Features of an Ideal Intro Stat Course (Response to comments by Bob Hayden) To: EdStat E-mail List and sci.stat.edu Newsgroup From: Donald B. Macnaughton <donmac@matstat.com> Date: Sunday July 23, 2000 Cc: Bob Hayden <hayden@oz.plymouth.edu> ----------------------------------------------------------------- Quoting a 99/5/9 post of mine, Bob Hayden writes (on 99/5/9) > ----- Forwarded message from Donald Macnaughton ----- >> < snip > >> It is not necessary to bring formal statistical procedures >> into the discussion to discuss relationships between vari- >> ables. I recommend that teachers capitalize on this fact and >> give students a strong sense of the concept of a relationship >> between variables before introducing ANY formal statistical >> procedures. >> < snip > >> > ----- End of forwarded message from Donald Macnaughton ----- > > Without getting too deeply into the main issue here of univari- > ate versus multivariate, I would like to comment on a couple of > details. > > I think the relationship between a measurement variable and a > categorical variable is best visualized with parallel boxplots > -- one for each category -- on the same scale. Indeed, such > plots are the main reason to learn boxplots. Many readers will agree that plots are essential tools for under- standing relationships between variables. Four standard types of plot for illustrating the type of relationship Bob describes are - parallel dot plot - parallel boxplot - graph (perhaps with standard-error-of-the-mean bars) and - parallel stem-and-leaf plot. To help with discussion of Bob's points, I show the same data plotted in each type of plot in the figure below.

FIGURE CAPTION: Four types of parallel plot, each re- flecting exactly the same data. The plots are called (clockwise from the top left) parallel dot plot, parallel boxplot, mean graph with standard-error-of-the-mean bars, and parallel stem-and-leaf plot. As can be seen on both the parallel dot plot and the parallel stem-and-leaf plot, the counts of the number of values of the response variable available for the three values of the predictor variable are (from left to right) 25, 26, and 24. (Appendix A describes how to obtain a higher-resolution copy of the figure.) The figure reflects a simple empirical research project in which a single "discrete" predictor variable is observed at or manipu- lated through three values in the research entities (or in the entities' "environment") and the values of a single "continuous" response variable are observed in the same entities. (Appendix B discusses the distinction between discrete and con- tinuous variables.) (I could have searched data archives to find an appropriate da- taset on which to base the figure. However, to save time and to get exactly what I wanted, I simply used the SAS normal random number generator to make up the values of the response variable in the figure. I specified nominal means of 28, 32, and 36 for the three groups and a nominal standard deviation [within each group] of 9.) If I were presenting the situation illustrated in the figure to students, I would make it very concrete, perhaps in part as fol- lows: The research entities are 75 AIDS patients who were ran- domly assigned to three groups. The predictor variable reflects three levels of a new drug that were (in double-blind fashion) administered to the three groups of patients -- a different level to each group. (To increase the power of the statistical tests, one of the levels of the drug was "zero".) The response variable is an appropriate measure of the healthiness of the patients af- ter six weeks of treatment with the drug. (I would tell students that in real AIDS research only two levels of the drug would normally be used because, when appropriate lev- els are chosen, this also helps increase the power of the statis- tical tests.) Also, if I were presenting the situation illustrated in the fig- ure to students, I would carefully discuss the important implica- tions of the figure for the treatment of AIDS patients, including how the implications are derived and the main caveats. * * * The four plots in the figure are quite different from each other, even though they all reflect exactly the same data. What are the advantages and disadvantages of each type of plot? Consider the parallel dot plot. Dot plots (Tukey 1977, p. 50; Wilkinson 1999) have the advantage that they are closer to the raw data than the other three types of plots -- dot plots picto- rially reflect the exact tabled values of both the response and predictor variables for each entity under study in the research. Because dot plots are close to the raw data, students find them easy to understand. Parallel dot plots can be easily drawn by any software that can draw scatterplots. (If many data values are present, it is help- ful to slightly offset the dots that lie atop one another, as shown in the figure. This offsetting is unfortunately not avail- able as a simple option in most plotting software, so the user must do it manually or write a program to do it semi- automatically. Appendix C discusses some offsetting algorithms.) Consider the parallel boxplots in the figure and consider any one of the three boxplots. To understand this boxplot a student must understand the notion of the quantiles of a distribution of nu- meric values (in particular, median and quartile) and a conven- tion that defines the length of the whiskers (Tukey 1977, pp. 39- 53). Although these technical concepts are not complicated, they make boxplots harder for students to understand than dot plots. Boxplots have the advantage over the other types