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Unit 0, Session 0.1 Pre-Course Biostatistics Math Review J. Jackson Barnette, PhD Professor of Biostatistics

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Unit 0, Session 0.1Copyright 2013 JJ Barnette2 Purpose How long has it been since you did much in the way of mathematics rather than balance your checkbook or fill out tax forms? For some, perhaps most of you, it has probably been a pretty long time It is likely that many of you have not done much in the way of applied mathematics since early college or even high school

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Unit 0, Session 0.1Copyright 2013 JJ Barnette3 Purpose There are many mathematical operations we use in statistics and a review of these may be useful You may be pleased to know that this is not a calculus-based course These four sessions ( ) provide overviews of many of the methods we will use in intro biostatistics and this review will include examples of how many of these methods are actually used in the course

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How Fortunate You Are When I started learning this stuff, we had very primitive tools compared to what you are able to use Hardware and software advances have made it possible to do extensive, accurate, and thorough computations instantly We will take advantage of these Unit 0, Session 0.1Copyright 2013 JJ Barnette4

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How Fortunate You Are The tools I used The abacus: Unit 0, Session 0.1Copyright 2013 JJ Barnette5 Did you think I was really this old?????

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How Fortunate You Are The tools I used The slide rule (I did use these) Good for basic math (could not add and subtract), but not much else Unit 0, Session 0.1Copyright 2013 JJ Barnette6

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How Fortunate You Are The tools I used Unit 0, Session 0.1Copyright 2013 JJ Barnette7 Desk-top calculators (early to mid 60’s) Great, but no Square-Root function

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How Fortunate You Are The tools I used Unit 0, Session 0.1Copyright 2013 JJ Barnette8 Hand-held calculators, late 60’s Still NO Square-Root Key, BUMMER!!!

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How Fortunate You Are The tools I used Finally, a square-root function: Early 70’s The big controversy was: “Should we allow students to use these to take tests?” Unit 0, Session 0.1Copyright 2013 JJ Barnette9

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Unit 0, Session 0.1Copyright 2013 JJ Barnette10 Here’s a calculator that one of my students had. Do you think she had problems whenever she wanted to put in a 6?

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How Fortunate You Are The tools I used Mainframe computers (mid 60’s, early 70’s) Very powerful compared to what we had before Punch cards were a pain and demanding (no backspace key), but the only way we had to get the data in for analysis (take to center, go back tomorrow), pray it worked Unit 0, Session 0.1Copyright 2013 JJ Barnette11

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How Fortunate You Are The tools I used The personal computer (late 70’s): WOW!!!!! Life did take a significant change!! Unit 0, Session 0.1Copyright 2013 JJ Barnette12

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How Fortunate You Are The tools I used And this doesn’t even touch on all the advances in the software (SAS, SPSS, Minitab, NCSS, Stata, Excel) which are incredible Thanks for enduring my journey down memory lane with all the tools I’ve used You do have life much easier than I did when it comes to doing data analysis Easier to do the math, but still need to know what it all means Unit 0, Session 0.1Copyright 2013 JJ Barnette13

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How Fortunate You Are But, you still must learn the concepts that these tools help you do, they only “crunch the numbers you put in”; you must decide what the results mean These tools have no ability to determine if data are reliable or valid or that they are being analyzed with the most appropriate methods ONLY the user can make those decisions Unit 0, Session 0.1Copyright 2013 JJ Barnette14

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Calculator for Course You probably don’t need to buy a calculator for this course; you are probably carrying the best (and cheapest) calculator you can get as: There are hundreds of apps, mostly free, that will do everything you need to do in this class and more Unit 0, Session 0.1Copyright 2013 JJ Barnette15

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Unit 0, Session 0.1Copyright 2013 JJ Barnette16 Session in Unit Symbolization and numbers and their relationships 0.2 More about numbers 0.3 Graphing 0.4 Useful calculator functions Handouts in pdf format are found on the course website in 3 and 6 per page formats

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Unit 0, Session 0.1Copyright 2013 JJ Barnette17 Topics for Session Symbolization 2.The five most important numbers in statistics 3.The number line 4.Using small numbers 5.Equality and inequality signs 6.The notion of an interval 7.Quotient relationships

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Unit 0, Session 0.1Copyright 2013 JJ Barnette18 1. Symbolization We will use a variety of extremely useful symbols as we deal with statistics Let’s take a look at some of these We use what we call variables These may be things like age, gender, height, weight, blood pressure, glucose level, cure status, survival rate, etc. The data we get are referred to as observations or scores on the variables

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Unit 0, Session 0.1Copyright 2013 JJ Barnette19 1. Symbolization For the value we observe for a single variable, we will label the value as X If we have a second variable, we would often label it as Y so we can keep it separate from X If we had several variables, we often label them as X 1, X 2, X 3, etc.

