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

Unit 0, Session 0.1 Pre-Course Biostatistics

Math Review

J. Jackson Barnette, PhD

Professor of Biostatistics

Unit 0, Session 0.1 Copyright 2013 JJ Barnette 2

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


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 (0.1 - 0.4) 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

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


How Fortunate You AreThe tools I used

The abacus:


Did you think I was really this old?????


The slide rule (I did use these)

Good for basic math (could not add and subtract), but not much else




Desk-top calculators (early to mid 60’s)

Great, but no Square-Root function



Hand-held calculators, late 60’s

Still NO Square-Root Key, BUMMER!!!


Finally, a square-root function:

Early 70’s

The big controversy was: “Should we allow students to use these to take tests?”



Here’s a calculator that one of my students had.Do you think she had problems whenever she wanted to put in a 6?


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


http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=nRpklWU4Jxy6qM&tbnid=aSK3NtiafKN7LM:&ved=0CAgQjRwwAA&url=http://www.computersciencelab.com/ComputerHistory/HistoryPt4.htm&ei=4Pt6Ub37KdGGqQG8vIGgDQ&psig=AFQjCNH7TBOr3Q19gBmJt-6_dcH5vxJsJQ&ust=1367100768732673


The personal computer (late 70’s):

WOW!!!!! Life did take a significant change!!



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


How Fortunate You AreBut, 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


Calculator for CourseYou 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



Session in Unit 00.1 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


Topics for Session 0.1

1. Symbolization2. The five most important numbers in

statistics3. The number line4. Using small numbers5. Equality and inequality signs6. The notion of an interval7. Quotient relationships


1. Symbolization

We will use a variety of extremely useful symbols as we deal with statistics

Let’s take a look at some of theseWe use what we call variablesThese 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


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 X1, X2, X3, etc.


1. Symbolization

There may be a list of symbols in your textbookn, when we want to indicate the number of

observations in a sample, we will use nN, when we want to indicate the number of

observations in a population, we will use Nf, when we want to indicate the number of

observations in given category or the frequency, we will use f


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:

n

ini X....XXXX

1321

X


1. SymbolizationThere 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

1. Symbolization


s2 sample variances sample standard deviationrxy correlation coefficient of variables X and Y

b0 Y-intercept (value of Y when X=0)b1 slope (change in Y when X increases by 1)

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 +/-1.96

0 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 0.1 Copyright 2013 JJ Barnette 24


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 0.05Unit 0, Session 0.1 Copyright 2013 JJ Barnette 25


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%



+/- 1.96 is related to our use of the normal probability distribution in our statistical decision-making

Between z values of -1.96 and +1.96 we have 95% of the area in the normal distribution

This relates to the 95% important number we just addressed



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


3. The Number LineGoes 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


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


3. The Number LineSome 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, 345.89Often the only limitation on how many decimal

places we have is the precision afforded by the measuring instrument

These are called continuous observations


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


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 ?>

0.006 and 0 ?>

–0.111 and 0 ?>

12 < 123

–6 < –40.002 < 0.01

–0.40 < –0.10

0 < 0.006

–0.111 < 0

Copyright 2013 JJ Barnette 34

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 0.005 less than 0.01? ?>

Is 0.055 less than 0.05? ?>

Yes, 0.04 < 0.05

Yes, 0.005 < 0.01

No, 0.055 > 0.05

Unit 0, Session 0.1

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 (<) of the inequality sign points toward the lower number and the open part (>) is toward the larger number


5. Equality and Inequality SignsWe 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




≈, A≈B, indicates the two numbers A and B are approximately equal

<, A<B, indicates the number in front (left) of this symbol (A) is less than the number that follows it (B)

>, A>B, indicates the number in front (left) of this symbol (A) is greater than the number that follows it (B)



≤, 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)

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 ≤ UWhere: 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



6. An IntervalThe interval is used to represent the likely

range of values of something we estimateWe 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


Interval ExamplesSay we have the following intervals and

questions:-3.2 ≤ X ≤ 4.6 Can X be 0 ?>12.63 ≤ X ≤ 22.14 Can X be 0 ?>-2.42 ≤ X ≤ -0.06 Can X be 0 ?>1.05 ≤ X ≤ 1.23 Can X be 1 ?>5.65 ≤ X ≤ 9.60 Can X be 10 ?> 49.62 ≤ X ≤ 63.42 Can X be 50 ?>0.98 ≤ X ≤ 1.15 Can X be 1 ?>

YES

NO

NO

NO

YES

NO

YES


7. Quotient RelationshipsIt 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

D

NX


7. Quotient RelationshipsIt 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

DN

X


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 0.3You 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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