Stuff You Should Know About ZScores Whats in
Stuff You Should Know About Z-Scores
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 rarely we use it.
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.
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.
Normal Curves �The point of inflection is exactly one standard deviation away from the mean. �In both directions. �Here’s the picture example:
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.
The 68 -95 -99. 7 Rule �In picture form:
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.
Formula �
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
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.
The 68 -95 -99. 7 Rule Recap �In picture form:
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.
So, how do I use my calculator? �We will need to know how to use the normalcdf and inv. Norm functions of our calculator. �When we want to get a percentage/area we will use normalcdf. �When we have a percentage/percentile we will use inv. Norm.
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.
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 normalcdf to get the area. �Step 4: We may need to turn our area into a percent. �Step 5: Phrase our answer for people.
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 inv. Norm to find a z-score. �Step 4: Turn our z-score into a data value. �Step 5: Phrase our answer for people.
How do I know if data is normal? This Will Be Question 5 On The Chapter 6 Quiz
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.
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.
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.
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.
Normality Plots
Normality Plots
Normality Plots
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 Wednesday. � Chapter 6: 1, 5, 10, 12, 15, 17, 24, 25, 29, 31, 42, 45, 46, 49. � Due on 10/3 (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.
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.
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