# Unit 0, Pre-Course Math Review Session 0.2 More About Numbers

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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

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

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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

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

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 Copyright 2013, JJBarnette

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

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

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

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

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 Copyright 2013, JJBarnette

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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

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

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

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

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

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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 Copyright 2013, JJBarnette

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