1. COM PER BIN AND MUT ATI ATI ONS ONS

2. WEEK 1 (hidden slides) WEEK 5&7 WEEK 1 WEEKS 8,9,10 WEEK 4 WEEK 11
Structure and Content of the Course
spatial
attribute
thematic
temporal
categorical
quantitative
discrete
continuous
WEEK 1 (hidden slides)
nominal
ordinal
ratio
interval
Population
data
Types
Levels
Relatives:
percents
contingency
index #s
coefficients
Seven Pillars
Data errors
WEEK
5&7
Information
Extraction
WEEK 1
Description
Sample
Inference
Relationships:
WEEKS
8,9,10
central
tendency
dispersion
correlation
Patterns:
forecasting
hypothesis
testing
graphs
WEEK 4
distributions
maps sd
var
Sampling
median
mean
error
z scores
range
ogives
mode
frequency distributions
WEEK 11
errors
1&2
WEEKS 2&3
theory
methods
histograms
normal curves SE
CLT
Bi-variate & multivariate PARAMETRIC
techniques
non-normal
curves
shape
Chebyshev
symmetry
WEEKS
12&13
Probability
Binomial
Poisson
Problem solving
NON-PARAMETRIC
TECHNIQUES
Combinatorics
relationships
differences
WEEK 1
Significance tests
1,2,..n samples
α and p values
Confidence tests

3. Bi-variate & multivariate
Structure and Content of the Course
spatial
attribute
thematic
temporal
categorical
quantitative
discrete
continuous
nominal
ordinal
ratio
interval
Population
data
Types
Levels
Relatives:
percents
contingency
index #s
coefficients
You Have Been Here
Data errors
Information
Extraction
Description
Sample
Inference
central
tendency
Patterns:
dispersion
hypothesis
testing
graphs
distributions
You Are Here
maps
var sd
Sampling
median
mean
error
You Will Be Going Here
(Bwahaha)
ogives
mode
frequency distributions
range
errors
1&2
theory
methods
histograms
normal curves SE
CLT
Bi-variate & multivariate
techniques
non-normal
curves
shape
Chebyshev
symmetry
Problem solving
significance tests
NON-PARAMETRIC
TECHNIQUES
differences
relationships
confidence tests
1. 2,..n samples
February 23, 2020

4. Because they couldn’t lose.
The Drinking Problem
In 1982 Schlitz Brewing Company took a $4.2 million gamble that they couldn’t lose, because they hired people who knew their statistics.
At halftime during the Super Bowl in front of 100 million people they did a live taste test between Schlitz beer and their closest competitor, Michelob.
But crazier still, they pitted their brand against Michelob drinkers!
And Budweiser drinkers. And Miller drinkers.
And they had an NFL referee oversee the taste test
Crazy?
Like a fox.
Because they couldn’t lose.

5. This does not mean they cheated.
How To Gamble and Win
Schlitz won their gamble because they didn’t gamble. They knew, statistically, what the outcome would be before they started.
And it wasn’t because they produced an amazing beer because all the beers taste the same.
They knew what the outcome would be because they produced simple statistics and manipulated the sample!
This does not mean they cheated.
It means they thought about what research design would give them the result they wanted.
Consider…

6. So you can imagine what the marketing conclusion would be:
How To Gamble and Win
First, they carefully picked their sample – 100 each of Michelob, Miller and Budweiser drinkers an not just a mix of beer drinkers.
Then they calculated the probabilities of how many of them might choose Schlitz.
If you think about this, even without calculating the odds you might say that there was a 50/50 chance of either beer being chosen.
So you can imagine what the marketing conclusion would be:
“Look! Fifty percent of Michelob (or Miller or Bud) drinkers prefer Schlitz!”

