z score | Statistical Sage Blog

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.