Statistics and Distributions

Question 1

Distributions

Answer 1

• Representation of the way values tend to vary across a single attribute
• Usually presented as a histogram
• Where is the data concentrated? Which values are less likely? Which is most likely?

Question 2

Which single value best represents the data?

Answer 2

Central Tendency
Context dependent
- On a histogram: affects the location on the x-axis

Question 3

Mean

Answer 3

arithmetic mean:
sum of values/number of values

Question 4

Median

Answer 4

Middle value of sorted data
- Resistant to outliers and skew

Question 5

Variability

Answer 5

How far does the data spread away from the mean?
Affects the width of the histogram

Question 6

Standard Deviation

Answer 6

This is the average distance from the mean
If we pick a random value from the data, how far should we expect it to be from the mean?

sd = sqrt(sum(x-mu)^2 / N)

Question 7

Percentiles and Quartiles

Answer 7

25th Percentile : 1st Quartile
50th Percentile : 2nd Quartile
75th Percentile : 3rd Quartile

Question 8

IQR and Outliers

Answer 8

Interquartile Range : Q3-Q1
Lower/Upper Fences: [Q1 - (3/2) * IQR, Q3 - (3/2) * IQR]
Outlier: A value that falls outside of the fences.

Question 9

Boxplots

Answer 9

Excellent tool to display and compare measures of variability

They display:
- Median
- IQR
- Fences
- Outliers
- Range

Question 10

Normal Distribution

Answer 10

• Gaussian Distribution or Bell Curve
  Fundamental to statistics
  Countless occurrences in nature
  Has a number of useful properties

Question 11

Normal Distribution Properties

Answer 11

1. Symmetric
   Mean = Median = Mode
2. 68-95-99 Rule
3. Foundation of the Central Limit Theorem

Question 12

Random Experiment

Answer 12

A process that results in an outcome

Question 13

Outcome

Answer 13

The value of the result of a single experiment

Question 14

Sample Space

Answer 14

The set of all possible outcomes for an experiment

Question 15

Event

Answer 15

A subset of the sample space

Question 16

Probability

Answer 16

A number between 0 and 1 that dictates the chance of an event occurring

Question 17

Sample Space

Answer 17

A sample space of an experiment is the set of all possible outcomes
Ex: Sample Space of a single die roll is: {1, 2, 3, 4, 5, 6}

Question 18

Event

Answer 18

An event usually denoted by a single capital letter, is a subset of the sample space.
Ex: If you roll two dice, some possible events include:
- (1,1), (1,2), (2,1), (1,6), (6,6)

Question 19

Probability

Answer 19

For a single event A, the probability of A occurring, P(A), is denoted as:

P(A) = number of outcomes in which A occurs/ total possible outcomes

Question 20

Addition Rule

Answer 20

Addition Rule states:
P(A or B) = P(A) + P(A) - P(A and B)

Question 21

Multiplication Rule

Answer 21

Two events are said to be independent if the outcome of one does not depend on the outcome of the other. Otherwise, they are dependent.

The multiplication rule states:
P(A and B) = P(A) * P(B, given that A occurred) = P(A) * P(B|A)

For independent events, this is simply:
P(A and B) = P(A) * P(B)

Question 22

Complements

Answer 22

P(A) + P(not A) = 1

Question 23

Deterministic Sampling

Answer 23

Rather than randomizing, you take the first people that walk by or choose the people deterministically

Question 24

Uniform Random Sampling

Answer 24

Use software to assign and pick off an n’th group of people to choose

Question 25

Random Sampling

Answer 25

Randomly select

Question 26

From random sampling, what do we know about the sample mean?

Answer 26

The sample mean is the mean of the data sampled, and approximates the true mean.

Question 27

Probability Distribution

Answer 27

the calculated likelihood of each possible event occurring without simulation or conducting the experiment

Question 28

Empirical Distribution

Answer 28

the proportion of times a value is observed in a simulation or experiment, relative to the number of possible values

Question 29

Law of large numbers

Answer 29

As our sample size grows larger, the data represents the population more accurately

Question 30

statistic

Answer 30

a calculated number which describes a characteristic of a sample

Question 31

parameter

Answer 31

value that estimates a characteristic of a population

Question 32

statistical inference

Answer 32

a conclusion made based on data from multiple random samples.

Question 33

Central Limit Theorem

Answer 33

This theorem states: Upon taking sufficiently large samples, the distribution of the sample means will approximate a normal distribution, regardless of the distribution sampled from.