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This example illustrates how to use WSS to determine an appropriate number, k, of clusters, the following example uses R to perform a k-means analysis. The task is to group 620 high school seniors based on their grades in three subject areas: English, mathematics, and science. The grades are averaged over their high school career and assume values from 0 to 100.   # The following R code establishes the necessary R libraries > library(plyr) > library(ggplot2) > library(cluster) > library(lattice) > library(graphics) > library(grid) > library(gridExtra) > library(cowplot)   # Import the CSV file containing the grades > grades<-read.csv("grades.csv", header=TRUE, sep=",") > grades<   # Let's take a look at the structure of the dataset > str(grades)   R output: 'data.frame': 620 obs. of 4 variables: $... read more

Math Student's Civil Rights   I have the right to learn Math (Math is learnable like other subjects) I have a right to make mistakes, erase then, and try again (Failure points to what I have not learned yet) I have the right to ask for help (asking for help is a great decision) I have the right to ask questions when I don't understand (understanding is the primary goal) I have the right to ask questions until I understand (perseverance is priceless) I have the right to receive help and not feel stupid for receiving it (asking for help is natural) I have the right to not like some math concepts or disciplines (i.e. trigonometry, statistics, differential equations, etc.) I have the right to define success as learning no matter how I feel about Math or supporters I have the right to reduce negative self-talk & feelings I have the right to be treated as a person capable of learning I have the right to assess a helper's ability to... read more

Meta-cognition is thinking about thinking, and higher levels tend to be associated with intelligence and - in the parlance of the times - high income and job satisfaction. So, to keep things simple: if you think that you might struggle with learning concepts in statistics or research methods, then get in touch with a tutor early. It's important to establish some type of relationship so you can evaluate their style, methods, and how likely it is that you'll be successful within their approach. If you wait too long, or right before an assignment is upon you, you may be stuck with a limited number of tutors that you are forced to work with.    Prevention through establishing early contact will help you iron out details, assess a match in learning and tutoring styles, and prepare for mastering any difficult concept.   Don't wait - contact tutors before it's too late!

As with most college students, statistics is a really hard subject to grasp, even for somebody that is good in math (i.e. algebra,calculus, etc.).   What one needs to realize is that although a good foundation in mathematics is needed to truly understand statistics, the concepts behind the use of statistics is actually quite simple.   Since I have always enjoyed hunting and fishing, I always love bringing up the example of the mathematicians and statistician that went hunting.   So, 2 mathematicians and a statistician went hunting for deer.  Both were standing in the same stand and talking about all the fun things in life, when all of a sudden the biggest and nicest buck on the property staggered into the area.   The first mathematician pulled his bow back and shot towards the deer.  Unfortunately the arrow missed it's target high and the mathmatecian became really sad.  Reveling in this new found opportunity,... read more

The general form for a box-and-whisker plot is really easy. Let's take a simple data set.   8.2, 15.9, 12.8, 7.4, 24.7, 23.2, 9.6, 7.9, 8.3, 10.2   First, we need to take those data and put them in numerical order.  When we do that, this is what the data set looks like: 7.4, 7.9, 8.2, 8.3, 9.6, 10.2, 12.8, 15.9, 23.2, 24.7   [Note: Any computer program that runs spreadsheets or statistical analysis will probably accept the data in any sequence.  Ordering the data is only necessary when doing this process by hand.] Once ordered, we need to find the median of the set. The median means the "middle" value. In this case, the set has 10 values, so there's no singular "middle" value of the set when ordered least to greatest. To create one, we'll take the two middle values and average them. (9.6 + 10.2)/2 = 9.9 [The only reason we took an average is because there is not "middle"... read more

When conducting a group of statistical comparisons, it is important to understand the concept of familywise error or the 'familywise error rate'. The concept of familywise error describes the probability of incorrectly rejecting the null hypothesis. Perhaps more informally, familywise error represents the odds of reporting a statistical difference when no such difference exists. To understand why familywise error is important we must revert back to concepts learned in a basic introductory statistics class. First, we generally compare t-values (or other test statistics) to a known distribution of scores (t-scores in this case). Like any distribution, some values near the center of the distribution are more likely to occur than others. By understanding the odds that a given t-score will occur, we are then able to designate values that define regions of the distribution that are unlikely to occur as "critical values". That is, values above or below these critical values... read more

Statistics Class is one of those oddball math classes because it is one of the first university level math class that a student will take. See most students coming into statistics have gotten into the    1. Attendance 2. Speech 3. Syllabus 4. Done   Stats does not necesserily act like that because stats is more labor intensive the first week. Most of the information though is only required on the first test but is necessary for the rest of the semester. For instance Statistics is the math where students COLLECT, ORGANIZE, ANALYZE AND SUMMARIZE large amounts of data.    Hmm...    If students studied that definition just for a second, they know what the rest of the semester is going to bring them, but we will continue that in a different posting.    First important thing to notice is the title of my blog. PPSS This is probably the most important item you need for class because it is arranged... read more