of plots that they highlight outliers -- points that lie well away from the other points on the plot. For example, note the solitary outlier in the upper tail of the rightmost boxplot in the figure. Consider the graph in the lower-right quadrant of the figure. Graphs showing the mean (or median) values of the response vari- able for each value of the predictor variable (possibly with standard-error-of-the-mean bars) are often used in reports in the empirical research literature and in the popular press. Like boxplots, graphs showing the mean or median with error bars are harder for students to understand than dot plots because these graphs are based on technical concepts (i.e., a measure of the central tendency of a distribution and a measure of the spread). Furthermore, graphs with standard-error-of-the-mean bars hide the extent of the distribution because (as dictated by the formula for the standard error of the mean) the height of each bar is strongly (inversely) dependent on the number of values of the re- sponse variable available for the given value of the predictor variable. On the other hand, graphs with standard-error-of-the-mean bars are useful if we wish to focus on the "average" relationship be- tween the two variables under study. We can thus focus on a nar- rower range of values of the response variable, as is reflected by the difference between the vertical axis scale on the plot in the lower-right quadrant of the figure and the vertical axis scales on the two plots in the upper half of the figure. Furthermore, graphs with standard-error-of-the-mean bars are use- ful because they enable an experienced researcher to quickly per- form a "visual t-test". This gives one a visual confirmation of what takes place mathematically in the t-test. Appendix D de- scribes the visual t-test. (Standard-error-of-the-mean bars enable a visual t-test because the bars are scaled to reflect the number of values of the re- sponse variable available for a given value of the predictor variable. Boxplots cannot be used for visual t-tests because they are not so scaled.) Both parallel boxplots and graphs have important advantages over parallel dot plots: Boxplots and graphs SUMMARIZE the univariate distribution of the values of the response variable for a given value of the predictor variable. Thus boxplots and graphs hide some of the detail that is present in the corresponding parallel dot plot. Also, boxplots and graphs are often easier to draw and generally take up less horizontal space on a page than dot plots. Although in certain situations boxplots and graphs have advan- tages over dot plots, students should learn that before they use a summary plot they should study a dot plot of the raw data to ensure that the summary plot is not hiding some important feature of the distribution of the values, as illustrated by Tukey (1977, pp. 49-50). Consider the parallel stem-and-leaf plot in the figure. This type of plot is useful when we need to display details of the ac- tual values of a variable (Tukey 1977, pp. 6 - 16). On the other hand, when these details are not needed, this type of plot has a significant disadvantage: The extra textual detail distracts the viewer from the overall sense of the distribution of the values. The overall sense is often more important than the mostly unsub- stantial specific numerical differences that are reflected in the digits in the "leaves" of the plot. Also, stem-and-leaf plots are inferior to dot plots at highlight- ing gaps in the distribution of a set of values. This can be seen by studying the gaps in the dot plot and stem-and-leaf plot in the figure, especially the gap for the outlier in the upper tail when the predictor variable is at level 3. Appendix E discusses some other approaches to displaying the data in the figure. Because I believe dot plots are the easiest of the various types of plots for students to understand, I recommend that discussion of parallel plots in the introductory statistics course begin with parallel dot plots. I recommend that this discussion be followed by discussion of parallel boxplots and graphs because the latter two types of plots are often used in reports of em- pirical research. * * * Bob's example studies a relationship between variables in which the response variable is continuous, but the predictor variable is discrete. Bob may be suggesting that we use this type of ex- ample as the FIRST detailed example of a relationship between variables in an introductory statistics course. However, other types of example are also possible. In particular, instead of using a discrete predictor variable we could use a continuous one. Which type of relationship is best for the first detailed example of a relationship between variables at the beginning of an introductory course? I recommend that the first detailed example of a relationship use response and predictor variables that are BOTH CONTINUOUS for the following reasons: - To facilitate student understanding, the first example of a re- lationship should be as simple as possible. This suggests us- ing an example of an observational research project as opposed to an example of an experiment. This is because with experi- ments students must understand the concept of random assignment and the concept of "manipulation" of the values of a predictor variable. These concepts are not needed if we use an example of an observational research project. - It is desirable (when possible) to use continuous variables in empirical research because a continuous variable almost always carries more information in its values than a discrete variable measuring