Category Archives: Sampling Distribution

THE most critical concepts in applied statistics: Treating students like family

There is nothing like having a child preparing to learn statistics that really gets a mother to focus on … what are THE most critical concepts in applied statistics. I’ll be honest; I’m not basing this posting off of research, as sadly, no such research exists. It is, instead, based off of my experience in teaching and research coupled with the reality, I only have a few hours to cover the most important material to my son and sons and daughters of a few of my dearest friends. You see, they are all preparing to take a math statistics class either this summer or this fall. We all want our children to understand math stats in the larger concept of applied statistics.

In this posting, I will cover the outlines of what I have deemed most critical, then over the course of the next few weeks, I will detail the lessons, activities, and homework assignments.  Each session is equivalent to one weeks’ worth of work during a typical semester for the type of students I teach. As with everything … there may be some variability in how much time it takes to cover this material depending on your class size and student type.

#1: Making Sense of Variability

  • Introduction to Epistemology — the four ways of knowing, with a focus on the dance between rationalism (forming hypotheses) and empiricism (gathering observations in the form of data).
  • 4 Uses of Statistics: Describe, Infer, Test Hypotheses, Find Associations
  • Introduction to research methods (just the experiment, and appropriate terms).
  • Brief review of mean, median, and mode

Session #2: Capturing Variability

  • Conceptually understanding variability (deviation) and the sum of squares
  • Finding the Sum of Squares
  • Obtaining the average Sum of Squares — the variance
  • Understanding why we need the standard deviation (as it makes conceptual sense, where the variance doesn’t)
  • Population Variance and Standard Deviation and Sample Variances and Standard Deviations used to infer the population

Session #3: Normal Distribution

  • Review population vs. sample/ parameter vs. statistic
  • Normal Distribution as a type of a population
  • Properties of the Normal Distribution
  • Area under the curve of a normal distribution
  • Z-scores as a means of identifying location of an observation on the normal distribution

Session #4: Sampling Distribution of the Means and Standard Error

  • Conceptually understanding a sampling distribution
  • Exploring the variability in sample mean and understanding why
  • Sampling Distribution and the Central Limit Theorem
  • Standard Error of the Mean (actual and estimated)
  • Introduction to the z-test as a means of finding the location of a sample mean on the sampling distribution of the means
  • Comparing and Contrasting the Normal Distribution with the Sampling Distribution of the Means

Session #5: Understanding Hypothesis Testing

  • Statistical Hypotheses
  • Decisions/ Assumptions/ and Consequences (outside of statistics: common examples, selecting a college & deciding to go on a date).
  • Steps of Hypothesis Testing: Research Hypothesis; Statistical Hypothesis; Creation of Sampling Distribution of the Means, and identification of rejection region; Gather Data/Calculate Statistic; Make a decision from data; Draw a Conclusion from data
  • Errors in Statistical Decision Making

Now, by understanding all of these concepts, I believe my son and my friends’ children will be prepared to learn any calculation in statistic and better understand what is happening, and how they can interpret the results.

My hope for their classes is that the profession teaching the mathematical statistics class informs the students: Where in the formula the sampling error is calculated or estimated; the times when the statistic can and cannot be used; the assumptions underlying the statistic and what happens to the results when they are violated. I would like my son and my friends’ children to learn about basic parametric and nonparametric statistics, and a little about statistical computing.

Over the next few weeks, I will lay out detailed activities and homework assignments that align with these critical concepts.

Please let me know if you feel I missed a critical component or overstated a concept that you feel isn’t as critical.

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Filed under Core Concepts, Curriculum, Hypothesis Testing, Normal Distribution, Sampling Distribution, Standard Error, Variability, variance / standard deviation, z score

Difficult Concept: Teaching Sampling Error and Sampling Distribution of the Means

I am currently teaching sampling distribution of the means and sampling error to my students. They are difficult concepts to convey to students, and unlike much of my teaching, where lecture comprises a fair portion of my teaching time, I find myself “slowing down” the progress at this point by putting more of the activities in the hands of the students, forcing   them to participate in activities during class time, and requiring them to generate ideas in and out of class.

There are three activities that I use to help students learn the concept of the sampling distribution of the means and sampling error.