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Unit 0, Session 0.1Copyright 2013 JJ Barnette20 1. Symbolization There may be a list of symbols in your textbook n, when we want to indicate the number of observations in a sample, we will use n N, when we want to indicate the number of observations in a population, we will use N f, when we want to indicate the number of observations in given category or the frequency, we will use f

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Unit 0, Session 0.1Copyright 2013 JJ Barnette21 1. Symbolization We often use Greek symbols for operations and values One of the most common symbols is the summation notation, capital sigma, It represents adding a series of values We add the scores from score i= 1 through score n However, we usually just see this as:

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Unit 0, Session 0.1Copyright 2013 JJ Barnette22 1. Symbolization There are other common symbols we will use: (Alpha) represents two things: the level of significance (risk of a Type I error) and Cronbach’s reliability coefficient (Beta) represents risk of a Type II error and a regression weight (Mu) represents a population mean 2 (Sigma-squared) represents a population variance (Sigma) represents the population standard deviation

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1. Symbolization Unit 0, Session 0.1Copyright 2013 JJ Barnette23 s 2 sample variance s sample standard deviation r xy correlation coefficient of variables X and Y b 0 Y-intercept (value of Y when X=0) b 1 slope (change in Y when X increases by 1)

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2. The Five Most Important Numbers There are five numbers we will see more than any other numbers They are: 0, 1, 0.05, 95%, and +/ and 1 are both used mostly to indicate no difference We can talk about X-Y= 0 to represent no difference We can talk about X/Y= 1 to represent no difference Unit 0, Session 0.1Copyright 2013 JJ Barnette24

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2. The Five Most Important Numbers 0.05 is used as what we call the level of significance, the alpha level, or the risk of committing a Type I error It is a value we often set for making decisions about the results we observe It is a standard used in many if not most statistical decisions Sometimes we may use a different value such as 0.10 or 0.01, but mostly we use 0.05 Unit 0, Session 0.1Copyright 2013 JJ Barnette25

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2. The Five Most Important Numbers 95% will be used as a standard for confidence in our estimates We often want to be 95% confident in our estimates Sometimes we may want to be 90% or 99% confident, but most of the time we will want to set our standard of confidence at 95% Unit 0, Session 0.1Copyright 2013 JJ Barnette26

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2. The Five Most Important Numbers +/ is related to our use of the normal probability distribution in our statistical decision-making Between z values of and we have 95% of the area in the normal distribution This relates to the 95% important number we just addressed Unit 0, Session 0.1Copyright 2013 JJ Barnette27

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Unit 0, Session 0.1Copyright 2013 JJ Barnette28 3. The Number Line It may seem real simple, but remembering what we mean by the number line is important in statistics We often need to be able to visualize where a number is on the number line when we need to decide whether numbers are smaller or larger than each other

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Unit 0, Session 0.1Copyright 2013 JJ Barnette29 3. The Number Line Goes from : The values on the number line can theoretically be any value either negative, zero, or positive with any whole number or decimal number - to + Negative Values 0 Positive Values

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Unit 0, Session 0.1Copyright 2013 JJ Barnette30 3. The Number Line Some of the values we have from the variables we use can only fall on whole number points such as –12, –3, 0, +2, + 15 (these are in order from low to high) If an observed value can only fall on a number that is a whole number such as these, we call these discrete observations

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Unit 0, Session 0.1Copyright 2013 JJ Barnette31 3. The Number Line Some of the values we use will be in non-whole number or decimal form where there theoretically may not be any limit to the number of decimal places We may see numbers such as: –45.67, –0.023, –0.003, 2.256, Often the only limitation on how many decimal places we have is the precision afforded by the measuring instrument These are called continuous observations

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Unit 0, Session 0.1Copyright 2013 JJ Barnette32 Higher or Lower? We will often compare values to see which value is higher or lower than the other one This involves using the number line Values to the left are always lower than values to the right