the same property. (An important exception is that the "manipulated" variables in experiments are almost always discrete because appropriately used discrete manipulated vari- ables provide more powerful statistical tests.) - Many examples of observational research projects are available that have both a continuous response variable and a continuous predictor variable. These points suggest that the first detailed example of study of a relationship between variables in an introductory course should be an example of an observational research project that studies the relationship between two continuous variables. I recommend the following example: The response variable is the mark (say, out of 100) that each student obtained in a particular course of study. The predictor variable is the total amount of time (in minutes) each student spent working on the course during the term, as tracked by student time diaries. You can pique stu- dent curiosity by using the data for the students in the preced- ing term of your present course. Appendix F discusses the logis- tics of tracking student time spent on a course. Studying the relationship between study-times and course-marks is effective because this relationship is of serious direct interest to most students. Also, the example provides an easily under- stood basis for discussing several important general concepts of statistics and empirical research such as measurement accuracy, weak relationships between variables, alternative explanations, the need for hypothesis testing about the presence of a relation- ship, causation, multiple causation, observational versus experi- mental research, and bivariate regression. In an introductory course that follows the recommended approach and begins with an example with two continuous variables, the first graphic that students see is a scatterplot rather than a parallel plot. After students understand how scatterplots illus- trate the relationship between two continuous variables, we can THEN introduce the parallel dot plot as a special type of scat- terplot that illustrates a new type of relationship between vari- ables -- a relationship in which the predictor variable is no longer continuous, but is instead discrete. * * * Let us return to Bob's comments. Recall that he says above that certain relationships between variables are best visualized with parallel boxplots. He continues > However, I see many texts that focus on the mechanics of con- > structing a single boxplot, but then never go on to use them to > visually compare several groups. Perhaps this is the extreme > in being adamantly univariate. I agree. > On the other hand, I do think it is useful for students to > learn to make boxplots without a computer, and for purposes of > teaching this, there is an advantage in concentrating on one > boxplot at a time. I agree that students can best understand boxplots if they con- centrate on one boxplot at a time. However, as discussed above, if a teacher wishes to use a discrete predictor variable in the first detailed example of a relationship, I recommend NOT start- ing with boxplots, but with dot plots. Under this approach I be- lieve it is not necessary to begin with discussion of a dot plot of a single distribution. Instead, after introducing the concept of a relationship between variables (which is what all the paral- lel plots illustrate), we can immediately introduce a parallel dot plot to students as a useful tool for illustrating certain relationships. > HOWEVER, as soon as the students understand what a boxplot IS, > you can immediately put the boxplots to good use by having a > computer generate parallel boxplots comparing several groups. As noted, I agree with Bob that parallel plots (dot plots, box- plots, or graphs) are fundamental tools for illustrating certain relationships between variables. However, an issue on which Bob and I may disagree concerns the ORDER in which a teacher should introduce the ideas of (a) relationships between variables and (b) parallel plots. For students who are not majoring in statistics or mathematics, I recommend introducing relationships between variables FIRST, be- fore we introduce individual or parallel plots (or scatterplots). On the other hand, Bob may be recommending that we introduce re- lationships between variables SECOND, after we have introduced individual (and possibly parallel) plots. Clearly, the approach of introducing individual or parallel uni- variate plots (or scatterplots) before we introduce relationships between variables has SOME appeal. In particular, if we follow this approach, when the time comes in the course to illustrate a relationship between variables with plots the students will al- ready be familiar with the plots. However, as I discuss elsewhere - almost all the commonly used statistical procedures can be rea- sonably viewed as procedures for studying relationships between variables (1999, sec 4.3) and - almost all formally reported empirical research projects can be reasonably viewed as studying relationships between variables (1999, app. B). Thus the concept of 'relationship between variables unifies al- most all statistical procedures and almost all empirical research projects. Therefore, I recommend that teachers center the intro- ductory statistics course on the fundamental unifying concept of 'relationship between variables'. I illustrate in two papers how a teacher can easily introduce the concept of 'relationship between variables' in an introductory course without having to first cover univariate plots (1996, 1999). The 1999 paper also discusses how concepts related to