or
Observations cases form the rows; common attributes or variables form the
columns
Categorical Data
Also called qualitative or nominal variables
Identify group membership
Numerical Data
Also called quantitative or continuous variables
Describe numerical properties of cases
Have measurement units
Size of bike (cm) and Amount spent ($) are examples
Measurement Scales
Nominal name categories without implying order (categorical). E.g. gender.
Ordinal name categories that can be ordered (categorical). E.g. the good, the
bad and the ugly.
Interval numerical values that can be added or subtracted (arbitrary
zero).E.g. temperature.
Ratio numerical values that can be added, subtracted, multiplied or divided
(makes ratio comparisons possible, zero is not arbitrary). E.g. height.
Likert Scale (Ordinal 5 to 7 Categories)
Recode: building a new variable from another (recoding price into expensive
or inexpensive)
Aggregate: reduce rows in a data table by counting or summing values within
categories
Time series data recorded over time
Timeplot graph of a time series showing values in chronological order
Frequency regular time spacing of data in a time series (e.g., daily, monthly,
etc.)
Cross-sectional data observed at the same time
The distribution of a categorical variable is a list of values with its associated
count (frequency)
A frequency table summarizes the distribution of a categorical variable
A relative frequency table shows the proportion (or percentage) in each
category
Bar Chart
Uses horizontal or vertical bars to show the distribution of a categorical
variable
Is called a Pareto chart when the categories are sorted by frequency (popular
in quality control)
Becomes cluttered with too many categories
Is appropriate for ordinal categorical variables
The Pie Chart
Uses wedges of a circle to show the distribution of a categorical variable
Median
Mode
Commonly chosen to illustrate market shares or sources of revenue for a
company
Less useful than bar charts if we want to compare actual counts (easier to
compare bars than angles of wedges)
Not appropriate for nominal data
Data must be ordinal
It is the category label of the middle observation in ordered data
Grades of 13 students B,C,A,A,A,C,C,B,B,B,B,A,D. Which is the median
grade? What if there were 12 students?
Category with the highest frequency
The longest bar in a bar chart
The widest slice in a pie chart
Two or more categories can tie with the highest frequency (bimodal or
multimodal)
Histograms
The distribution of a numerical variable is collection of possible values and their
frequencies.
Plot the distribution of a numerical variable by showing counts of values
occurring within adjacent intervals (similar to bar charts, but for continuous data)
A histogram with all bars about the same height is uniform
Symmetry and Skewness
A distribution is symmetric if the two sides of its histogram are mirror images
A distribution is skewed if one tail of the histogram stretches out farther than the
other
The Interquartile Range (IQR), a measure of spread:
A measure of variation based on quartiles
The Range
Range = Maximum - Minimum
Maximum Value = 100th Percentile
Minimum Value = 0th Percentile
Another measure of variation; not preferred because based on extreme values
The Five Number Summary
Minimum
Lower Quartile
Median
Upper Quartile
Maximum
The Variance (s2)
Is a measure of variation based on the mean.
Is a measure of risk for financial analysts. Lower is better.
How far a value is from the mean is known as its deviation; the variance is the
average of the squared deviations.
The Standard Deviation (SD)
Is the square root of the variance
Is a measure of variability in the original units of the data (the variance results in
squared units)
Bell-Shaped Distributions and Empirical Rule
A bell-shaped distribution is symmetric and unimodal
The empirical rule uses the standard deviation to describe how data with a bellshaped distribution cluster around the mean
Standardizing
Z- scores measure distance the from the mean in standard deviations
Z-scores are dimensionless and help identify the relative (to the mean) of each
observation.
The Law of Large Numbers (LLN)
The relative frequency of an outcome converges to a number, the probability
of the outcome, as the number of observed outcomes increases.
Note: The pattern must converge for LLN to apply.
Venn diagrams are graphs for depicting the relationships among events
The probability of an event A is denoted as P(A).
Sample Space
Set of all possible outcomes
Subsets of samples spaces are events; denoted as A, B, etc.
Rule 1: Since S is the set of all possible outcomes, P(S) = 1
Rule 2: For any event A, 0 P(A) 1.
Rule 3: Addition Rule for Disjoint Events
Disjoint events are mutually exclusive;
i.e., they have no outcomes in common.
The union of two events is the collection of outcomes in A, in B, or in both (A
o r B)
If A and B are disjoint events, then
P (A or B) = P(A) + P(B).
Rule 4: Complement Rule
The complement of event A consists of the outcomes in S but not in A
Denoted as Ac
P(A) = 1 P(Ac)
General Addition Rule
The intersection of A and B contains the outcomes in both A and B
Denoted as A B read A and B
: General Addition Rule
P (A or B) = P(A) + P(B) P (A and B).
Two events are independent if the occurrence of one does not affect the
chances for the occurrence of the other
Two events A and B are independent if the probability that both A and B
occur is the product of the probabilities of the two events.
P (A and B) = P(A) X P(B)
Booles Inequality
Also known as Bonferronis inequality
The probability of a union is less than or equal to the sum of the probabilities
of the events.
Conditional Probability
We calculated P(A | B) in the following fashion:
P(A | B) = P(A B) / P(B)
The general multiplication rule for two arbitrary events is
P(A B) = P(B) * P(A | B)
= P(A) * P(B | A)
As a special case, if two events A and B are independent i.e. they do not
influence each other, then P(A | B) = P(A) and therefore P(A B) = P(A)*
P(B).
Bayed theorem ;
P( A | B) =
P( B | A) * P( A)
P( B | A) * P( A) + P( B | Ac ) * P( Ac )
Random variable: Takes on different numerical values determined by the
outcome of an experiment
Discrete random variable: Possible values can be counted
Continuous random variable: May assume any numerical value in one or
more intervals
Continuous random variable: May assume any numerical value in one or
more intervals
If X is a discrete RV, expected value is
E(X) is a weighted average of possible values of X. Weights are probabilities
of those values occurring.
E(X) is the long run average value of X if experiment is repeated many
times. May be thought of as the theoretical mean of the random variable X.
E(X) is mean of population of values of X obtained over many repetitions of
the experiment.
E(X) = .
Variance of X is expected value of the RV (X )2.
V(X) = Mean Squared Deviation of X from its mean, .
Standard Deviation (SD): Standard deviation of X is
measures amount of fluctuation in X over a large number of repetitions of
an experiment.
E(X c) = E(X) c where c is a constant
E(c*X) = c*E(X)
V(X c) = V(X)
SD(X c) = SD(X)
V(c*X) = c2 * V(X)
SD(c*X) = c*SD(X)
Normal Distribution
1. f (x) has a bell shape, is symmetrical about , and reaches its maximum at .
2. and determine center and spread of distribution.
3. Tails, or ends, of curve extend out to and never actually touch zero.
4. Empirical rule holds for all normal distributions:
68% of area under curve lies between ( , + )
95% of area under curve lies between ( 2 , + 2 )
99.7% of area under curve lies between ( 3 , + 3 )
Normal Distribution
Normal random variable with = 0 and 2 = 1 is said to have a standard
normal distribution.
There are infinitely many normal distributions, there is only one standard
normal distribution.
Standard normal distribution has been standardized:
center is at 0
spread (distance from center to inflection points) is 1
often denote a standard normal RV by Z
x
z=
Z-scores measure how many SDs X is from its mean.