7. How To Gamble, Win and Know You’ll Win
But you don’t have to rely on a guess that “50 out of 100 people will pick a Schlitz beer”.
You can actually calculate the probabilities quite easily, which the Schlitz statisticians did using what is called a Bernoulli Trial and binomial statistics.
This type of statistics can tell you what the probability is of 1, 2, 3, … or even 100 people choosing or not choosing a type of beer would be.
Binomial statistics is a very, very, important statistical method used widely in studies where you want to know what the probability is of ‘k’ things happening out of ‘n’ things and this is how you do it…

8. How To Gamble, Win, and Know You’ll Win
What’s the probability that 40 out of 100 Michelob drinkers will choose the Schlitz beer? Note that this would not be a good outcome if all beer drinkers were asked!
Well, first lets assuage the fears of the marketing people by asking what are the odds of the worst possible outcome - all 100 Michelob drinkers choosing Michelob?
1 in 1,267,650,600,228,229,401,496,703,205,376
or 1 in about 1.3 million trillion trillion.
And of all 100 choosing Schlitz (i.e. no-one choosing Michelob)?
1 in

9. How To Gamble, Win, and Know You’ll Win
Now back to a more realistic question?
What’s the probability that 40 out of 100 Michelob drinkers will choose the Schlitz beer? Note that this would not be a good outcome if all beer drinkers were asked!
The answer is that the probability is 98% that at least 40 of the 100 Michelob drinkers would choose the Schlitz and an 86% chance that 45 would, and a 54% chance that 50 or more would!
And even if it had been Michelob doing the taste test with Schlitz drinkers, or Pepsi with Coke the same results would occur!
It wouldn’t matter what the products taste like because they pretty much taste the same anyway!
It was all about Combinatorics and statistics.

10. Combinatorics

11. Combinatorics
Combinatorics is the name given to the branch of mathematics that calculates how many groups of things you can choose out of a larger group of things and is generally written as 𝑛 𝑟 .
There are two ways to answer this question and it depends on whether the order in which you choose the ‘r’ matters.
If order matters you are finding a permutation and if order does not matter you are finding a combination.
That is, when you say “order matters” or not, you are asking whether the letters AB and BA are different sets because they are in a different order (permutation) or the same set because order doesn’t differentiate them (combination).
The other aspect of choosing a set of ‘r’ objects from a set of ‘n’ objects is whether you can repeat any of the ‘r’ objects.

12. Permutations, Combinations: Order and Repetition
How many groups (2P2 and 2C2) of r=2 can you get from another group of n=2, where group n is the letters A and B?
Permutation with Repetition (4)
All possible groups are allowed because order matters and repeats don’t.
Permutation without Repetition (2)
No repeated letters among the groups are permitted, so AA and BB are not allowed. A B AA AB BA BB A B AA AB BA BB
Combination with Repetition (3)
Because order doesn’t count, BA is the same as AB so counts as one group, and repetition allows repeats of letters.
Combination without Repetition (1)
Because order doesn’t count and no repetition is allowed, all permissible combinations count as the same group. A B AA AB BA BB A B AA AB BA BB

13. Permutations, Combinations: Order and RepetitionAA NO
Permutation with Repetition: Order in which members appear is unique and letters can repeat.
Combination with Repetition: Order in which members appear is not duplicated in other groups and letters can repeat.
Permutation without Repetition: Order is unique, but members repeat a letter.
Combination without Repetition: Order is not duplicated in any form, but members repeat a letter. AB
Permutation with Repetition: Order in which members appear is unique.
Combination with Repetition: Order in which members appear is not duplicated in other groups and no letter repeats.
Permutation without Repetition: Order in which members appear is unique and no letter repeats.
Combination without Repetition: Order in which members appear is not duplicated in other groups and no letter repeats. BB
Combination without Repetition: Order in which members appear is not duplicated in other groups and letters can repeat. BA
Combination with Repetition: Order does not matter so BA and AB are the same.
Combination without Repetition: Order does not matter so BA and AB are the same.