univariate distributions are boring for students because the con- cepts have no obvious practical value (sec. 6.9). In view of these points, I recommend introducing relationships between variables first. However, shortly after introducing re- lationships between variables, I recommend that teachers intro- duce the various types of plots that help us to ILLUSTRATE rela- tionships between variables. Such plots are essential tools for understanding relationships. * * * In my 99/5/9 post I discuss why I believe teachers continue to discuss univariate distributions at the beginning of introductory statistics courses even though it is no longer necessary to dis- cuss this topic. As part of that discussion I say >> In the past, before the arrival of good statistical computing >> packages, a person performing a statistical analysis had to >> understand the mathematics of statistics in order to carry out >> the (necessarily manual) computations. (It is almost impossi- >> ble to perform statistical computations manually if one does >> not properly understand them.) Quoting this passage Bob writes > I would have to disagree that carrying out statistical computa- > tions "by hand" requires or demonstrates statistical under- > standing. It only demonstrates that the steps in the computa- > tion have been mastered. Computers grind out statistical com- > putations all the time without understanding them. Programmers > implement statistical formulas all the time with little or no > understanding of why anyone wants to calculate this or what it > means. In the days before students mindlessly pushed buttons > on their calculators, they mindlessly pushed pencils across > pages of paper. I agree with Bob that some people learn to perform statistical computations without understanding what they are doing -- my point above does not contradict this point. My point is that in the days before we had good computer software to perform statis- tical computations, if one wished to perform a responsible sta- tistical analysis, one had to understand the underlying mathemat- ics. This was necessary to ensure that the computations were performed correctly. Nowadays, as Bob implies, the need for understanding is still very much present. But for students who are not majoring in sta- tistics or mathematics, it is no longer necessary to attain MATHEMATICAL understanding. This is because a computer can do all the standard mathematical computations of statistics, and generally do them very well. What students need instead of mathematical understanding is "conceptual" understanding. As I discuss in the 1999 paper, I believe we can give students a thorough conceptual understanding of the role of the field of statistics by showing them that statistics helps us to study variables and relationships between variables as a means to accu- rate prediction and control. A student need not understand the underlying mathematics of statistics to understand these simple ideas. ------------------------------------------------------- Donald B. Macnaughton MatStat Research Consulting Inc donmac@matstat.com Toronto, Canada ------------------------------------------------------- APPENDIX A: HOW TO OBTAIN A HIGHER-RESOLUTION COPY OF THE FIGURE A higher-resolution copy of the figure is available in Adobe Portable Document Format. To view or print files stored in this format you can download a free reader (Adobe Acrobat) from Adobe Systems. To view the figure, click here. APPENDIX B: CONTINUOUS VERSUS DISCRETE VARIABLES In the body of this post I refer to the concepts of "continuous" and "discrete" variables. I propose the following definition: A variable is a CONTINUOUS variable if and only if (1) it has numeric values and (2) it is capable of assumingallvalues within its range of allowable values. If a vari- able is not a continuous variable, it is a DISCRETE vari- able. As suggested by Cox (1999), no real-life variable is truly con- tinuous according to this definition because we can always dream up values within the range of a variable that the variable cannot assume -- in particular, values with more significant digits than the associated measuring instrument is capable of delivering. Thus any given real-life "continuous" variable is generally inca- pable of assumingallpossible values within its range, but may only be capable of assuming several thousand different values, or perhaps a hundred or so different values, or perhaps only twenty or so different values. However, the breakdown of the definition is usually not a problem in practice because the statistical techniques for handling con- tinuous variables do not require that the variables be "truly" continuous -- they generally only require that the ordering of the values be meaningful and the error term in the model have an "adequate"appearanceof coming from a certain (continuous) dis- tribution. APPENDIX C: OFFSETTING OVERLAPPING POINTS ON DOT PLOTS Various ways are available to offset overlapping points on dot plots. In particular, on a parallel dot plot such as the one shown in the upper-left quadrant of the figure, we can offset dots in the direction of the predictor variable, in the direction of the response variable, or in both directions. Statisticians have suggested the following ways of offsetting dots: - If necessary to avoid overlap, offset the dots in the direction of the predictor variable in increments of one dot width, and offset the dots in the direction of the response variable so as to form bins, with the center of each bin being independently placed to be as close as possible to the mean value of the dots it contains, possibly