2. Variances Add Chapter 16 – Day 2

3. In chapter 4, we learned that what effect transformations had on shape, center and spread. –When adding (or subtracting) a constant, only measures of position and location (mean, 1 st and 3 rd quartiles, etc.) change. –When multiplying (or dividing) by a constant, all measures of position, location and spread change. How are probability models affected by transformations?

4. Adding (or Subtracting) a Constant There is a correlation to how probability models behave and other measures of location, spread, etc. that you have already learned. Adding a constant will shift the mean (Expected Value), but will not affect any measure of spread – standard deviation. This is the same as what you learned in Chapter 4! Please note that the shape of this distribution will not change.

5. Multiplying (dividing) by a Constant More Good News!!! Multiplying by a constant will change both measures of location, and measures of spread….just as before. Also, just as before, the shape remains unaffected.

6. To Re-Cap:

7. Basic Rules to Remember

8. a c b

9. What if you don’t have X+Y, but have X-Y instead? The equation does not change at all…..you STILL add variances!! For a more detailed explanation – complete with proofs, go to the AP Central website and read an article written by our very own Mr. David Bock. The article is called “Why Variances Add and Why It Matters”

10. If this doesn’t make sense to you…. Think about: You mix a quart of lemonade. There should be 32 ounces, but your measuring process isn’t perfectly accurate so, While 32 ounces is the expected amount, there remains some variability. You pour a 12 ounce glass. Of course that measurement is not perfect either, so it’s actually somewhere around 12 (give or take a little). How much is left in the pitcher? It should be around 20 ounces or so, but given the uncertainty about the initial amount and the variability in how far you filled the glass, you can’t say exactly how much is left. You are less sure about the amount left because you removed an unknown amount. Subtracting some lemonade has increased the variability in the amount remaining. VARIANCES ADD! (even when you are subtracting)