(1)    Generating hypothesis, then identifying “individual differences in extraneous variables”

  • First, I model for them, using the Socratic method (asking them questions as a means of leading them to the answer), how to identify individual differences. I first do this when introducing extraneous variables, during the first week or two of class, and periodically do so throughout the first half of the semester, anytime I speak of Independent, Dependent, or Subject Variables, I have students generate the extraneous variables as well. This task, repeated early on, and especially as we approach sampling error, not only helps students to understand sampling error, but it makes the teaching of confounds easier as well. (Sampling error are random variations in extraneous variables, while confounds are systematic variations in extraneous variables.)
  • I assign for homework, that students have to generate a hypothesis (by this point, they have been doing this throughout the semester), then generate a list of 10 individual differences in extraneous variables.
  • During class time, they form groups, to discuss and critique each others’ list, then generate another list, as a group, that gets graded as a quiz. Truthfully, I have too many students (and no TA)  to grade all 80 of these assignments, by working in groups of 5, I have little trouble grading the list.

Notice how much time I spend on the concept of individual differences and extraneous variables. But, as a critical concept, it is time well spent. Truthfully, it comprises about 50 minutes, but it typically takes place over the course of weeks, helping build students’ thinking.

(2)    M&E creation of a pseudo empirical distribution of the means.

  • I formally model sampling distribution in class with the M&M demonstration.  Though I’ve described this activity before, I’ll describe it again here.
  • I get plain M&M’s whose proportion by color is: 24% blue, 14% brown, 16% green, 20% orange, 13% red, and 14% yellow.
  • Each color receive a value (e.g., 1 – 6).
  • I calculate what the mu would be given the stated proportions.
  • I have students randomly sample N=X (that value depends on how many M&M’s I have to share with the students, 10 should be the smallest value).
  • Students then calculate the mean for their sample.
  • Then I have them report their sample means, I enter them into Excel and do a very quick (and sloppy) empirical sampling distribution, and tell them what mu is.
  • We compare our mean of the mean to the mu, and talk about the variability in the rest of the sample means.  
  • We talk about how their individual sample means differ from mu and why.
  • It seems so obvious to the students, that I can then switch over to other examples, like dog weight or performance on at recall for a list of words. 
  • Students generate the extraneous variables that serve as sampling error, just as the colors of the M&M’s can serve as sampling error.

(3)    I end with having students participate in a Mathematica Demonstration, both in and outside of class.  If you haven’t used Mathematica Demonstrations, start with  reviewing this prior blog https://statisticalsage.wordpress.com/2011/01/08/before-the-semester-starts-im-playing-with-pictures/ or this one https://statisticalsage.wordpress.com/2011/05/24/using-mathematica-deomnstrations-to-visualize-statistical-concepts/.

If you have used Mathematica, this demonstration works well in helping students to understanding the sampling distribution of the means

http://demonstrations.wolfram.com/SamplingDistributionOfTheSampleMean/.

This year, I am requiring that student answer a series of questions about each mathematic demonstration to see if focusing them on the activity will increase what they are gaining from it.

For this demonstration the questions are as follows:

1. Try three different sample sizes. Which ones did you select? Draw the sampling distribution of the means by each N. What happens to the shape of the sampling distribution of the mean as N gets larger? Explain why this happens.

2. Using N = 15, change mu. What happens to the shape of the sampling distribution of the means as mu changes? Explain why this happens.  

3. Write the symbol for standard error. Change the standard deviation. What happens to the standard error as sigma gets larger? Explain why this happens.

4. Define Sampling Distribution of the Means. Define sampling error. What value do we calculate to find sampling error. Write down that formula. Why is this such an important part of statistics?

As with all of our difficult concepts. If you have any recommendations, I encourage you to  first work on getting it published in http://www.teachpsychscience.org/ and then let us know about it!

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Filed under Engaging students, Sampling Distribution, Technology

Before the semester starts … I’m playing with pictures!

I am sure I’m not alone in wanting to use the time between semesters to make adjustments to what I am teaching or how I am teaching it. By now, you probably recognize that I am a fan of learning about new pedagogical techniques. I am dedicated to helping students to truly understanding the concepts of statistics. Often, having visuals when you teach is useful for students.