14. The Importance of Permutations and Combinations
Permutations and combinations are very important in statistics because they underlie the Binomial and Poisson distributions that are widely used to calculate the probabilities of ‘r’ events happening in ‘n’ trials.
In fact, these types of ‘n choose r’ research problems (or 𝑛 𝑟 as it is sometimes written), and the binomial method especially, are probably the most widely used statistical methods of all.
The notation for permutations is nPr but you may also see nPr , P(n,r), or P(n,r)= 𝑛! 𝑛−𝑟 ! which is its computational formula.
The notation for combinations is nCr but you may also see nCr , C(n,r), 𝑛 𝑟 or C(n,r)= 𝑛! 𝑟! 𝑛−𝑟 ! which is its computational formula.
In math and engineering you may also see ‘k’ instead of ‘r’ but it means the same thing.

15. Examples of Permutations and Combinations
Permutation with repetition
Number of different 4 digit codes on a 0-9 digit combination lock.
(You can repeat any number on any dial but the numbers must sequence).
Permutation without repetition
Number of ways 8 runners could place in a race.
(If you’re 1st you’re 1st, but you cannot also come 2nd).
Combination with repetition
Number of triple scoop ice cream flavours you could get from 31 flavours.
(You can have V,V,C or V,C,V, etc. and order doesn’t matter).
(And, by the way, there are 5,456 different combinations).
Combination without repetition
Lotteries, e.g. 6/49
(The 6 numbers are chosen from 1 to 49 in any order but none can repeat).

16. Permutations and Combinations
In Summary
The only thing that describes the difference between a permutation and a combination is the order in which things can be selected.
Permutation = Order Matters
Combination = Order Doesn’t Matter
Either one can have repetition or not.
Commit this distinction to memory.

17. Calculating
Permutations
and
Combinations

18. Permutations With Repetition
(Remember: order matters with permutations)
These are the easiest to calculate and understand, and are what you do with a “combination” lock – which should actually be called a permutation lock.
If you can re-use any member of the usually larger group ‘n’ to form your smaller group ‘r’, you can basically choose any ‘n’, ‘r’ times, thus:
n*n* … (‘r’ times) or mathematically stated as nr.
An example is a combination lock with a 3 number code and choosing any 3 numbers (r) from 10 (n) for your “combination”:
10*10*10 or 103 = 1,000 possible codes because any number can be used over again – the code could be 3,3,3 or any other 3 numbers such as 1,2,3 – but in that order only.

19. Permutations Without Repetition
(Remember: order matters with permutations)
Now we have to reduce the number of choices because you cannot repeat a number.
An example would be a race with, say, 8 runners numbered 1 to 8, the question being how many ways are there for 8 runners to place?
The reason we have to reduce the number of choices is that whoever comes in 1st place cannot be 2nd as well, nor can 2nd be 3rd and so on for all runners.
So, for all 8 runners the arithmetic is:
8*7*6*5*4*3*2*1 = 40,320
possible ways for 8 runners to place 1st through 8th.

20. Permutations Without Repetition
But calculating 8*7*6*5*4*3*2*1 = 40,320 every time is tedious – what if you had the 27,000 runners in the NY City marathon?
There’s an easier mathematical way to calculate such reducing series of numbers by using factorials.
A factorial is just a mathematical way of writing “multiply this number by all whole numbers below it as a descending series” and it is written as 8! for the example we are using.
Thus 8! means the same as 8*7*6*5*4*3*2*1
And for the NY marathon?
27,000! is effectively infinite!
Ask Google.

21. Permutations Without Repetition
What if you wanted just the first three placings? Now you are asking how many ways are there for 8 runners to place 1st, 2nd, and 3rd?
That’s easy. There are 8*7*6 = 336 ways for 8 runners to place 1st, 2nd, and 3rd.
How do you write that as a factorial? That’s not so easy, but is pretty clever because it involves dividing by the 5 runners who don’t place:
(8*7*6*5*4*3*2*1) / (5*4*3*2*1) = 336 or
8! / 5! = 336

