Summary containing all the relevant theory discussed during the lectures of the course Business Statistics given in the first year of International Business Administration at the Vrije Universiteit Amsterdam. By learning this summary I personally passed the final exam.
Lecture 1: Data, visuals and descriptives
The data matrix or data frame:
Data are put into a Data Matrix or Data Frame (Excel sheet)
- Columns: variables
- Rows: subjects/cases
- Cells: observations of a variable for that specific subject/case
Data types and example:
Determining the measurement level:
Missing data:
Missing data can be dealt with in various ways in
statistical analysis
- Delete missing cases: easy, but loses information
- Impute (cleverly guess) missing cases: for
instance,
o by filling out the mean income if income is
missing
o by filling out the most frequent video
category (if category is missing). This
retains more observations / cases, but
hinges on the correctness of the
imputation assumptions
,Population vs sample:
The population is the collection of all possible data points: typically, we do *not* have it! (e.g.,
the population of ALL 1st year VU business students)
A sample is a subset of data taken from the population. (e.g., the students present today in
this session are a sample of all VU 1st year business students)
- We use this sample to infer something about the population:
o e.g., is there sufficient support for increasing expat subsidies under low-
income residents
o A sample always has an aspect of randomness to it: it could have been a
different sample
Categorical data:
#occurrences
-Summary measures for categorical data: Proportion: 𝑝 = 𝑛
-Sample proportion = p, population proportion = , population size = N
-Skewness is a measure of asymmetry
-Kurtosis is a measure of tail flatness/fatness → if kurtosis is large, more outliers/huge
outcomes compared to normal cases
Correlation is a standardized (scale free) analogue of the covariance: both should
have the same sign.
, Lecture 2: Probability
Event: A is an event (A’ denotes not event A)
Examples: event A can be “heads” in a coin toss (and A’ is then “tails”), or A can be throwing
4 with a fair dice, or having a goal outcome (149,0)
- An event must be inside the sample space, otherwise it cannot occur (it will have
probability zero; in a coin toss throwing “telephone” is impossible)
Probability: P(A) the probability of event A
Notation:
-𝑃 (𝐴 ∪ 𝐵) means probability of either A or B or both A and B happening
-𝑃 (𝐴 ∩ 𝐵) means probability of both A and B happening jointly
-Disjoint: events A and B are disjoint if they cannot happen at the same
time (i.e., probability of A and B together is zero, or 𝑃 𝐴 ∩ 𝐵 = 0)
𝑃(𝐴) 1−𝑃(𝐴)
-Odds for 𝐴: 1−𝑃(𝐴)
; odds against 𝐴: 𝑃(𝐴)
• General law of addition: 𝑃 (𝐴 ∪ 𝐵) = 𝑃 (𝐴) + 𝑃 (𝐵) − 𝑃 (𝐴 ∩ 𝐵)
• Conditional probability: 𝑃 (𝐴 |𝐵) = 𝑃(𝐴 ∩ 𝐵)/𝑃 (𝐵)
• General law of multiplication: 𝑃 (𝐴 ∩ 𝐵) = 𝑃 (𝐴|𝐵) 𝑃 (𝐵) = 𝑃 (𝐵|𝐴) 𝑃(𝐴)
Types of probability:
-Classical: P (event) =
number of elementary outcomes in event
number of possible elementary outcomes
-Empirical: P (event) =
number of elementary outcomes in event
number of observations
Important properties of a probability function P(A)
- For every event A in the sample space: 0 P(A)1
- For entire sample space S, we have P(S) = 1: the probability of obtaining some
outcome out of the set of all possible outcomes is 1
- For disjoint events A and B, we have we have 𝑃 (𝐴 ∪ 𝐵) = 𝑃 (𝐴) + 𝑃 (𝐵)
- However, if events are not disjoint, then 𝑃 (𝐴 ∪ 𝐵) = 𝑃 (𝐴) + 𝑃 (𝐵) - 𝑃 (𝐴 ∩ 𝐵)
The complement of an event 𝐴 is denoted by 𝐴% and consists of everything in the sample
space 𝑆 except event 𝐴 → Since 𝐴 and 𝐴% have no overlap and together comprise the entire
sample space 𝑆, 𝑃 (𝐴) + 𝑃 (𝐴’) = 1 or 𝑷 (𝑨’) = 𝟏 − 𝑷(𝑨)
-The empty set denoted as ∅ contains no elements: 𝑃 ∅ = 0.
𝐴∪B
-The union of two events consists of all elementary outcomes in the
sample space that are contained either in event 𝐴 or in event 𝐵 or in
both
- denoted by 𝐴 ∪ B
- pronounced as “𝐴 or 𝐵” (“or” meaning here “and/or”)
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