Unit 5: The Normal Distribution

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Connections

Models
  • 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 distribution.

  • The cumulative distribution function of the normal distribution is related to the logistic growth curve (one could present applications relevant to student interests).
Probability
  • 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.
Comparing Groups
  • 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 deviation.