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Unit 0, Session 0.1Copyright 2013 JJ Barnette33 Some Examples Which is lower than the other of: 12 and 123?> –4 and –6 ?> 0.01 and 0.002?> –0.10 and –0.40?> and 0?> –0.111 and 0?> 12 < 123 –6 < – < 0.01 –0.40 < – < –0.111 < 0

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Copyright 2013 JJ Barnette34 4. Using Small Numbers We will often use numbers that are between the values of 0 and 1 We often will use small numbers less than 0.10 and it is important to be able to express them and compare them Is 0.04 less than 0.05??> Is less than 0.01??> Is less than 0.05??> Yes, 0.04 < 0.05 Yes, < 0.01 No, > 0.05 Unit 0, Session 0.1

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5. Equality and Inequality Signs In the course, we will often use inequality signs to identify whether numbers are equal to, larger, or smaller than each other Often this will also include the equal sign Remember that the point ( ) is toward the larger number Unit 0, Session 0.1Copyright 2013 JJ Barnette35

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5. Equality and Inequality Signs We will use: =, , +/-, ,, ≤, AND You should be familiar with most of these =, A=B, indicates the value to the left (A) is equal to the value to the right (B) of the symbol ≠, A≠B, indicates the value to the left (A) is NOT equal to the value to the right (B) of the symbol +/-, A +/- B, indicates that the value to the right of the symbol (B) is subtracted from the value to the left (A) AND B is added to the value on the left (A) of the symbol, resulting in two values, A– B and A+B Unit 0, Session 0.1Copyright 2013 JJ Barnette36

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Unit 0, Session 0.1Copyright 2013 JJ Barnette37 5. Equality and Inequality Signs ≈, A≈B, indicates the two numbers A and B are approximately equal <, A**, A>B, indicates the number in front (left) of this symbol (A) is greater than the number that follows it (B)
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Unit 0, Session 0.1Copyright 2013 JJ Barnette38 5. Equality and Inequality Signs ≤, A≤B, indicates the number in front (left) of this symbol (A) is less than or equal to the number that follows it (B) ≥, A≥B, indicates the number in front (left) of this symbol (A) is greater than or equal to the number that follows it (B)

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The Notion of an Interval We will be using many intervals in the course and it will be good to make sure we understand their use An interval has the form: L ≤ X ≤ U Where: L is the LOWER LIMIT U is the UPPER LIMIT X is the value that is in the interval We say that X is less than or equal to the LOWER LIMIT AND X is greater than or equal to the UPPER LIMIT Unit 0, Session 0.1Copyright 2013 JJ Barnette39

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Unit 0, Session 0.1Copyright 2013 JJ Barnette40 6. An Interval The interval is used to represent the likely range of values of something we estimate We may estimate a value such as the mean and have an interval of likely possible mean values We may estimate a proportion and have an interval of likely possible proportion values We may estimate a value such as a difference in means and have an interval of likely possible difference values

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Unit 0, Session 0.1Copyright 2013 JJ Barnette 41 Interval Examples Say we have the following intervals and questions: -3.2 ≤ X ≤ 4.6Can X be 0 ?> ≤ X ≤ Can X be 0 ?> ≤ X ≤ -0.06Can X be 0 ?> 1.05 ≤ X ≤ 1.23Can X be 1 ?> 5.65 ≤ X ≤ 9.60Can X be 10 ?> ≤ X ≤ 63.42Can X be 50 ?> 0.98 ≤ X ≤ 1.15Can X be 1 ?> YES NO YES NO YES

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Unit 0, Session 0.1Copyright 2013 JJ Barnette42 7. Quotient Relationships It will be useful to predict what result will happen when a numerator (N) or denominator (D) changes in a quotient To DECREASE X: If N decreases and D stays the same, X decreases If D increases and N stays the same, X decreases

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Unit 0, Session 0.1Copyright 2013 JJ Barnette43 7. Quotient Relationships It will be useful to predict what result will happen when a numerator (N) or denominator (D) changes in a quotient To INCREASE X: If N increases and D stays the same, X increases If D decreases and N stays the same, X increases

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Unit 0, Session 0.1Copyright 2013 JJ Barnette44 Conclusion I hope this review has been helpful in reacquainting you or introducing you to many of the mathematics operations used in introductory statistics There is more in sessions 0.2 and 0.3 You may want to copy these slides as pdf file handouts from the Course Website and refer to them occasionally as we go through the course

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