See if there is one thing that I cannot stand, is seeing a student rush into a Stats class and stating that this class is remarkably easy because my friend said so. Too many times has a student come up to me and asked is statistics easy, and I reply "it most certainly is... for myself, because I studied my content for two years before retaking the course.". If you know that you are going to start a statistics class anywhere at any time your going to need the following items.   1. TI-83 or 84 preferably Now why one of these calculators? Students, if you are going to take a stats class be aware that there is a lot of data or numerical values that can be used to find the measures of central tendency and measures of dispersion. These calculators excel for data entry and double checking your answers on the very first test.    *If your teacher says you cannot use this calculator, then get its cheaper cousin. TI-30XII Hint there usually... read more

You can find some really good resources for math test prep in the used bookstores in a college town. Some examples that I like are: (1) Humongous Book of ______________ Problems (fill in the blank with your math topic);  (2) the REA Problem Solvers series; and (3) the Schaum's Outlines. If you don't live near a college town it might be worth a Saturday trip just to buy books. Alternately, all of these are available (used) through the Amazon Marketplace sellers at really low prices.   You should preview each title of these book series that you might be considering to be sure you like the authors style. Each one is different. You may like one series' treatment of Pre-Calc but prefer a different series for Calculus.   So how do you use these books ?   They are an alternate resource for explanations of basic concepts and problem solving techniques. You should use them as 'hint mills' and sources of problems to make... read more

This is a response to:  How did you master a subject or concept that challenged you in school?    For me, the biggest academic challenge in school occurred when I took a required graduate statistics course.  In this program, there were two semesters of stats required, and I did just fine in the first semester. In fact, I really don't remember been scared at all.  However, there was a big fear among all the first year grad students going into the second semester of the course with a very famous faculty member who actually authored the textbook we used.  The man seemed nice enough, yet I was terrified.  In class, I took down every note, but after class, it appeared to be written in Chinese.  I tried recording the lecture; still, there was little progress. I joined a study group, and it was clear that everyone was equally lost. In fact, most of their fears INCREASED my anxiety.  Going into the midterm, I experience a family crisis.... read more

Many statistics students have likely heard their instructor say the phrase, "correlation does not mean causation."  As a statistics student, you may have even recited this phrase during a discussion or included the phrase in a written assignment.  But what does this phrase truly mean and why do we care?   Consider this scenario: A researcher wants to examine the impact that watching the show "The Jersey Shore" has on IQ test performance for a class of Introductory Statistics students. To investigate the relationship, the instructor surveys the classroom and asks students to 1) answer some questions that will be used to calculate their IQ and 2) indicate the amount of time (in hours per week) they spend watching The Jersey Shore. The instructor compares the responses for hours spent watching The Jersey Shore to student performance on the IQ test and finds a strong, negative relationship, r(30) = -.45. Remember, a strong negative... read more

Before I go run a marathon, play with my family at the pool, ride a roller coaster, head to the beach, or eat some serious amounts of ice cream, I will look back on the successful school year I have had. I got to tutor over 15 students in Middle School, High School, SAT, and College Math...and even Chemistry! I watched GPAs rise for everybody-some were happy just to pass that College Math course to graduate, others enjoyed their hard earned As that were brought up from the D level. I must say, that things did get crazy with exams at the end of the year, but it all worked out amidst our busy-ness. My tutoring schedule is light for the summer, and I am hoping nobody waits until December exams to contact me! I want things to be done the right way...after all of the swimming, adventure, and ice cream, of course! :) -Becky

SUMMER OPPORTUNITIES Now that students, teachers, parents and tutors have had a chance to catch their breath from final exams, it's time to make use of the weeks we have before school starts back. Consider all that could be accomplished in the next few weeks: Areas of math that students NEVER REALLY GRASPED could be fully explained. This could be elementary skills like adding fractions, middle school topics like systems of equations, or high school areas like sequences and series. Students could have a TREMENDOUS HEAD STARTon topics that will be covered in the first few weeks of school. Imagine your son or daughter being able to raise their hand to answer a question in the first week of school because they had worked several problems just like the ones that the teacher is demonstrating. ENORMOUS PROGRESS could be made in the area of preparation for the standardized tests (PSAT, SAT, ACT and more) that are so important to getting into a great college. STUDY SKILLS... read more

I have been working with a few students who are ready to learn math much, MUCH faster than allowed by the traditional classroom model in which math is taught over 6 to 8 years. Based on this experience I believe that many students as young as 4th grade and as old as 8th grade (when starting in the program) can master math in 2 years from simple addition through the first semester of Calculus, with Arithmetic, Algebra 1, Geometry, Algebra 2, Precalculus, Probability, Statistics, and Trigonometry in between. This is significantly faster than the traditional approach and is enabled by a combination of one-on-one teaching and coaching and a variety of media that I assign to students to complete in between our sessions. This is a "leveraged blended learning" approach that makes use of online software, selected games, and selected videos with guided notes that I have created that ensure that students pick up the key points of the videos, and which we discuss later. The result... read more

Congratulations to Niesha for getting an "A" on her Operations Management final exam. That was a challenging class, and I know you worked hard for your good grades. I am so proud of you for your dedication and resulting success. Keep up the good work, and before you know it, you will be graduating!

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