Chapter 2

This chapter parallels the previous chapter (considering of data collection issues, numerical and graphical summaries, and statistical inference from empirical p-values) but for quantitative variables instead of categorical variables.  The themes of considering the study design and exploring the data are reiterated to remind students of their importance.  Analyses for quantitative data are a bit more complicated than for categorical data, because no longer does one number suitably summarize a distribution by itself, and we also need to focus on aspects such as shape, center, and spread in describing these distributions.  This also leads to heavier use of Minitab for analyzing data (e.g., constructing graphs and calculating numerical summaries) as well as for simulating randomization distributions.  If your class does not meet regularly in a computer lab, you might want to consider having students work through the initial study questions of several investigations, saving up the Minitab analysis parts for when you can visit a computer lab.  Or if you do not have much lab access, you could use computer projection to demonstrate the Minitab analyses.  Keep in mind that there are a few menu differences if you are using Minitab 13 instead of Minitab 14 (see the powerpoint slides for Day 8 of Stat 212).  One thing you will want to discuss with your students is the best way to save and import graphics for your computer setting.  Some things we’ve used can be found here.

Section 2.1: Summarizing Quantitative Data

Timing/Materials:

This section covers graphical and numerical summaries for quantitative data, and these investigations will take several class sessions.  Students will be using Minitab in Investigations 2.1.3 (oldfaithful.mtw), 2.1.5 (temps.mtw), and 2.1.6 (fan03.mtw, the ISCAM webpage also provides access to the most recent season’s data.).  Instructions for replicating the output shown in Investigation 2.1.2 (CloudSeeding.mtw) are included as a Minitab Detour on p. 111.  Excel is used in Investigation 2.1.7 (housing.xls).  Investigations 2.1.1 and 2.1.2 together should take about 50-60 minutes.  Investigations 2.1.3, 2.1.4, and 2.1.5 together should take another 60 minutes or so.  Investigation 2.1.6 could take 40-50 minutes, and Investigation 2.1.7 could take 50-60 minutes.  You might consider assigning Investigation 2.1.6 as a lab assignment that students work on in pairs and complete the “write-up” outside of class.  Or you can expand on the instructions for Practice Problem 2.1.7 as the lab-writeup assignment.  Investigation 2.1.7 explores the mathematical properties of least squares estimation in this univariate case and can be skipped or moved outside of class, perhaps as a “lab assignment.”

Investigation 2.1.1 is meant to provoke informal discussions of anticipating variable behavior.  You may choose to wait until students have been introduced to histograms (in which case it could also serve to practice terminology such as skewness).  One goal is to help students get used to having the variable along the horizontal axis with the vertical axis representing the frequencies of observational units.  Furthermore, we want to build student intuition for how different variables might be expected to behave.

Students usually quickly identify graphs 1 and 6 as either the soda choice or the gender variable, the only categorical variables.  Reasonable arguments can be made for either choice.  In fact, we try to resist telling students there is one “right answer” (another habit of mind we want them to get into in this statistics class that some students may not be expecting, as well as that writing coherent explanations will be an expected skill in this class).  We tell them we are more interested in their justification than their final choice, but that we see how well they support their answers and the consistency of their arguments.  A clue could be given to remind students the name of the course these 35 students were taking.  This often leads students to select graph 1 as the gender variable, assuming the second bar represents a smaller proportion of women in a class for engineers and scientists.  Students usually pick graphs 2 and 3 (the two skewed to the right graphs) as number of siblings and haircut cost.  We do hope they will realize that graph 3, with its gap in the second position and its longer right tail (encourage students to try to put numerical values along the horizontal scale) is not reasonable for number of siblings.  However the higher peak at \$0 (free haircuts by friends) and the gap between probably \$5 and \$10 does seem reasonable.  (In fact, students often fail to think about the graph possibly starting at 0.)  We also expect students to choose between height and guesses of age for graphs 4 and 5.  Again, reasonable arguments could be made for either, such as a more symmetric shape for height, as expected for a biological characteristic?  Or one could argue for a skewed shape for height (especially if they felt the class had a smaller proportion of women)?  Again, we evaluate their ability to justify the variable behavior, not just their final choice.  This investigation also works well as a paired quiz but the habits of mind that this investigation advocates were part of our motivation for moving it to first in the section.

