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The Standard Normal Probability Distribution

The big one. The book does an excellent job of introducing each key part of the normal distribution. I try to keep students away from the equation that describes the distribution for as long as I possibly can. By accessing students prior experiences with things that are normal, and then looking at the key parts of the distribution graph, students will have a better chance of a strong conceptual understanding that can be better applied for a variety of problems. Really, the equation of the distribution is mostly helpful only after calculus.

A key idea that must be stressed is the movement away from discreet distributions and into continuous distributions. This can be tough, because there are many things that we measure in a discreet manner, but model with the continuous curve. SAT scores is one, as there is no possibility of getting half of a question right (well, technically fractional credit is possible with the penalty for errors, but this still does not create a continuous score range) and no one gets half points on the scale after normalizing scores. The key here is the language that I used: we model behavior with the continuous distribution. Students will be well served to remember this little step to avoid thinking that normality requires continuity.

As a personal note, I really get personally confused by Z-scores. I suspect this is because I had many calculus classes before my first stats class, and therefore used calculus to solve the problems. The only reason why I bring this up is that reading a standard normal chart, finding Z-scores and relating normal distributions that might not be standard to the standard normal curve is a critical skill for a non-calculus based stats class. For my high school and university classes I would spend significant time on the algorithmic process of finding scores and relating them. It’s one of the few times in a probability and stats class where a purely algorithmic skill is needed to be practiced (and practiced and practiced!) Plan extra time just for this skill.  

The Density Curve of the Normal Distribution

A key idea that I talk about when introducing the standard normal curve, empirical rules and the standard deviation is where the inflection point is placed. This helps with students drawing good curves as well. The inflection point is always going to be a single standard deviation from the mean in each direction. This means that the inflection is going to “pull in” or “spread out” proportionally with the rest of the curve. I often have my students practice drawing generic curves with the same mean, and placing the inflection point in the same place. I don’t focus on it yet, but I also want them to understand that the areas (under the whole curve, between two inflection points) must remain equal, therefore establishing the connection between the height of the peak and the spread of the graph.

I’d like to take an additional moment to stress the importance of sketching the graph and the area you are inspecting when working problems involving normal probability density. Because the direction of the “tail” changes the way things are looked at, or the additional steps needed to find an area that is in the middle, or split on both ends, the sketch and shading is critical to staying on track. Another thing that I have my students get in the habit of is labeling the area once they find it. For instance, if they are looking at the area between one standard deviation and the mean, they would shade the area and then label it “.34”.

This is one of the key areas where calculators can really help with making things easy and fast. Because the calculator will find the continuous density, regardless of the location of the mean. This is a calculator skill I would make sure all students are comfortable with.

It would be wise to get a copy of the tables that will be provided in the AP examination. I don’t have my students’ use the charts in the text, but copies of the tables that they have for the examination.  

Applications of the Normal Distribution

The lessons at the start of this section are designed primarily as an intermediate step. Rare is the occasion where those specific types of questions are asked. However, trying to solve questions in context requires both the mechanical understanding of how to find the missing information as well as the contextual understanding of what information is given, what is needed and how to set it all up. Trying to teach both at the same time might be too much all at once, hence these initial exercises. If these lessons are used, then I strongly recommend stopping before the section using real data, and having the students work problems 1-3 at the end of the section. Then after everyone is on the same page for those questions you can move on to the real data.

There are a couple of real data problems included in this section, but I would add a bunch more. Interpreting normal data is a huge part of what students will be asked to do. Fortunately, most of the data that is out there is normal, so finding data sets is pretty easy. I have included an appendix of sources from the internet at the end.

The big task here for students is making sure they know what quantity they are being asked to find. Most of the time this is straightforward because it is directly asked for. However, they might not be comfortable “translating” to the variable names that we have been using, like percentile, z-value, mean and standard deviation. For instance the last example asks “How rare is it that we find a female marine iguana...” This isn’t using any term that we have before. Students, through the practice of doing many guided problems, need to develop a lexicon of the different ways that quantities can be asked for. In this case, how rare, what is the chance of, and other similar statements are looking for the percentage under the curve beyond a certain marker, in the case of this example below 400g.  

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