# Category Archives: variance / standard deviation

## 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.

## 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.

(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.