allowing partial overlapping of dots in adjacent bins, which is similar to a procedure described by Wilkinson (1999). - Offset the dots in the direction of either the response or pre- dictor variable (or both) with "jittering" (Chambers, Cleveland, Kleiner, and Tukey 1983; Cleveland 1993) in which the locations of overlapping dots are perturbed by small amounts of random noise. - Offset the dots in a systematic manner, which (according to Wilkinson 1999) was originally proposed by Tukey and Tukey (1990). The method I used to draw the dot plot in the figure was to off- set the dots systematically. This method has the following ad- vantages: - The method avoids the artificial appearance of bins on the plot and instead allows the viewer to see the actual values of the response variable. - The method ensures that all the dots are completely visible on the plot and thus one does not have to wonder how many dots are in a clump, which may (due to the random element) occur if one uses jittering. The algorithm I used to draw the dot plot in the figure operates (for each level of the predictor variable) as follows: Add the dots to the plot one at a time in increasing order of the value of the response variable. For each dot, keep they-coordinate for the dot fixed at its correct value but, if necessary, move the dot in thex-direction out from its nominal position (in al- ternating directions) small amounts (e.g., a quarter-dot-width) until the dot is sufficiently far away from all the previously- placed dots. A disadvantage of plots generated by this algorithm is that it can generate slight patterns in the columns of dots -- a branch- ing upward and outward of some dots as one moves vertically up a column of dots. (The branching will be downward and outward if one chooses the top of the plot as the arbitrary starting point for the placement of dots instead of the bottom. One might also start at a central point of the distribution and work both up and down from that point. A final perhaps best approach is to treat the points in a random order, since this should minimize patterns appearing in the dots.) The dot plot in the figure shows the dots distributed roughly evenly on both sides of an imaginary vertical line. It is also possible to draw dot plots with the dots distributed on only one side of the line, perhaps the right side, which makes the dot plot look more like a stem-and-leaf plot or like a histogram (with the long dimension of its rectangles horizontal). I recom- mend that programs that draw dot plots be able to draw both types. The offsetting method I used for the parallel dot plot does not work well for some scatterplots, since "unused" space in the horizontal (or vertical) direction may be unavailable on scatter- plots. Thus on scatterplots it often makes more sense to use jittering to offset overlapping dots. I recommend that all scatterplot-drawing software have built-in algorithms for offsetting overlapping points on parallel dot plots and scatterplots. APPENDIX D: THE VISUALt-TEST Consider the graph in the lower-right quadrant in the figure. Suppose we wish to perform at-test for a significant difference between two of the three group means of the values of the re- sponse variable shown on the graph. (This is a test for the presence of a relationship between the response variable and the predictor variable.) It can be easily shown that if the stan- dard-error-of-the-mean bars of two means show a "sufficient" lack of overlap on the graph, thet-testp-value will be less than .05. This implies that (assuming no reasonable alternative ex- planation is present) we can easily obtain good evidence of a re- lationship between the variables by merely scanning the graph. I call this approach the visualt-test. To illustrate the visualt-test I performed threemathematicaltwo-groupt-tests on the data behind the figure above. That is, I performed thet-test to test the (null) hypothesis that pairs of means of the response variable are the same (in the popula- tion) for pairs of values of the predictor variable. This yielded the following threep-values: -----------------------------t-test Predictor Variablep-value Values Compared (2-tail) ----------------------------- 1 vs. 2 .0354 1 vs. 3 .0003 2 vs. 3 .1621 ----------------------------- Note how thesep-values relate to the amount of vertical overlap shown by the standard-error-of-the-mean bars on the graph in the lower-right quadrant of the figure -- the less the vertical over- lap, the lower thep-value. (The method I describe for performing a visualt-test is impre- cise because the necessary amount of lack of overlap for ap- value of, say, .05 still depends somewhat on the number of values of the response variable for each of the two values of the pre- dictor variable [because these numbers determine the degrees of freedom for thet-statistic]. But, under certain reasonable as- sumptions, it is easy to show that the two means must be at least 2.77 standard errors apart for ap-value of .05. Sall describes a precise method for performing visualt-tests with "comparison circles" [1992].) APPENDIX E: OTHER METHODS FOR DISPLAYING THE DATA IN THE FIGURE Another method for displaying the data in the figure is with par- allel histograms. A parallel histogram looks like the parallel stem-and-leaf plot in the figure except that the rows of numbers are replaced by (less distracting) rectangles. Interestingly, I have been unable to find examples of parallel histograms in the statistical literature or in various statistical software prod- ucts I am familiar with. Histograms have an artificial air to them when compared