I use the chalk and a board (OK, more like 8 boards that move). I draw a lot of pictures. However, a mathematics professor (who is both a great colleague and friend) has been bugging me about using Mathematic in addition to chalk (a delivery system she also loves).

With Mathematica, it is my hope that I will not only be able to present my students with a visual image of certain concepts during class time (like how a normal distribution changes when the size of the standard deviation gets larger or smaller) but by making these demonstrations available electronically to students for them to explore these concepts on their own, I am hoping students will gain a greater conceptual understanding of critical statistical concepts.

Mathematica is a software package, that among other things, provides demonstrations of statistical concepts. Each demonstration was designed by an instructor. For it to be published, it is my understanding that it goes through a rigorous peer-review process. As such, if it’s printed for use, you know it will work. The down side is that your university would have to pay for a subscription to Mathematica for the demonstrations to be useful. http://www.wolfram.com/solutions/education/higher-education/uses-for-education.html

As I stated last week, in my list of resolutions, my goal is to find five different demonstrations this semester. Why five? It seemed like a reasonable number … not too challenging.

This was really easier than I anticipated. I started by indentifying the concepts that would most benefit from being able to visualize and manipulate variables. Then I visited the Mathematica web site and searched the topics. Each search yielded anywhere from 5 to 25 demonstrations, some were appropriate, others weren’t. I looked through the demonstrations and selected the ones I liked.

Here are the concepts and the demonstrations I identified as being potentually useful this semester.

(1) The Normal Distribution, where students get to input mu and sigma, would make a nice visual demonstration.

http://demonstrations.wolfram.com/TheNormalDistribution/

This Normal Distribution also shows the area under the curve (i.e., you can manipulate the z-score)

http://demonstrations.wolfram.com/AreaOfANormalDistribution/

(2) Another good demonstration would be the Sampling Distribution of the Means, where students can see the impact of changing mu, sigma, or sample size on its shape.

http://demonstrations.wolfram.com/SamplingDistributionOfTheSampleMean/

I’m also going to throw in a demonstration on the Central Limit Theorem, as how can we talk about the Sampling Distribution of the Means without mentioning the Central Limit Theorem?

http://demonstrations.wolfram.com/TheCentralLimitTheorem/

(3) Of course, what changes in the Sampling Distribution of the mean is the standard error, thus showing how a standard error changes due to changes in the sample size and/or variability makes a great deal of sense. I was really hoping that a demonstration on the standard error would already be available, unfortunately, it doesn’t seem to be. A similar concept is the confidence interval, though even with this demonstration the writer of the Mathematica code for this demonstration did not include how variability (i.e., standard deviation) impacts the size of the “margin of error.” However, it still could be a useful demonstration.

http://demonstrations.wolfram.com/ConfidenceIntervalsConfidenceLevelSampleSizeAndMarginOfError/

Though not as clean looking at the one above, this demonstration also includes the size of the standard deviation. http://demonstrations.wolfram.com/ConfidenceIntervalExploration/

I would expect that the two demonstrations would be necessary for student to get a richer understanding of confidence intervals.

That having been said, I believe that two new Mathematica Demonstrations are in order … one dealing with the size of the standard error based on changes in sample size and variability and a possibily a new CI demonstration that merges the best of these two demonstrations.

(4) The effects of the sample size and population variance on hypothesis testing with the t-test seems like a great visual demonstration.

http://demonstrations.wolfram.com/HypothesisTestsAboutAPopulationMean/

(5) How changes in the variables impact correlation’s (depending on how they are calculated) should be useful for my students.

http://demonstrations.wolfram.com/CorrelationAndRegressionExplorer/

(6) Those of you who know me, are probably not surprised that I can’t just stop at 5 examples for this first semester … so here is a great demonstration on Power. Though I can get students to define power, and identify threats to power, I am never fully certain that they truly get the beauty (and hassle) of power. This demonstration may help.

http://demonstrations.wolfram.com/ThePowerOfATestConcerningTheMeanOfANormalPopulation/

Of course, without proper instruction during class time and an accompanying explanation following class instruction, these demonstrations may end up being little more than pretty pictures to students.

In a few weeks, especially after I actually try these demonstrations with my students, I will provide for you the information I attached with the demonstrations as well as feedback as to what worked and what didn’t. After all … anyone who has taught long enough knows, even the best planned lessons and demonstrations some times flop.