In Investigation 2.1.2 students are introduced to some of the common graphical and numerical summaries used with quantitative data, while still in the context of comparing the results of two experimental groups. We present these fairly quickly, and we emphasize the subtler ideas of comparing distributions, because we don't really want to pretend that these mathematically inclined students have never seen a histogram or a median before!  (Note: the lower whisker on p. 108 extends to the value 1.)  After the Minitab detour (which they can verify outside of class), this investigation concludes by having students transform the data. While not involving calculus, transforming data is an idea that mathematically inclined students find easier to handle than their more mathematically challenged peers.  This part of the investigation can be skipped but there are later investigations that assume they have seen this transformation idea. You might also consider asking students to work on these Minitab steps outside of class.

Practice 2.1.1 may seem straight-forward, but some students struggle with it, and it does assess whether students understand how to interpret boxplots.

Investigation 2.1.3 formally introduces measures of spread and histograms.  The data concern observations of times between eruptions at Old Faithful.  We have in mind that spread is a more interesting characteristic than center for this distribution, because spread relates to how consistent/predictable the time of the next eruption is.  In fact, the 2003 distribution is worse for tourists because the average wait time is much longer, but it is better in that the wait times are much more predictable.  You may have students go to the website to look at current data and/or pictures of geyser eruptions. One thing to insist on in their discussions of the data is that they treat “IQR” as a number measuring the spread, not as a range of values as many students are prone to do.

Investigation 2.1.4 asks students to think about how measures of spread relate to histograms.  This is one of the rare times that we use hypothetical data, rather than real data, because we have some very specifics points in mind.  This is a “low-tech” activity that can really catch some students in common misconceptions and you will definitely want to give students time to think through (a)-(d) on their own first.  The goal is to entice these students in a safe environment to make some common errors like mistaking bumpiness and variety for variability (as explained in the Discussion) so they can confront their misconceptions head on. Our hope is that the resulting cognitive dissonance will deepen the students' understanding of variability.   It will be important to provide students with immediate feedback on this investigation.  We encourage taking the time to have students calculate the interquartile ranges by hand as doing so for tallied data appears to be nontrivial for them.  This is a very flexible investigation that you could plug into a 20-minute time slot wherever it might fit in your schedule.

The actual numerical values for Practice 2.1.3 are below:

 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Raleigh 39 42 50 59 67 74 78 77 71 60 51 43 SF 49 52 53 56 58 62 63 64 65 61 55 49

Investigation 2.1.5 aims to motivate the idea of standardized scores for "comparing apples and oranges."  While students may realize you are working with a linear transformation of the data, we hope they will see the larger message of trying to compare observations on different scales through standardization.  This idea of converting to a common scale measuring the number of standard deviations away from the mean will recur often.  Practice 2.1.5 should help drive this point home.  The empirical rule is used to motivate an interpretation of standard deviation (half-width of middle 68% of a symmetric distribution) that parallels their understanding of IQR.

Investigation 2.1.6 gives students considerably more practice in using Minitab to analyze data.  Students will probably need some help with questions (n)-(p) especially if they are not baseball fans.  These questions can be addressed in class discussion where those that are baseball fans can be the "experts" for the day.  Still, we also want students to get into the mental habit of playing detective as they explore data.  We find Practice 2.1.7 helps transition the data set to one that applies more directly to individual students.  We encourage you to collect students' written explanations (perhaps in pairs) to provide feedback on their report writing skills (incorporating graphics and interpreting results in context).  If this practice problem is treated more as a lab assignment, you might consider a 20 point scale:

Defining MCI: 2 pts; Creating dotplots: 2 pts; Creating boxplots: 2 pts; Producing descriptive statistics: 2 pts; Discussion: 8 pts (shape, center, spread, outliers); Removing one team and commenting on influence: 3 points.

Exploration 2.1 leads students to explore mathematical properties of measures of center, and it also introduces the principle of least squares in a univariate setting.  As we mentioned above, this investigation can be skipped or used as a group lab assignment.  Questions (a) and (b) motivate the need for some criterion by which to compare point estimates, and questions (c)-(h) reveal that the mean serves as the balance point of a distribution.  Beginning in (k), students use Excel to compare various other criteria, principally for comparing the sum of absolute deviations and the sum of squared deviations.  Students who are not familiar with Excel may need some help, particularly with the “fill down” feature.  Questions (o) and (p) are meant to show students that the location(s) of the minimum SAD value is not affected by the extremes but is affected by the middle.  Students will be challenged to make a conjecture in (q), but some students will realize that the median does the job.  Questions (t)-(w) redo the analysis for the sum of squared deviations, and in (x) students are asked to use calculus to prove that the mean minimizes SSD.  This calculus derivation goes slowly for most students working on their own, so you will want to decide whether to save time by leading them through that.  Practice 2.1.8 extends the analysis to an odd number of observations, where the SAD criterion now has a unique minimum at the median.    Practice 2.1.9 asks students to create an example based on the resistance properties of these numerical measures and is worth discussing even if Exploration 2.1 is not assigned.