to dot plots because the data are hidden inside the rectangles, rather than appearing in their raw form. Another method of displaying the data in the figure is with com- parison circles which, as noted above, allow one to perform a precise visualt-test of the differences between the means. Sall (1992) gives examples of comparison circles. Another method of displaying the data in the figure is with a "diamond plot". This plot resembles a mean graph with standard- error-of-the-mean bars except that the bars are replaced by "dia- monds", which are actually pairs of congruent isosceles triangles that share the same base. The base of each triangle is horizon- tal at the vertical height of its respective mean value (as re- flected by the scale on the vertical axis). The width of the base is proportional to the number of measurements that were used in computing the mean. Two triangles are erected on the base -- an upper triangle with the apex above the base and a lower trian- gle that is the reflection of the upper triangle on the other side of the base. The heights of the triangles indicate the standard error of the mean, or some other measure of dispersion of the values. Sall (1992) gives examples of diamond plots. Another method of displaying the data in the figure is with a violin plot, in which a "density trace" is fitted to the points (Hintze and Nelson 1998). This trace estimates the underlying distribution function of the values. In a violin plot both the density trace and its mirror reflection are shown in the plot, making a symmetrical figure that may resemble the silhouette of the body of a violin with its axis vertical and with the plane of the body perpendicular to the line of sight of the viewer. Violin plots are harder to understand than dot plots because stu- dents must understand the idea of fitting a density trace to the data. Also, the density trace reflects an assumption (which changes if we change the "tuning parameter") while the dot plot makes no assumptions, showing only the raw data values in an easy-to-understand layout. Violin plots can be useful in cases when a large number of data points (i.e., greater than 30 or 40) are available for each group of points on the plot because then it makes more sense to fit a density trace to the data. (Violin plots are effectively smoothed histograms that display both the assumed distribution and its mirror reflection.) In situations in which it is reasonable to use a violin plot, it may be useful to show only half of each "violin" because showing a fitted distribution trace and its mirror reflection seems more complicated than merely showing the fitted distribution trace alone. (One would not normally show a histogram anditsmirror reflection, so why do so with a violin plot?) Hintze and Nelson justify using both the density trace and its mirror reflection by saying that this "gives a symmetric plot which makes it easier to see the magnitude of the density." I am unable to see how the symmetric plot makes it easier to see the magnitude -- but this is an aesthetic matter -- a matter of taste. I recommend that programs that draw violin plots be able to draw both symmetric and non-symmetric plots. All the plots I have discussed have the response variable plotted on the vertical axis and the predictor variable plotted on the horizontal axis. All the plots could be drawn with the assign- ment of the variables reversed -- that is, with the response variable plotted on the horizontal axis and with the predictor variable plotted on the vertical axis. However, it is a general convention in statistics and empirical research that the response variable in a relationship between variables is shown on the ver- tical axis of a plot because this helps viewers to rapidly orient themselves to the plot. Lee and Tu (1997) discuss some other similar approaches to plot- ting the data in the figure. APPENDIX F: THE LOGISTICS OF TRACKING STUDENT TIME SPENT ON A COURSE To help students study the relationship between the time they spend working on a course and their marks (or grades) it is nec- essary to collect course work-time data. To collect useable data one needs a reliable data-capture system and careful instruc- tions. To help with the data capture, a week-at-a-glance data- capture form is available over the web in Adobe Portable Document Format (PDF). (To view or print PDF files you can download a free reader [Adobe Acrobat] from Adobe Systems.) The data-capture form is available here. (The form works best on 8.5 x 14-inch paper. However, you can check the "Fit to page" box in the Acrobat "Print" dialog to print the form on another size of paper if 8.5 x 14-inch printing is unavailable.) To emphasize the importance of the data collection, you may wish to include a notice in your course description telling students that collection and submission of work-time data is a prerequi- site for passing the course. Perhaps the best way to show students how to use the form is to complete a small portion of it on the board or on an overhead in class. Also, written instructions for the form are available here. If you decide to use the form, I recommend that you distribute a fresh copy of it to students each week, even if students have ready access to an appropriate printer. Weekly distribution of the form in class increases the chance that students will use it. I recommend that you collect last week's data from students at the beginning of each week. You could collect the weekly data from students through e-mail, via a paper-based system, or over the web. (I recommend against asking students to hand in their forms because