1. Unit 0, Pre-Course Math Review Session 0.2 More About Numbers
Introductory Biostatistics
Unit 0, Pre-Course Math Review Session 0.2 More About Numbers
J. Jackson Barnette, PhD
Professor of Biostatistics
Copyright 2013, JJBarnette

2. Topics for Session 0.2 Discrete or continuous Scientific notation
0.2 More about Numbers
Introductory Biostatistics
Topics for Session 0.2
Discrete or continuous
Scientific notation
Rounding rules
Operations with negative numbers
Operations with zero
Order of operations
The deviation score
The factorial
The combination and permutation
Logs and anti-logs (inverses)
Unit 0, Session 0.2
Copyright 2013, JJBarnette
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
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4. 1. Discrete or Continuous
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
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5. 1. Discrete or Continuous
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
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6. 0.2 More about Numbers
Introductory Biostatistics
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 x (the decimal place is moved 13 times to the LEFT)
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

7. 2. Scientific Notation, Especially 10-x
0.2 More about Numbers
Introductory Biostatistics
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
– 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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

8. 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
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9. 2. Scientific Notation Examples
0.2 More about Numbers
Introductory Biostatistics
2. Scientific Notation Examples
Convert to Scientific Notation
?>
15,400,000,000 ?>
?>
10,450, ?>
?>
4.5 x 10-4
1.54 x 1010
6.7 x 10-7
1.045 x 107
6.9 x 10-3
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

10. 2. Scientific Notation Examples
0.2 More about Numbers
Introductory Biostatistics
2. Scientific Notation Examples
Convert to Decimal Notation
2.45 x 10-5 ?>
3.33 X107 ?>
1.11 x ?>
7.897 x ?>
3.5 x ?>
33,300,000
7,809,000,000,000
0.0035
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

11. 2. Scientific Notation, Especially 10-x
0.2 More about Numbers
Introductory Biostatistics
2. Scientific Notation, Especially 10-x
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
We 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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

12. 2. Scientific Notation, Especially 10-x
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: X 10-5 which would be 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<
Unit 0, Session 0.2
Copyright 2013, JJBarnette

13. 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
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14. 0.2 More about Numbers
Introductory Biostatistics
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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

15. 0.2 More about Numbers
Introductory Biostatistics
3. Rounding Examples
Round the following to the next highest decimal place:
?>
?>
?>
– ?>
?>
0.007
12.38
–0.0002
0.16
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

16. 4. Operations with Negative Numbers
0.2 More about Numbers
Introductory Biostatistics
4. Operations with Negative Numbers
We will often use negative numbers in statistics and there are some rules we use for:
Adding with negative numbers
Subtracting with negative numbers
Multiplying and squaring with negative numbers
Dividing with a negative numerator
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

17. 4. Operations with Negative Numbers
0.2 More about Numbers
Introductory Biostatistics
4. Operations with Negative Numbers
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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

18. 4. Operations with Negative Numbers
0.2 More about Numbers
Introductory Biostatistics
4. Operations with Negative Numbers
Adding two negative numbers
–5 + (–6)= ?
Add the two numbers together and give the
result a negative sign
(–)5 + (–)6= –11
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

19. 4. Operations with Negative Numbers
0.2 More about Numbers
Introductory Biostatistics
4. Operations with Negative Numbers
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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

20. 4. Operations with Negative Numbers
0.2 More about Numbers
Introductory Biostatistics
4. Operations with Negative Numbers
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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

21. 4. Operations with Negative Numbers
0.2 More about Numbers
Introductory Biostatistics
4. Operations with Negative Numbers
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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

22. 4. Operations with Negative Numbers
0.2 More about Numbers
Introductory Biostatistics
4. Operations with Negative Numbers
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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

23. 4. Operations with Negative Numbers
0.2 More about Numbers
Introductory Biostatistics
4. Operations with Negative Numbers
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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

24. 4. Operations with Negative Numbers
0.2 More about Numbers
Introductory Biostatistics
4. Operations with Negative Numbers
In the following, find the result:
–72= ?>
–8 + (–14)= ?>
–7 – (–7)= ?>
12 – (–12)= ?>
8 x (–9)= ?>
–77 / 11= ?>
–12 x (–10)= ?>
+49
–22
+24
–72 –7
120
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

25. 5. Statistical Operations with Zero
0.2 More about Numbers
Introductory Biostatistics
5. Statistical Operations with Zero
A 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)
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

26. 6. Order of Operations
There 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:
Square-root and exponents
Multiplication and division
Addition and subtraction
Unit 0, Session 0.2
Copyright 2013, JJBarnette

27. 6. Order of Operations Here are a few examples: 5 x 6+13=30+13=43
/2 – 12= – 12= 56 – 12= 44
3∗9 +8 = = 35 =5.92
3∗ = = =4.6
Unit 0, Session 0.2
Copyright 2013, JJBarnette