22. Permutations Without Repetition
The factorial solution is written as:
𝑛! 𝑛−𝑟 !
Also seen as: P(n,r) or nPr or nPr
One more example to clarify this. How many ways can 1st and 2nd place be awarded to our 8 runners?
8! 8−2 ! = 8! 6! = 40, = 56
And our original 1st, 2nd and 3rd places to our 8 runners?
8! 8−3 ! = 8! 5! = 40, = 336

23. Permutations Without Repetition
The important thing here is not that you understand why factorials (or even factoring) works to give us answer to permutations (and combinations) questions, but that it does work.
More importantly, as we’ll see shortly, permutations and especially combinations, form the heart of a widely used class of statistics that looks at the probability of getting ‘r’ results in ‘n’ trials.
These are called the Binomial and Poisson distributions.
But for now, I want to continue by looking at combinations.

24. Combinations Without Repetition
(Remember: order doesn’t matter in combinations.)
This is how lotteries work. Numbers are drawn one at a time and if you have them, no matter in what order they were drawn, you win.
For Lotto 6/49, 6 numbers are drawn out of 49 numbers and might be chosen in the order:
And the winning number would be:
In fact the number of combinations of 3 numbers out of 3 (3 choose 3) or 3C3) is only one: 123.
But the number of permutations of 3 choose 3 (3P3) is six:
So these two series are considered the same!
Combinations don’t care about order.
So this…
…is considered the same as all these.

25. Combinations Without Repetition
So how do we turn this into a combinations formula?
We know that the permutations formula is:
𝑛! 𝑛−𝑟 !
So to remove the choices we don’t want (the various orders 1,2,3 could also be put into but aren’t in combinations) we multiply it by the choices:
𝑛! 𝑛−𝑟 ! * 1 𝑟! = 𝑛! 𝑟! 𝑛−𝑟 ! = 𝑛 𝑟
We have now reached the reason for doing all this permutations/combinations stuff, and it is this formula,
called the Binomial Coefficient or often simply “n choose r”.

26. Combinations and the Binomial Coefficient
This formula, the Binomial coefficient…
𝑛! 𝑟! 𝑛−𝑟 ! or 𝑛 𝑟
is one of the most important in statistics because it allows you to calculate the probability of ‘r’ events occurring in ‘n’ trials.
For example, what are the odds of getting 5 heads out of 9 throws? A better example is this:
You sell real estate. 70% of people like houses and 30% like condos. What is the probability that you will sell 2 houses to the next 3 customers?
We’ll get to the answer in a little while, but hopefully you now see why combinations and permutations is important.
For now, we’ll finish up with combinations with repetition.

27. Combinations With Repetition
These are the hardest to calculate and certainly to understand, so I'm going to focus on calculation because the formula here is also used widely to solve the “combinations with repetition problem”.
If I gave you 31 ice cream flavours (your n) and said choose 3 (your r) and don’t worry about order or repetition (you could choose 3 of the same flavour) I would be telling you to choose r out of n+r-1.
It doesn’t matter why!
So the factorials formula for combinations with repetition is:
(𝑛+𝑟−1)! 𝑟! 𝑛−1 !

28. Permutations and Combinations Summary
This is what you’ve learned, in case you didn’t know.
Combinations and permutations are about choosing ‘r’ things from ‘n’ things (e.g. 6 #s from 49 #s).
Combinations don’t worry about the order of the ‘r’ things but permutations do.
Both combinations and permutations can allow or not allow repeats of the ‘r’ things.
The calculation formulas for both involves factorials and factoring.
Combinatorics is very important because it underlies Binomial and Poisson statistics.

29. Permutations and Combinations Models
Below are the formulas for calculating permutations and combinations, with and without repetition. Simply plug in the ‘r’ (size of the group you want), and ‘n’ (size of the population from which the group is being selected).
Combinations without repetition are also called the binomial coefficient.
Without
repetition
With
Permutation
𝑛! 𝑛−𝑟 ! nr
Combination
𝑛! 𝑟! 𝑛−1 !
(𝑛+𝑟−1)! 𝑟! 𝑛−1 !
Note that other notational formulas exist and say the same thing. The ones above are the simplest from which to calculate answers.

