unit 0, pre-course math review session 0.2 more about numbers j. jackson barnette, phd professor of...

Unit 0, Pre-Course Math ReviewSession 0.2

More About Numbers

J. Jackson Barnette, PhD

Professor of Biostatistics

Unit 0, Session 0.2 Copyright 2013, JJBarnette 2

Topics for Session 0.21. Discrete or continuous2. Scientific notation3. Rounding rules4. Operations with negative numbers5. Operations with zero6. Order of operations 7. The deviation score8. The factorial9. The combination and permutation10.Logs and anti-logs (inverses)

Copyright 2013, JJBarnette 3

1. Discrete or Continuous

We will need to be able to classify the variables we use as being discrete or continuous

A discrete variable can only take on whole number values, no fractional values

Discrete variables would be variables such as gender, political preference, marital status, disease status, etc.

Unit 0, Session 0.2



Continuous variables can take on values that are fractional or points on a continuum that (theoretically) has no specific stop-end points

Our measures are limited by the ability of our measuring instrument to measure to specific levels of precision

Unit 0, Session 0.2



We sometimes have continuous variables that are measured using discrete points such as age to the nearest year, weight to the nearest pound, knowledge measured using number correct on an exam

It is important to be able to distinguish between the actual variable and how it is measured relative to discrete/ continuous

Unit 0, Session 0.2


2. Scientific Notation

Occasionally, we need to express very large or very small numbers in an easier form than the actual number

A number such as 23,456,000,000,000 might be better expressed in what is referred to as scientific notation

We use a power of ten to express this number in the form of 2.3456 x 1013 (the decimal place is moved 13 times to the LEFT)


2. Scientific Notation, Especially 10-x

In statistics, we are more likely to see decimal numbers less than 1 put in this form

0.1 is the same as 1 x 10-1

0.011 is the same as 1.1 x 10-2

0.003 is the same as 3 x 10-3

–0.00014 is the same as –1.4 x 10-4

As the decimal place is moved one place to the RIGHT, the exponent number increases by –1


2. Scientific Notation

Every time the decimal place moves to the LEFT, the exponent of 10 increases by +1

Every time the decimal place moves to the RIGHT, the exponent of 10 decreases by -1

Unit 0, Session 0.2


2. Scientific Notation Examples

Convert to Scientific Notation

0.00045 ?>

15,400,000,000 ?>

0.00000067 ?>

10,450,000 ?>

0.0069 ?>

4.5 x 10-4

6.7 x 10-7

1.045 x 107

1.54 x 1010

6.9 x 10-3


2. Scientific Notation Examples

Convert to Decimal Notation

2.45 x 10-5 ?>

3.33 X107 ?>

1.11 x 10-8 ?>

7.897 x 1012 ?>

3.5 x 10-3 ?>

0.0000245

33,300,000

0.00000001117,809,000,000,000

0.0035



Most of the time, we will carry values out to three decimal places

However, computer programs will often give us small values in scientific notation form and we will need to interpret them

Such as 3.2 x 10-4 and this would be 0.00032We often will need to be able to determine if

this number is higher or lower than values such as 0.05 or 0.01, <0.05, <0.01



Where we are most likely to see the need for doing something related to this situation is when we have what’s called a p-value that a computer might present in this form such as:

4.445 X 10-5 which would be 0.00004445.

We usually only need to know that it is less than a given value (such as 0.05) that does not need this level of precision.

So, we might report this as p< 0.0001

Unit 0, Session 0.2


3. Rounding Rules

In general, most of our computations need to be carried out to two or three decimal places.

What we typically do is perform the mathematics using one more decimal place than we really want to report, we find our final answer and the we round it down to one less decimal place

Unit 0, Session 0.2


3. Rounding Rules

When we have results in decimal form, we will may want to round off values

In the course, when we compute values using decimal values, we will usually compute values to four decimal places and then we round off the final result to three decimal places

If a value is less than 5, we round down If a value is 5 or greater, we round up


3. Rounding Examples

Round the following to the next highest decimal place:

0.0065 ?>

12.378?>

456.3333 ?>

–0.00021 ?>

0.155 ?>

0.007

12.38

456.333

–0.0002

0.16


4. Operations with Negative Numbers

We will often use negative numbers in statistics and there are some rules we use for:

1. Adding with negative numbers

2. Subtracting with negative numbers

3. Multiplying and squaring with negative numbers

4. Dividing with a negative numerator



Adding a negative and positive number

+3 + (–4)= ?