28. 6. Order of Operations How these are done (order) is very important:
𝑋2 (Square each score, them sum them up)
( 𝑋)2 (Sum scores, then square the result)
𝑿𝟐  ( 𝑿)𝟐
Unit 0, Session 0.2
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29. 6. Order of Operations Find the following: 4∗8 +(6/2)= ?>
∗2 = ?>
X= 5, 7, 9, 𝑋= ?>
X= 3, 5, 6, 𝑋2= ?>
x= -3, -2, 0, 1, 𝑥 = ?>
x= -3, -2, 0, 1, 𝑥2 = ?>
Using 𝑥2 from before, 𝑥2 5−1 = ?>
32+3= 35
𝟏𝟔𝟗+𝟓𝟎 = 𝟐𝟏𝟗 = 14.8
= 31
= 170
-3+(-2)+0+1+4= 0
= 30
𝟑𝟎 𝟒 = 𝟕.𝟓 =𝟐.𝟕𝟒
Unit 0, Session 0.2
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30. 0.2 More about Numbers
Introductory Biostatistics
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
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

31. 0.2 More about Numbers
Introductory Biostatistics
7. The Deviation Score
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 ( 𝑋 )
We could then find the deviation score for each of the scores by subtracting the mean from each score
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

32. 7. The Deviation Score If the score is higher than the mean, the
0.2 More about Numbers
Introductory Biostatistics
7. The Deviation Score
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 +
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

33. 0.2 More about Numbers
Introductory Biostatistics
7. The Deviation Score
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)
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

34. 0.2 More about Numbers
Introductory Biostatistics
8. The Factorial
When we compute probabilities, we occasionally need to find a factorial
It 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= 720
You may have a factorial key stroke on your calculator
Unit 0, Session 0.2
Copyright 2013, JJBarnette
Copyright 2013, JJBarnette

35. 8. The Factorial
What are the following factorials? 3!= ?> 10!= ?> 5!= ?> 0!= ?>
3x2x1= 6
10x9x8x7x6x5x4x3x2x1=
5x4x3x2x1= 120
0 (this goes against basic math theory, but in practice, it has to be 0 for the probabilities to work)
Unit 0, Session 0.2
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36. 8. The Factorial
There 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 objects
Unit 0, Session 0.2
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37. 9. The Combination 𝑛𝐶𝑟= 𝑛! 𝑛−𝑟 !𝑟!
In a combination, order is not considered
Thus, would be the same combination as and , counted just once
The combination is symbolized as:
𝑛𝐶𝑟= 𝑛! 𝑛−𝑟 !𝑟!
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
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38. 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
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39. 9. The Combination
Here’s how we would find out how many combinations we would have: 𝑛𝐶𝑟= 𝑛! 𝑛−𝑟 !𝑟! = 10! 10−4 !4! = !4! = ∗ 24 = = 210 There are 210 different combinations of 4 ingredients out of a possible 10
Unit 0, Session 0.2
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40. 9. The Permutation 𝑛𝑃𝑟= 𝑛! 𝑛−𝑟 ! In a permutation, order is considered
Thus, would not be the same as or , this is three possible permutations
The permutation is symbolized as:
𝑛𝑃𝑟= 𝑛! 𝑛−𝑟 !
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
Copyright 2013, JJBarnette

41. 9. The Permutation
Let’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
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42. 9. The Permutation
Here’s how we would find out how many permutations we would have: 𝑛𝑃𝑟= 𝑛! 𝑛−𝑟 ! = 10! 10−4 ! = ! = =5040 There are 5040 different permutations of 4 ingredients out of a possible 10 There are 5040 ways of entering any 4 out of 10 ingredients into a salad bowl
Unit 0, Session 0.2
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43. 9. Permutations and Combinations
You may want to use your calculator to find these (see session 0.4 for examples) 8𝐶3= ?> 12𝐶5= ?> 8𝑃3= ?> 12𝑃5= ?> 56
792
336
95040
Unit 0, Session 0.2
Copyright 2013, JJBarnette

44. 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
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45. 10. Logs and Anti-logs
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: …… The natural log of a number X is symbolized as ln(X)
Unit 0, Session 0.2
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46. 10. Logs and Anti-logs
For example, here are some natural logs (values must be positive numbers, but can be less than 1: ln(20) = ln(0.65) = ln(0.001)= 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
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47. 10. Logs and Anti-logs
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= 𝑒-X where X is the natural log and e is the constant we just identified (2.718….)
Unit 0, Session 0.2
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48. 10. Logs and Anti-logs
For example, here are some anti-logs (inverses): Found as: INV= 𝑒 𝑙𝑛 ln= 1.5, Inv= ln= 0.5, Inv= ln= -0.75, Inv= ln= 1, Inv= … (𝒆) 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
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49. Conclusion
I 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
Copyright 2013, JJBarnette