30. And Which Gives You The Largest ‘r’?
How many groups of 3 (r) objects can you choose from a larger group of 5 objects (n) if you used:
Permutation with repetition allowed:
125
Permutation without repetition allowed: 60
Combination with repetition allowed: 35
Combination without repetition allowed: 10
Commit this order to memory.

31. Good Online Combinatorics Calculators
You don’t actually have to calculate by hand of course because there are plenty of on line Combinatorics calculators available, and these are three of the best.
Combinations and Permutations Calculators:
(Good one for showing you the actual groups you get –within reason).

32. Fun With Permutations & Combinations – Lotto 6/49
Last example: how many sets of 6 numbers can you get out of out of 49 numbers if you don’t care about the order in which they come?
Answer is 13,983,816 million) and that’s what Lotto 6/49 is all about – solving for 49C6.
What if the order mattered – solving for 49P6?
That is, rather than OLGC rank ordering the six numbers that fall out of the machine, what if you had to get the order they fell in as well?
Answer is that there are 10,068,347,520 billion) ways 6 numbers can fall out of a machine with 49 numbers in it.

33. Binomial
And
Poisson
Distributions

34. Bernoulli Trial
A Bernoulli trial (or binomial trial) is an experiment whose outcome is random and can have either of two possible outcomes, "success" or "failure”.
In practice it refers to a single experiment which can have one of two possible outcomes. These events can be phrased into "yes or no" questions:
Did the coin land heads?
Was the fish caught a small mouthed bass?
Was the newborn child a girl?
Therefore success and failure are labels for outcomes, and should not be construed literally. The term "success" in this sense consists in the result meeting specified conditions, not in any subjective judgment.

35. Examples of Bernoulli Trials
Flipping a coin, where "heads" conventionally denotes success and "tails" denotes failure.
A fair coin has a probability of success of 0.5 by definition.
Rolling a die, where a six is "success" and everything else a "failure".
With a fair die the probability of “success” is 1/6th or 0.167
Conducting a political opinion poll to ascertain whether voters chosen at random will vote "yes" in an upcoming referendum.
Each voter has a 0.5 probability of voting yes and being a “successful” outcome, or no and not being a successful outcome.
Remember that “success” and “failure” are labels and have NO subjective connotation and should NEVER be used to suggest a favoured outcome.

36. Poisson and Binomial Distributions
The Binomial and Poisson distributions and the statistics that go with them are used very widely in research because they can answer the question “what's the chance that I will get ‘r’ events out of ‘n’ trials.”
Obviously, this is the same question you ask with combinations and permutations (and the math is similar), so hence knowing how to use them is important.
Here are some examples of research questions that require you to know about Binomial and Poisson distributions and calculations.
Hopefully you’ll see the usefulness of this type of stats and why they’re so widely used.