Though not specifically having to do with teaching statistics … I found a nice article at Chronicle of Higher Education on Iphones, Blackberries, etc … and apps that could help professors. The attendance and learning students’ names apps look promising. http://chronicle.com/article/College-20-6-Top-Smartphone/125764/

I look forward to hearing from any of you who have used Mathematica Demonstrations (or others) during class and for homework.

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Filed under confidence intervals, correlation, Hypothesis Testing, Pedagogy, Sampling Distribution, Significance (Statistical/ Practical), Statistical Hypothesis Testing, t test, Uncategorized, Variability, variance / standard deviation

How wonderful and I wish, I wish …

As I type this, I have one fifty minute class left to teach, and my time with my statistics class will be over. As with anything, each semester is varied. Some semesters I cover more information than other semesters. I liken this semester to driving through the city and hitting all green lights! As such, I believe my students were able to master additional information based on what is probably mostly good fortune.

So, here is my list of things I’m so thrilled I covered:

(1) Effect size statistics, like eta squared: Sure effect size statistics are not used that much, and lets face it, they are super easy to calculate, but my biggest reason for wanting to teach effect size statistics is it helps students to understand what a t-test or F-test can tell us (is there a difference) and what it can’t tell (how big is the effect). In fact, by spending about 20 minutes on the teaching of effect size statistics, students were better able to understand why the “p-value” for an observed t or F score provides us with no information. All we need to know is, did we pass the threshold.

(2) We find the critical value BEFORE calculating the observed value: This discussion helps focus student on the logic of statistical hypothesis testing. Specifically, statistical hypothesis testing works because we assume that the null hypothesis is true, that there is no effect of the independent variable on the dependent variable. With this assumption, we are able to generate the sampling distribution that provides us with information on the standard error. Now, if our sample mean is too extreme, we reject our initial hypothesis, the null, and accept the alternative hypothesis, that is the means are different. By finding the critical value prior to calculating the statistic, it helps focus students on that “line in the sand” to say … my observations are too extreme for me to stay with my current hypothesis. Students are far less likely to fall victim to equating p-value with the strength of the effect of the independent variable, or to conclude … the data is trending because I have a p-value of .07 or some other funky thing far too many people do with null hypothesis testing. By spending a bit more time on the steps involved in hypothesis testing, I think students are less likely to fall victim to the common misconceptions surrounding Statistical Null Hypothesis Testing.

(3) Though not a specific concept, I am pleased that for almost every concept I taught this semester I used new examples. Sure, I’m still a sage in training, no grey hair and all, but I was beginning to find myself using the same examples. As this is the third semester my supplement instructor, Amy, is taking notes in class, I felt I owed it to her, at least, to “keep it fresh.” I also found thinking about this blog helped spur my mind toward different examples. In doing so, I found some worked even better than my “old stand by” examples, but the great things was, when the new example flopped, I just quickly switched to the example I knew helped students.

Now for my Wish List of things I always wished I could have covered, but didn’t.

(1) Though I do get to cover the concepts of the F-test. I teach a three credit class, and only have time to cover the one-factor between subject ANOVA. If only I could cover a two-factor between subject ANOVA and a one-factor within subject ANOVA, I would feel my students would really understand the F-test (and as such, be less incline to misuse or over use it).

(2) Yet, I feel if I could cover non-parametrics, students would better understand the role of the assumptions in parametric tests, and issues like Power and random error could be even better understood. Plus they would get the benefit of learning about a really important class of statistics. Sadly, another semester has passed without me being able to cover this topic with the depth I think it deserves.

(3) I fear I don’t emphasize the weakness of statistics, and that they are only as good as the quality of the theories being tested in the design. They are also only as good as the quality of the sample and the quality of the measure. At least the latter two concepts get covered in classes that will follow the statistics class. But so few people speak of the topic of equifinity, that the same outcome can have multiple explanations. Again, though I touch on this, the idea of developing the alternative rival hypotheses that could explain the same empirical evidence is one I simply don’t have time to cover to the extent I would like. If you have a weak theory or haven’t taken into account the alternative rival hypotheses when designing your study, cool statistics will not improve the quality of your findings.