Section 2.2: Statistical Significance

Timing/Materials:  Students are asked to conduct a simulation using index cards in Investigation 2.2.1, followed-up by creating and executing a Minitab macro.  This macro is used again in Investigation 2.2.2 and then modified to carry out an analysis in Investigation 2.2.3.  This section might take 75-90 minutes.

This section again returns to the question of statistical significance, as in Chapter 1, but now for a quantitative response variable.  Students will use shuffled cards and then the computer to simulate a randomization distribution.  However this time there will not be a corresponding exact probability model (as there was with the hypergeometric distribution from Chapter 1), because we need to consider not only the number of randomizations but also the value of the difference in group means for each randomization, which is very computationally intensive.  We encourage you to especially play up the context in Investigation 2.2.1, where students can learn a powerful message about the effects of sleep deprivation. (It has been shown that sleep deprivation impairs visual skills and thinking the next day, and this study indicates that the negative effects persist 3 days later.)  The tactile simulation will probably feel repetitive, so you may try to streamline it, but we have found students still need some convincing on the process behind a randomization test.  It is also interesting to have students examine medians as well as means.  In question (h) we again have students add their results to a dotplot on the board.

Students then use Minitab to replicate the randomization process.  They do this once by directly typing the commands in Minitab (question j), where you might want to make sure that they understand what each line is doing. (One common frustration is that if students mis-type a Minitab command, they cannot simply go back and edit it; they need to re-type or copy-and-paste the edited correction at the most recent MTB> prompt.)  But then rather than have to copy-and-paste those line 1000 times, they are then stepped through the creation of a Minitab macro to repeat their commands and thus automate that process.  This is the first time students create and use a Minitab macro, in which they provide Minitab with the relevant Session commands (instead of working through the menus).  Some students will pick up these programming ideas very quickly, others will need a fair bit of help.  You may want to pause a fair bit to make sure they understand what is going on in Minitab.  If a majority of your students do not have programming background, you may want to conduct a demonstration of the procedure first.  The two big issues are usually helping students save the file in a form that is easily retrieved later and getting them into the habit of using (and initializing!) a counter.  We suggest using a text editor (rather than a word processing program) for creating these macro files so Minitab has less trouble with them.  In fact, Notepad can be accessed under the Tools menu in Minitab.  It is also a nice feature that this file can be kept open while the student runs the macro in Minitab.  In saving these files, you will want to put double quotes around the file name.  This prevents the text editor from adding ".txt" to the file name.  The macro will still run perfectly fine with a .txt extension but it is a little harder for Minitab to find the file (it only automatically lists those files that have the .mtb extension - you would need to type *.txt in the File name box first to be able to see and select the file if you don't use the .mtb extension).  On some computer systems, you also have to be rather careful in which directories you save the file.  You might want students to get into the habit of saving their files onto personal disks or onto the computer desktop for easier retrieval.  Remembering to initialize the counter (let k1=1 at the MTB> prompt) before running the macro is the most common error that occurs; students also need to be reminded that spelling and punctuation are crucial to the functionality of the macro.  We encourage students to get into the habit of running the macro once to make sure it is executing properly before they try to execute it 1000 times.  These steps may require some trial and error to smooth out the kinks.

In this investigation, you will want to be careful in clarifying in which direction to carry out the subtraction (in fact for (k), we suggest instead using let c6(k1)=mean(c5)-mean(c4)and then in part (m), using let c8=(c6>=15.92), then consider the area to right of +15.92 in the graph on p. 148).  Indicator variables, as in (m), will also be used extensively throughout the text.  We do show students the results of generating all possible randomizations in (q) to convince them of the intractability of this exact approach and to sow the seeds for later study of the t distribution.

Investigations 2.2.2 and 2.2.3 provide further practice with simulating a randomization test while focusing on two statistical issues: the effect of within group variability on p-values (having already studied sample size effects in Chapter 1) and the interpretation of p-values with observational data.  Question (b) of Investigation 2.2.2 is a good example of how important we think it is to force students to make predictions, regardless of whether or not their intuition guides them well at this point; students struggle with this one, but we hope that they better remember and understand the point (that more variability within groups leads to less convincing evidence that the groups differ) through having made a conjecture in the first place.  The subsequent practice problems are a bit more involved than most, so you may want to incorporate them more into class and/or homework.