1. About Z-Scores

2. What’s in a Unit? A unit by any other name would convey the same quantity. It would likely label the number differently though. For example, Mr. Sanford is 5.7621 x 10 -17 Parsecs tall. The abbreviation for parsec is pc, which goes to show how rarely we use it.

3. Changing Scales We can change our scales by switching to a new unit. Some units make things easier to understand. Some make them stupid and hard to understand. Z-scores feel like the second but are really the first.

4. Normal Curves A point of inflection is a point in a curve where the concavity changes. Those of you in AP Calculus will come to understand this. For most of us, this is where it goes from bowl to umbrella.

5. Normal Curves The point of inflection is exactly one standard deviation away from the mean. In both directions. Here’s the picture example:

6. Normal Curves The best part is that the same percentages of data are found between the standard deviations of all normal curves. Specifically, this is explained in the Empirical Rule. It is also known as the 68-95-99.7 Rule.

7. The 68-95-99.7 Rule In picture form:

8. Stereotyping Normal Curves It turns out that normal curves are all so exactly the same that we can scale our data by simply measuring in standard deviations. So we figure out how much a data point is above or below the mean, and then divide it by the size of the standard deviation. This produces a z-score.

9. Formula

10. Recap Normal distributions all look exactly the same. The only thing that sets them apart is which specific mean and which specific standard deviation they have. To make things easy we will standardize data so our normal curve will have μ = 0 and σ = 1. The formula for this is

11. The 68-95-99.7 Rule Recap This rule means that – within 1 standard deviation of the mean, you will find about 68% of the data if the data are normally distributed. – Within 2 standard deviations of the mean, you will find about 95% of the data if the data are normally distributed. – Within 3 standard deviations of the mean, you will find about 99.7% of the data if the data are normally distributed.

12. The 68-95-99.7 Rule Recap In picture form:

13. What if your z-score is something else? Then we have to use the Z-Score Table™. This is a table which has frustrated and tortured statistics students for decades. Or…we could use our calculators. This involves less torture, which makes Mr. Sanford a little sad inside. Mr. Sanford will get over it.

14. So, how do I use my calculator? Hahahahahaha! (This is Mr. Sanford laughing at you, in case you are reading this outside of class.) We will learn the table today! Because. Seriously though, because it helps teach information literacy by making you read tables.

15. Molasses, To Rum, To Slaves Ignoring the golden triangle above (which was the basis of early American economic growth) we will focus on a golden triangle within statistics. Data, To Z-Score, To Area This trio is useful, but is not mentioned in the musical 1776. Or any other musical to my knowledge.

16. To find out the percentage When we have a data value and we want to know what percent are above or below it, the procedure is: Step 1: Draw and label “normie”. Step 2: Find a z-score for the data value. Step 3: Use the table to turn the z-score into an area. Step 4: We may need to turn our area into a percent. Step 5: Phrase our answer for people.

17. To find a percentile value Sometimes we might want to find out what the 25% mark is, or the 4% mark, or the cutoff for the highest 10% of the data. The procedure is: Step 1: Draw and label “normie”. Step 2: We may need to turn our percent into a decimal. Step 3: Use the table to find the z-score for the percentile. Step 4: Turn our z-score into a data value. Step 5: Phrase our answer for people.

18. How do I use the table then? Mr. Sanford will do some examples now. You will learn how to do this on a calculator before the test. You will learn how to do the “by hand” steps by calculator before the final/AP Test.

19. This Will Be Question 5 On The Chapter 6 Quiz

20. Normality Assumption Q: How do we know that our data is normal? A: We don’t. Q: Why would we assume it is normal? A: We won’t assume it is normal without checking it somehow. Q: How do we check? A: In this chapter we will learn 2 methods, but there are others taught in the class.

21. When Good Data Goes Abnormal Q: What if it is not normal? A: Then we will either normalize it with methods we learn later or we will use a different model. This class will teach you only a small amount of the models that are out there, although the most common ones are included. You will not be expected to use an alternative model until you have actually learned some.

22. 2 Methods to Check Normality The first one is intuition. Ask yourself…does this look like “normie”? If you are not sure if it looks enough like “normie”, then we will use our second method. The second method is a normality plot. This is a more reliable and objective method and in a formal or professional setting will be used to replace intuition altogether. Also called a Normal Probability Plot.

23. Normality Plots If the data is normal, the points will be in a straight line. The straighter the line, the closer to normal it is. Keep in mind we only need to achieve roughly normal…or good enough. It is common for the ends of the line to curve, but if they do, this typically means skew.

24. Normality Plots

27. Assignments Chapter 5: 5, 9, 12, 13, 14, 17, 18, 19, 21, 25, 29, 33, 34, 37, 41, 45. Read all of them, and then do eight of them. At least 3 from the first half (through 19) and at least 3 from the second half (from 21 on). Due 9/22 Chapter 6 quiz Friday. Unit 1 Test next week on Tuesday. Chapter 6: 1, 5, 10, 12, 15, 17, 24, 25, 29, 31, 41, 42, 45, 46, 49. Due on 10/6 (After midterms) So obviously you can ignore it until after the test, right, since why would chapter 6 material be even slightly related to the unit 1 test? In case you are not catching the sarcasm in that previous question, I would definitely take a peek at the questions even before the test, since their material is actually on the test.

28. More Unsupporting Cast Now we will focus on the last two members of our unsupporting cast. We will talk about the Dissembler and the Competitive Debater. These are differently flawed archetypes, but what they have in common is that they are both liars.

29. The Dissembler Dissembling is the process of saying something that is literally true but deliberately encourages the audience to jump to a conclusion which is false. This is Mr. Sanford’s favorite form of deception. There are actually multiple reasons why we need to be aware of this. For starters, some people will use statistics in misleading ways. The area principle is an example of something which some people *coughfoxnewscough* might use to mislead others.

30. The Dissembler Also, however, some people do not realize that they have been dissembled at. The reason why the hasty generalizer and the casual observer are so dangerous is that the strategies of the dissembler work on them. The deliberate dissembler is pretty much one of the main villains in statistics. Statistics is a powerful tool and can easily be misused for personal gain and this lures some into temptation. It is important to be aware of this possibility so you can make an informed decision on what you will do about statistics.

31. The Competitive Debater Some people prize winning an argument over the quality of truth contained in that argument. It is important to be somewhat skeptical of things which people say because sometimes people will cite “research” which is not actually legitimate research. Sometimes it is even made up. It is important to consider the source that is speaking as well as the source they are speaking about, or you could be mislead by what someone claims is science. Please remember that open-mindedness is also important along with the skepticism.

32. Chapter 6 Quiz Bulletpoints Be able to find and compare two z-scores by saying which one is more unusual. Be able to find the IQR of a normally distributed variable. Be able to give the percent of data found in a specific region of the normal curve. Be able to find a z-score or data score that corresponds to a given percent of the data. Be able to determine whether or not a graph is roughly normally distributed.