Unit 5: The Normal Distribution

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 Teaching Tips

• Make sure students connect the population proportions to the area under the normal curve. Encourage them to sketch the curve and shade the appropriate regions to reinforce this concept. They should also clearly label their sketches, clearly differentiating between raw scores (x), z-scores and areas under the curve (probabilities).

• Two important technical skills are (1) to go from x values to z-scores to areas under the normal curve and (2) in the other direction: from areas back to x values.

• The normal probability plot or the normal quantile plot (they are slightly different but for our purposes the differences are neglible) is a useful and commonly used tool for assessing whether or not a sample is plausibly from a normally distributed population. However, these plots are rather complex and difficult to understand, and there's no need to get caught up in the "why" at this point. Learning to interpret the plots is enough.

• It helps to do simulations to get a sense of when a normal probability plot is "straight." There is a lot of variability in the tails, and so normal probability plots that stray from a straight-line out at the ends – even quite a bit of straying – might still be from normal distributions, particularly if the sample size is not too big.

Student Misconceptions and Confusions

• The 68-95-99.7 rule of thumb is a very useful tool for understanding distributions. However, beware of students who will try to apply this rule to everything. First, remind students it doesn’t apply to skewed or multi-modal distribution. Second, if we know the distribution is normal, we can find more precise answers than this rule provides.

• When comparing different groups, students will tend to compare distances of actual observations from means rather than standardized distances of observations from means. Remind students that they must consider not just the mean (center), but also the standard deviation (spread). Similarly, students will often interpret the z-score as how many units an observation is from the mean rather than how many standard deviations an observation is from the mean.

Resources

SOCR Normal Distribution Applet

SOCR Normal Distribution Calculator