37. Research Questions Requiring Poisson or Binomial Statistics
You throw a fair die 9 times. What is the chance of getting 5 heads?
[Binomial: finite ‘n’ (9).]
What is the chance of getting at least half of 30 multiple choice questions correct by guessing?
[Binomial: finite ‘n’ (30).]
A road intersection has an average of 1 collision every 4 days. What is the probability of there being 2 collisions in 1 month?
[Poisson: unknown ‘n’ (average (called λ) =1 in 4/365)]
In a given lake there are twice as many Asian carp as small mouthed bass. Four fish are caught at random. What are the probabilities that none, 1, 2, 3, or all 4 will be small mouthed bass?
[Poisson, cumulative: unknown ‘n’ (4 of many)]
These are all solvable with binomial or Poisson statistics.
Poisson or Binomial distribution?
Read the following questions and decide whether the Poisson or the Binomial distribution should be used to answer it. Calculate the required probabilities. As a guide,
If a mean or average probability of an event happening per unit time/per page/per mile cycled etc., is given, and you are asked to calculate a probability of n events happening in a given time/number of pages/number of miles cycled, then the Poisson Distribution is used.
If, on the other hand, an exact probability of an event happening is given, or implied, in the question, and you are asked to caclulate the probability of this event happening k times out of n, then the Binomial Distribution must be used.
A typist makes on average 2 mistakes per page. What is the probability of a particular page having no errors on it?
A computer crashes once every 2 days on average. What is the probability of there being 2 crashes in one week?
Components are packed in boxes of 20. The probability of a component being defective is 0.1. What is the probability of a box containing 2 defective components?
ICs are packaged in boxes of 10. The probability of an ic being faulty is 2%. What is the probability of a box containing 2 faulty ics?
The mean number of faults in a new house is 8. What is the probability of buying a new house with exactly 1 fault?
A box contains a large number of washers; there are twice as many steel washers as brass ones. Four washers are selected at random from the box. What is the probability that 0, 1, 2, 3, 4 are brass?
Answers
We have an average rate here: lambda = 2 errors per page. We don't have an exact probability (e.g. something like "there is a probability of 1/2 that a page contains errors"). Hence, Poisson distribution. (lambda t) = (2 errors per page * 1 page) = 2. Hence P0 = 2^0/0! * exp(-2) =
Again, average rate given: lambda = 0.5 crashes/day. Hence, Poisson. (lambda t) = (0.5 per day * 7 days) = 3.5/week and n = 2. P2 = (3.5)^2/2! * exp (-3.5) =
Here we are given a definite probability, in this case, of defective components, p = 0.1 and hence q = 0.9 = Prob. not defective. Hence, Binomial, with n = 20. Expand (q + p)^20 to get q^ q^19 p + 20(20-1)/2! q^18 p^ No. faulty So P(2) = 20(20-1)/2! q^18 p^2 =
We have a probability of something being true and the same thing not being true; in this case, an ic being faulty. Hence, Binomial distribution. p = Prob. faulty = 0.02, q = Prob. not faulty = n = 10. Expand (q + p)^10 to get q^ q^9 p + 10(10-1)/2! q^8 p^ No. of faulty ics. So, Prob of a box containing 2 faulty ics P2 = 10(10-1)/2! q^8 p^2 =
Here we have an average rate of faults occurring: 8 per house. Hence, Poisson, with (lambda t) = (8 faults/house * 1 house) = 8. [1 house because we're only buying one new house.] n = 1 too, so P1 = 8^1/1! * exp(-8) =
Here too we have a probability of brass (1/3) and of not brass --- i.e. steel --- which is 2/3. Hence, use the Binomial distribution with p = 1/3, q = 2/3 and n = 4 to get (p + q)^4 = p^4 + 4 p^3 q + 6 p^2 q^2 + 4 p q^3 + q^ No. of brass so P(0) = (2/3)^4 = 0.197, P(1) = 4 (1/3)(2/3)^3 = 0.395, P(2) = 6 (1/3)^2 (2/3)^2 = 0.296, P(3) = 4 (1/3)^3 (2/3) = and P(4) =
Back to index.
January 2004

38. More examples: Learn how to switch out the specifics in these research questions to solve just about any problem requiring binomial or Poisson statistics.
Example 1: For security purpose, a bank is interested in studying the number of people who use a particular ATM late at night. On average, 1.6 customers walk up to the ATM during any 10 minute interval between 9pm and midnight. What is the probability of exactly 3 customers using the ATM during any 10 minute interval? What is the probability of 3 or fewer people? Example: 2 Wal-Mart is trying to cut down on overtime among its cashiers by ensuring they do not have long line ups before a shift change. The store policy is to close checkouts 15 minutes before a shift ends (in this case at 4:45) so that the cashier can finish checking-out customers already in their line and leave on-time. By examining overhead cameras, store data indicates that between 4:30pm and 4:45pm each weekday, an average of 10 customers enter any given checkout line. What is the probability that exactly 7 customers enter a line between 4:30 and 4:45? What is the probability that more than 10 people arrive?
Example: 3 Suppose a fast food restaurant can expect two customers every 3 minutes, on average. What is the probability that four or fewer patrons will enter the restaurant in a 9 minute period?
Example 4: The Ontario Ministry of Transportation is concerned about the number of deer being struck by cars between Huntsville and Bracebridge. They note the number of deer carcasses and other deer-related accidents over a 1-month period in 2-mile intervals. What is the probability of zero deer strike incidents during any 2-mile interval between Huntsville and Bracebridge?