(4) Though I tell students the hypothesis drive everything, from the selection of the measure and research design, to the specific statistic one would select, and though there are example problems in the textbook (Integrating Your Knowledge) that students have to complete, I really wish we could spend more time on this.

Maybe next semester, I can find a way to reach my wish list … maybe!

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Filed under ANOVA Analysis of Variance, Core Concepts, Curriculum, effect size, Hypothesis Testing, Hypothesis Testing, non-parametric, Sampling Distribution, Statistical Hypothesis Testing, Statistical Tests, t test

Core Statistical Concepts

I have been spending the week thinking about what I consider to be the “core concepts” that need to be covered in an applied statistics class, be it in psychology, health, business, or education. However, before I post my personal thoughts, I felt it necessary to see what other applied statisticians had to say. In my search, I found http://www.statlit.org/pdf/2004McKenzieASA.pdf . This work was conducted by John McKenzie (2004), Conveying the Core Concepts, is from the Proceedings of the ASA Section on Statistical Education, pages 2755-2757.

In reading what  McKenzie, and several other professors of applied statistics identified as the core concepts in statistics, I must say … I concur. Listed below are the core concepts in applied statistics … the information that, in my opinion, simply has to be covered regardless of illness, snow days, or anything else that could interrupt a professors’ teaching schedule.

Variability: Students cannot understand the purpose of statistics unless they get the concept of variability. Within this, we can further talk about variability due to chance and variability due to effect. Including in the discussion of variability should be the difference between systematic and random variability. I would have to say that not a class period goes by without me spending at least a little time on helping students to focus on issues of variability (especially variability due to the individual differences of the subjects who just happen to be in our sample). 

Randomness: Though I would see randomness and variability as being part of the same large concept, McKenzie’s work identified the concept of randomness as not only separate from variability but also critical for students to master.

Sampling Distribution: Along with Hypothesis Testing, the teaching of sampling distribution is considered to be one of the most complicated to teach.  I would concur, which is why I spend an entire class period just on a single activity with M&M’s to demonstrate the concept of sampling distribution. (Please see a prior blog entry for details on this tactile activity).

Hypothesis Testing: The sages and I spent the month of October and much of November discussing whether Hypothesis Testing is critical and if so, how to best tackle the teaching of this complex topic. Not surprising, McKenzie identified the teaching of hypothesis testing as being one of the two most difficult concepts to teach in applied statistics (the other being sampling distribution). Though there may be several published articles on hypothesis testing no longer being a critical concept to teach, the individuals who were surveyed for McKenzie’s work, certainly consider it to be a critical concepts.

Data Collection Methods: Though I have said to my students more times that I can count, “the quality of our statistics is limited by the quality of our sample,” I must admit to being a bit surprised that this was considered critical by others, especially since when I look at many undergraduate statistics textbooks, data collection methods are barely mentioned. Kiess and Green’s (2010) Statistical Concept for the Behavioral Sciences, 4/e, certainly tackles the issue of data collection methods.

Association vs. Causality: This core concept makes me smile, as often when I meet someone for the first time, and they ask me what I do … my response is often met with one of two comments … “Oh, I hated statistics” or “Correlation does not mean causation.” It’s kind of like me recalling how to greet a person in German, a class that I had for three years, and yet recall so little. We, as applied statisticians, certainly engrave this concept into the minds of our students, but I’m sure most of you are like me, hoping student get more than a “pat phrase” out of our classes.

 Significance (Statistical vs. Practical): This is a critical concept in applied statistics and one that is probably not mentioned in theoretical statistics classes. Sure, we delineate a mark in which we have to say … these results are too extreme for us to attribute them to “chance” … but just because we found a statistically significant difference, doesn’t mean it’s a difference that truly matters. In applied statistics, it’s not enough to understand how statistical significance works, but to be able to interpret the results to determine practical difference. I must admit to not covering this core concept to the same extent I cover the others.

As I think of other “critical concepts” they tend to be a bit more specific and fall under the larger concepts listed above (e.g., understanding what a standard deviation can tell us, clearly falls under the concept of variability. I invite all of you, to comment on what concepts, if any, are missing from this list.

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Filed under Association vs. Causality, Core Concepts, Curriculum, Hypothesis Testing, Hypothesis Testing, Methods of Data Collection, Randomness, Sampling Distribution, Significance (Statistical/ Practical), Variability