- Similar to the relationship between scatter plots and the
theoretical regression line (linear model), students should
increasingly understand the relationship between histograms of data
sets that are bell-shaped and this theoretical continuous probability
distribution or model. In addition to comparing the shape of their
distributions, students could perhaps compare the percent of data in a
particular data set that lies within one, two, and three standard
deviations of the mean to the percents given by the theoretical normal
- The cumulative distribution function of the normal
distribution is related to the logistic growth curve (one could present
applications relevant to student interests).
• Comparing Groups
- This unit is students’ first experience with probability
models in the course. They will encounter other probability models in
much more detail in future units and will also refer back to the normal
model several times throughout the course.
- Specifically, students will later learn that we can use the
normal distribution to approximate other probabilities such as those
related to the binomial distribution and the distribution of sample
means and sample proportions when appropriate.
- We can use the normal curve as a descriptive summary of a
set of data or of a population. For example, by "heights follow a
normal distribution with mean 67 inches and standard deviation 3
inches" we are giving quite a bit of information with very few words.
In particular, you can now say approximately what percent of the
population/data are between any two values, say 65 and 68 inches, by
finding the appropriate area under the corresponding normal curve
(37.8%, in fact). Note that we are making a statement of fact; if the
model is a good fit to the population, then we are claiming that 37.8%
of the population falls between these heights.
- Later, we will talk about probabilities, and we will use
the same mathematical function – the normal curve – to describe
probability processes. In this context, the same mathematical function
serves a subtly different process. Now we would say that "if we were to
randomly select one person from this population, the probability that
he or she is between 65 and 68 inches is 0.378." This is not a
description of the population, but instead a prediction about what will
happen when we carry out a particular action.
- From previous units, students should be familiar with the
idea of comparing centers and spreads when comparing the distributions
of two groups. In this unit, students will learn to compare individual
observations from different groups or populations by comparing
standardized units (z-scores) or alternatively, probabilities beyond
those z-scores. In order to compute z-scores, students will need to use
measures of center and spread, specifically the mean and standard