39. Normal, Poisson & Binomial Distributions
A normal distribution describes continuous data which have a symmetric distribution with a characteristic 'bell' shape.
A binomial distribution describes the distribution of binary data (successes and failures) from a finite sample. Thus it gives the exact probability of getting ‘r’ events out of ‘n’ trials because you know what size ‘n’ is.
A Poisson distribution describes the distribution of binary data (successes and failures) from an infinite sample. Thus it gives the estimated probability of getting ‘r’ events in a population because you don’t know what size the population is.
Binomial and the Poisson distributions are called BINARY because they are concerned with the number of ‘successes’ and ‘failures’ you get from a series of ‘trials’.

40. Poisson & Binomial Distributions Attributes
A binomial distribution:
Comes from a finite sample size.
Describes the distribution of binary data.
Because data is discrete it uses a probability mass function.
Gives the exact probability of getting ‘r’ events out of ‘n’ trials.
A Poisson distribution:
Comes from an infinite sample size.
Gives an estimated probability of getting ‘r’ events in a population.
A normal distribution:
Describes the distribution of continuous data.
Because data is continuous it uses a probability density function.
Gives the estimated probability that calculated differences are statistically significant.

41. When to Use Binomial or Poisson Statistics
Binomial and Poisson are the most commonly used statistical tools.
Poisson is used for any research problem where you are using the average number of occurrences of an event in a given unit of time, space, area, distance, or volume (called λ or lambda) such as car accidents at an intersection or zebra mussels per volume of water.
If the events occur independently and the probability (p) of the event is constant in time and/or space, then ‘x’, the number of events in a fixed unit of time/space, has a Poisson distribution.
For any research problem that involves identical trials where each trial has only two possible outcomes (success or failure), and each trial is independent of the previous trial, and where the probabilities of success or failure on any given trial are constant and q=(1-p), then you can use a binomial statistic.

42. When Binomial and Poisson Are the Same
The binomial and Poisson distributions converge when the binomial n approaches infinity, p approaches 0, and np stays constant.
Thus, the binomial and Poisson statistics closely approximate one another if n is large and p is small.
The Poisson distribution is generally used as an approximation of the true underlying reality and is most useful when ‘n’ is undetermined, which can also mean when ‘n’ is very large and the p of an event is extremely rare.
It is usually quite difficult to ascertain whether a random variable has a Poisson distribution and thus, whenever possible, the binomial statistic is preferred.

43. Fun With Binomials

44. Yours to Discover
The following 15 slides are examples of binomial and Poisson experiments in action; types of problems and calculations.
You should review these because next term in inferential stats we will be using binomials and Poisson statistics so will return to them.
For now, well, we are done!

45. (Your) Statistics At Work
Overall Course Grades 2014 to 2019
Mean All
Mean Quiz
Mean Labs
2019
68.9
66.2
72.3
2018
67.4
64.4
71.2
2017
71.3
68.8
74.3
2016
63.2
64.1
62.2
2015
65.0
60.1
2014
57.8
51.5
65.8
65.6
62.5
69.5 s
4.7
6.1
4.6
Mean of All Years 65.5%
z Scores of Course Grades 2014 to 2019
YEAR
Mean All
Mean Quiz
Mean Labs
2019
0.70
0.61
2018
0.42
0.31
0.45
2017
1.24
1.03
1.16
2016
-0.48
0.26
-1.60
2015
-0.11
-0.40
0.44
2014
-1.65
-1.80
-0.78
-0.14
-0.12

46. Remember:
It takes only 1 person to make a statistically insignificant difference.
(hahahahahah)

47. I’m 95% sure this is The End ± maybe