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Aug 16

## The pearls of AP Statistics 14 Reasons to increase Math content in AP Statistics course

The definition of the standard deviation, to those who see it for the first time, looks complex and scary. Agresti and Franklin on p.57 have done an excellent job explaining it. They do it step by step: introduce deviations, the sum of squares, variance and give the formula in the end. The names introduced here will be useful later, in other contexts. Being a rotten theorist, I don't like the "small technical point" on p. 57 (the true reason why there is division by $n-1$ and not by $n$ is unbiasedness: $Es^2=\sigma^2$) but this is a minor point.

AP Stats teachers cannot discuss advanced facts because many students are not good in algebra. However, there are good methodological reasons to increase Math content of an AP Stats course. When students saw algebra for the first time, their cognitive skills may have been underdeveloped, which may have prevented them from leaping from numbers to algebraic notation. On the other hand, by the time they take AP Stats they mature. Their logic, power of observation, motivation etc. are better. The crucial fact is that in Statistics numbers meet algebra again, and this can be usefully employed.

Ask your students two questions. 1) You have observations on two stocks, $X$ and $Y$. How is the sample mean of their sum related to their individual sample means? 2) You have $s$ shares of stock $X$ ( $s$ is a number, $X$ is a random variable). How is the sample mean of your portfolio $sX$ related to the sample mean of $X$? This smells money and motivates well.

The first answer, $\overline{X+Y}=\bar{X}+\bar{Y}$, tells us that if we know the individual means, we can avoid calculating $\overline{X+Y}$ by simply adding two numbers. Similarly, the second formula, $\overline{sX}=s\bar{X}$, simplifies calculation of $\overline{sX}$. Methodologically, this is an excellent opportunity to dive into theory. Firstly, there is good motivation. Secondly, it's easy to see the link between numbers and algebra (see tabular representations of random variables in Chapters 4 and 5 of my book (you are welcome to download the free version). Thirdly, even though this is theory, many things here are done by analogy, which students love. Fourthly, this topic paves the road to properties of the variance and covariance (recall that the slope in simple regression is covariance over variance).

Agresti and Franklin don't have any theoretical properties of the mean, so without them the definition of the mean is kind of left hanging in the air. FYI: properties of the mean, variance, covariance and standard deviation are omnipresent in theory. The mode, median, range and IQR are not used at all because they have bad theoretical properties.