Subtract the lower number from the higher

number and give the result the sign of the

larger number

(–) 4 – 3 = –1



Adding two negative numbers

–5 + (–6)= ?

Add the two numbers together and give the

result a negative sign

(–)5 + (–)6= –11



Subtracting a negative from a positive number

+3 – (–4)= ?

Change the negative number to a positive

number and add the two positive numbers

+3 + (+4)= 3 + 4= +7



Subtracting a negative from a negative

number

–6 – (–4)= ?

Change the subtracted negative number to a

positive number, subtract the lower from the

higher number, give result the sign of the

higher number

–6 – (–4)= –6 + (+4) = –2



Multiplying a positive and negative number

4 x (–6)= ?

Multiply the two numbers as positive

numbers and give the product a negative

sign

4 x (–)6= –24



Multiplying two negative numbers

–4 x (–8)= ?

Multiply the two numbers as positive

numbers and give the product a positive sign

(–)4 x (–)8= +32

This rule also applies to squaring a negative

number –42 = +16



Division with a negative number

We will not be dividing any number by a

negative number so we only need to consider

dividing a negative number by a positive

number

A negative number divided by a positive

number results in a negative quotient

–5.4 / 9= –0.6



In the following, find the result:–72= ?> –8 + (–14)= ?>–7 – (–7)= ?>12 – (–12)= ?>8 x (–9)= ?>–77 / 11= ?> –12 x (–10)= ?>

+49

–220

+24–72–7

120


5. Statistical Operations with ZeroA number multiplied by 0 equals 0

9 x 0= 0

A number divided by 0 equals 0

9 / 0= 0

Zero divided by a number equals 0

0 / 5= 0

These are rules statisticians must use for things to work (we don’t mess with “imaginary” numbers)

6. Order of OperationsThere will be many times where we have to decide the order we do various mathematical operations and there are rules for doing this

First of all get a single value for anything within parentheses or under a square-root radical

Other than parentheses/radicals, The order goes:

1. Square-root and exponents

2. Multiplication and division

3. Addition and subtraction


6. Order of Operations

Here are a few examples:


5 x 6+13=30+13=43

7 2 + 14/2 – 12= 49 + 7 – 12= 56 – 12= 44

√ (3∗9 )+8= √27+8=√35=5.92


How these are done (order) is very important:


(Square each score, them sum them up)

(Sum scores, then square the result)



Find the following:= ?>

= ?>

X= 5, 7, 9,10 ?>

X= 3, 5, 6, 10 ?>

x= -3, -2, 0, 1, 4 = ?>

x= -3, -2, 0, 1, 4 = ?>

Using from before, = ?>

Unit 0, Session 0.2

32+3= 35 = = 14.8

5+7+9+10= 31 9+25+36+100= 170

9+4+0+1+16= 30 -3+(-2)+0+1+4= 0

=


7. The Deviation Score

One of the things we do often in statistics is comparing a score or variable value, symbolized as X with a standard such as the mean

We will use a term referred to as a deviation score and we will symbolize it as a small case x, It is found as

mean sample the is where XXXx



The deviation score (or variations of it) is used extensively in statistics

In several types of score distributions with several scores, we could find the mean ()

We could then find the deviation score for each of the scores by subtracting the mean from each score

scoredeviation XXx



If the score is higher than the mean, the

deviation score is positive and the square is +

If the score is lower than the mean, the

deviation score is negative and the square is +

3107 XXx

21012 XXx 4222 x

9322 x



If we add all the deviation scores, we ALWAYS get 0

If we square each deviation score and add these up, we will not get 0 (assuming the scores are not all the same value)

0x

02 x


8. The FactorialWhen we compute probabilities, we

occasionally need to find a factorialIt is symbolized as X!This represents the number that would result

if you continuously multiplied the number (say it is 6) times the sequence of the number –1 down to 1

6!= 6 x 5 x 4 x 3 x 2 x 1= 720You may have a factorial key stroke on your

calculator


8. The Factorial

What are the following factorials?