some students will find it useful to keep the forms as a record of their work.) A PDF form for a paper-based weekly data collection system is available here. Some students may be tempted to misrepresent their time spent working on the course. Some may be embarrassed about the low amount of time they are spending on the course and may thus re- port inflated times. Others may (mistakenly) feel that you will take account of their work-time in determining marks or grades so it will be to their advantage to report inflated times. To in- crease the likelihood that you find a relationship between work- times and marks, I recommend that you discuss these issues with students and assure them that - the reported times will definitely not be taken account of in assigning marks - a relationship may not be found if students report inaccurate times - if a student decides to work only a small amount of time on the course, this is perfectly reasonable, and not something to be embarrassed about, because students are subject to many pres- sures that determine where they must allocate their time. In addition, to distance yourself from the work-time data, you may wish to assign everything to do with the data collection and analysis to a teaching assistant Perhaps the easiest way to collect the weekly work-time data is to send an e-mail to each student early on Monday morning asking them to reply with the number of minutes they spent working on the statistics course in the preceding week. It is also useful to ask students whether their reported number of minutes reflects time that was actually tracked or represents an estimate. You could base the text in your e-mail on the text in the form avail- able at the link a few paragraphs above. In collecting the work-time data in the master data system, I recommend flagging values in the data that are only estimates. Then you can check whether differences exist in marks or differ- ences exist in times between students who tracked their time and those who only estimated it. One easy-to-handle structure for the master data system (which students will not have access to) is to have one record per stu- dent, with each record containing (at least) the following fields: - a student identifier - a field for each MARK that the student earns in the course (with the fields being filled in throughout the term as the marks become available from tests and assignments) - a field for each WEEK in the course to contain the number of minutes worked by the student on the course in that week (with the fields being filled in throughout the term as the data be- come available) - a field for each WEEK in the course containing an indicator (e.g., 0 or 1) whether the value for the time in the week rep- resents tracked time or is only an estimate (again filled in as the data become available). Whenever you generate a new major set of student marks (e.g., for a midterm test or important assignment) I recommend that you gen- erate a new data file containing one row for each student in the course, but with no student identifiers. Each row will contain two pieces of information about the associated student: - the sum of the number of minutes the student has worked on the course to date and - the mark the student obtained on the test or assignment. I recommend that you make this file available to students in the course as soon as the data are available. Students can then gen- erate a scatterplot to see if a relationship appears to exist be- tween the times they spent working on the course and their marks. Since students will likely be concerned about the use and confi- dentiality of their marks, you may wish to assure them that the published marks will never be published with student names or other identifiers. Thus it will generally be impossible to infer a student's mark from the published data. (But if student A tells student B the number of minutes student A worked on the course, student B may then be able to infer student A's mark from the published data file.) If you have comments about this time-tracking system or sugges- tions for improvements, I would be interested to hear them. You can reach me at donmac@matstat.com REFERENCES Cleveland, W. S. 1993.Visualizing data.Summit, NJ: Hobart Press. Chambers, J. M., Cleveland, W. S., Kleiner, B., and Tukey, P. A. 1983.Graphical methods for data analysis.Boston: Duxbury Press. Cox, D. R. 1999. "Variable, types of". InEncyclopedia ofStatistical Science, Update Volume 3ed. by S. Kotz. New York: John Wiley. Hintze, J. L. and Nelson, R. D. 1998. "Violin plots: A box plot - density trace synergism,"The American Statistician,52, 181-184. Lee, J. J. and Tu, Z. N. 1997. "A versatile one-dimensional dis- tribution plot: The BLiP plot,"The American Statistician,51, 353-358. Macnaughton, D. B. 1996. "The entity-property-relationship ap- proach to statistics: An introduction for students." Available at http://www.matstat.com/teach/ Macnaughton, D. B. 1999. "The introductory statistics course: The entity-property-relationship approach." Available at http://www.matstat.com/teach/ Sall, J. 1992. "Graphical comparison of means."American Statistical Association Statistical Computing and Statistical Graphics Newsletter,3, 27-32. Tukey, J. W. 1977.Exploratory data analysis.Reading, MA: Addison-Wesley. Tukey, J. and Tukey, P. 1990. "Strips displaying empirical dis- tributions: I. Textured dot strips." Technical Memorandum, Bellcore. Wilkinson, L. 1999. "Dot plots."The American Statistician53, 276-281. The URL of this page is http://www.matstat.com/teach/p0043.htm

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