3!= ?>

10!= ?>

5!= ?>

0!= ?>

Unit 0, Session 0.2

3x2x1= 6

10x9x8x7x6x5x4x3x2x1= 3628800

5x4x3x2x1= 120

0 (this goes against basic math theory, but in practice, it has to be 0 for the probabilities to work)


8. The FactorialThere are two types of counting techniques that use the factorial and are used to determine probabilities of events happening

One is the COMBINATION which is the number of different sets of non-ordered r objects that can be taken from a set of n possible objects

The other is the PERMUTATION which is the number of sets of ordered r objects that can be taken from a set of n possible objectsUnit 0, Session 0.2


9. The Combination

In a combination, order is not considered

Thus, 1-2-5-6-8 would be the same combination as 2-1-6-8-5 and 8-6-1-5-2, counted just once

The combination is symbolized as:

=

Where n is the total number of unique objects that can be selected and r is the number of unique objects selected out of n

Unit 0, Session 0.2


9. The Combination

To me, the easiest way to think about this is in preparing a salad. Let’s say we start with lettuce and then we want to know how many different combinations of 4 additional salad ingredients out of 10 possible ingredients.

Unit 0, Session 0.2


9. The CombinationHere’s how we would find out how many combinations we would have:

= = =

= = = 210

There are 210 different combinations of 4 ingredients out of a possible 10 Unit 0, Session 0.2


9. The Permutation

In a permutation, order is considered

Thus, 1-2-5-6-8 would not be the same as 2-1-6-8-5 or 8-6-1-5-2, this is three possible permutations

The permutation is symbolized as:

=

Where n is the total number of unique objects that can be selected and r is the number of unique objects selected out of n

Unit 0, Session 0.2


9. The PermutationLet’s continue to use the salad example. Let’s say we start with lettuce and then we want to know how many different permutations of 4 additional salad ingredients out of 10 possible ingredients.

This could be considered how many ways or orders the ingredients could be entered into the salad bowl.

Onion-pepper-radish-tomato would be counted as one and radish-pepper-tomato-onion would be counted as a different permutation because order is different

Unit 0, Session 0.2


9. The PermutationHere’s how we would find out how many permutations we would have:

= = ==

There are 5040 different permutations of 4 ingredients out of a possible 10There are 5040 ways of entering any 4 out of 10 ingredients into a salad bowl

Unit 0, Session 0.2


9. Permutations and Combinations

You may want to use your calculator to find these (see session 0.4 for examples)

?>

?>

?>

?>

Unit 0, Session 0.2

56

792

336

95040


10. Logs and Anti-logs

We will see the use of logs occasionally in the course

The need for this is that sometimes we will have variables that are not normally distributed (a very desirable property), but their logs will be approximately normally distributed

Thus, we may convert values to logs, find what we need to find, and then convert the logs back to values on the original variable scale

Unit 0, Session 0.2



Logs can be found in various forms depending on the base of the logs

We will be using what are called “natural” logs

Natural logs are on the basis of a constant referred to as “e”

e is the basis of the natural logs and is equal to: 2.718281828……

The natural log of a number X is symbolized as ln(X)

Unit 0, Session 0.2


10. Logs and Anti-logsFor example, here are some natural logs (values must be positive numbers, but can be less than 1:

ln(20) = 2.995

ln(0.65) = -4.31

ln(0.001)= -6.91

ln(45) = 3.81

In(1)= 0

A number higher than 1 will have a + ln

A number less than 1 will have a – ln

You can see how to do these with your calculator in Session 0.4

However, we will let the computers find these as we need them

Unit 0, Session 0.2



Once we have converted values to logs and do what we need to do with them, we will need to convert the log value back to the original variable scale

These are called the anti-log or inverse

For natural logs, we do this with this equation: Y= -X where X is the natural log and e is the constant we just identified (2.718….)

Unit 0, Session 0.2



For example, here are some anti-logs (inverses):

Found as: INV=

ln= 1.5, Inv= 4.482

ln= 0.5, Inv= 1.649

ln= -0.75, Inv= 0.472

ln= 1, Inv= 2.71828… ()

ln= 0, Inv= 1

Again, we will let the computer do these for us, but if you are compelled to find these on your own, this is how it works

Unit 0, Session 0.2


ConclusionI hope sessions 0.1 and 0.2 have provided a review of some of the terminology and mathematical methods you will see in this course.

You may want to print the handouts that are with these two sessions and refer to them as we use some of these methods in the course

Session 0.3 deals with a review of graphing methods used in the course

Unit 0, Session 0.2

unit 0, pre-course math review session 0.2 more about numbers j. jackson barnette, phd professor of...

Documents

discrete continuous

left slide

discrete points

jjbarnette5 slide

jjbarnette8 slide

jjbarnette3 slide

jjbarnette4 slide

decimal place