correlation.pptx

Applied Techniques for Economists
Josphat Omanga.
Copenhagen Business College.
1

Quantitative Methods, 4th edition, Swift & Piff
Describing Data Toolkit
Displays for numerical data:
• Frequency charts
• Relative frequency
• Histogram
Displays for categorical data:
• Bar charts
• Pie charts
• Contingency tables
Quantitative (numerical) data
• Mean
• Median
• Mode
• Range
• Variance
• Standard deviation
• Quartiles
• Index
Summarising Data Toolkit
Recap
Location
Spread
2

Main Objectives
Scatter Plot
Measures of Association
a) Covariance
b) Correlation
Contingency Table & Dependency/Independency
3

Why analyse two (or more) variables together ?
The main aim is to examine whether there is any pattern between
the responses of two or more variables:
- To understand the relationship better;
- To impact one variable by changing the other;
- To forecast;
Depending on the type of variable:
• Quantitative (numerical): scatter plot, covariance, correlation
coefficient
• Qualitative (categorical): contingency table and dependency
4

Scatter plot
As briefly mentioned before in Data and Graphing Data, a scatter plot can
show the relationship between two variables.
Returns of two shares in 9 consecutive months
5

Scatter plot examples
6

Scatter plot examples
7

Scatter plot
The owner of an ice cream store wants to
examine the relationship between daily sales
and atmospheric temperature.
A sample of 25 consecutive days is selected and
the data of consumption of ice cream per head
(in pints) and average temperature (in
Fahrenheit) is recorded as follows.
8

Scatter plot
• Does a relationship between daily sales and temperature exist?
• What kind of relationship exists?
• How to measure this relationship (if it exists)?
The distribution of the “cloud”
of data indicates a positive
relationship.
How to be more precise on
measuring this relationship?
9

Covariance
• Covariance is a measure of the linear relationship between two variables.
• A positive value indicates a direct or increasing linear relationship, while a
negative value indicates a decreasing linear relationship.
The equation for calculating the covariance is defined as:
Note that this is the formula for sample covariance, and for population covariance it
is divided by n. 10

Covariance
e.g. Data (no. of observations n=5 )
(smoking years) x: 0 5 10 15 20
(lung capacity) y: 45 42 33 31 29
𝑥 =
0 + 5 + 10 + 15 + 20
5
= 10
𝑦 =
45 + 42 + 33 + 31 + 29
5
= 36
𝐶𝑜𝑣(𝑥, 𝑦) =
0 − 10 ∗ 45 − 36 + 5 − 10 ∗ 42 − 36 + 10 − 10 ∗ 33 − 36 + 15 − 10 ∗ 31 − 36 + 20 − 10 ∗ (29 − 36)
𝑛 − 1
=
−10 ∗9 + −5 ∗6+0∗ −3 +5∗ −5 +10∗(−7)
5−1
=
−90−30+0−25−70
4
=
−215
4
= −53.75
11

Covariance
(lung capacity) y: 45 42 33 31 29
𝐶𝑜𝑣(𝑥, 𝑦) = −53.75
12

Practice question 1:
13
Person Age Days off work last year
A 19 21
B 21 18
C 32 15
D 27 17
E 45 8
The table below is the sickness records:
(a) Calculate the covariance of two variables according to question above.
- Compute it manually using the formula
- Compute it in Excel using spreadsheet.
- Step by step
- Using Excel formula

Covariance Properties
Note that when cov(x, y) = 0 , it indicates that these two variables are independent
(no relationship), whilst positive/negative relationship indicates dependent.
14

Correlation Coefficient
Covariance is a very useful index to verify if two variables are independent or not.
However, it fails to indicate how strong is this relationship.
Correlation coefficient is kind of a normalized index to measure the association.
The equation for calculating the correlation coefficient is defined as:
same as
15

Correlation Coefficient VS Covariance
The Correlation Coefficient has advantages over covariance
for determining strengths of relationships:
• Covariance can be any number with corresponding unit, while a
correlation coefficient is unit free and its range is limited between -1
to 1.
• Correlation is comparable index and it is useful for determining how
strong the relationship is.
16

Correlation Coefficient
(lung capacity) y: 45 42 33 31 29
𝑥 =
0+5+10+15+20
5
= 10 𝑦 =
45+42+33+31+29
5
= 36
𝑟 =
0 − 10 ∗ 45 − 36 + 5 − 10 ∗ 42 − 36 + 10 − 10 ∗ 33 − 36 + 15 − 10 ∗ 31 − 36 + 20 − 10 ∗ (29 − 36)
(0 − 10)2+(5 − 10)2+(10 − 10)2+(15 − 10)2+(20 − 10)2∗ (45 − 36)2+(42 − 36)2+(33 − 36)2+(31 − 36)2+(29 − 36)2
=
−10 ∗ 9 + −5 ∗ 6 + 0 ∗ −3 + 5 ∗ −5 + 10 ∗ (−7)
100 + 25 + 0 + 25 + 100 ∗ 81 + 36 + 9 + 25 + 49
=
−90 − 30 + 0 − 25 − 70
250 ∗ 200
=
−215
50000
= −0.9615
17

Correlation Examples
• The closer to -1, the
stronger the negative
linear relationship
• The closer to 1, the
stronger the positive
linear relationship
• The closer to 0, the
weaker the linear
relationship
18

Practice question 1 (cont.):
19
Person Age Days off work last year
A 19 21
B 21 18
C 32 15
D 27 17
E 45 8
The table below is the sickness records:
(a) Calculate the correlation of two variables according to question above.
- Compute it manually using the formula
- Compute it in Excel using spreadsheet.
- Step by step
- Using Excel formula

The contingency table can be used to study the relationships that may exist between
two qualitative variables.
i.e. The contingency table that was briefly introduced in Data and Graphing Data
20

The case of Titanic
Many well-known facts are reflected in the survival rates for various classes of
passenger. The British Board of Trade originally collected the data regarding
passengers in their investigation of the sinking, which makes it possible to
verify, i.e., if the “women and children first” or the “first-class passengers first”
policies have been entirely followed during the saving operations.
Here we will look into the “first-class passengers first” policy.
21

The case of Titanic
The data contain two qualitative variables: the class where each passenger was
travelling and if the passenger has survived or not to the disaster. (note that
unfortunately, no complete agreement among primary sources as to the exact
numbers on board, rescued, or lost.)
Just part of the long list of survivors:
22

The case of Titanic ---- univariate description by table and graphs
23

The case of Titanic ---- contingency table
By frequency By relative frequency
24

25

26

The case of Titanic ---- contingency table ---conditional relative distribution
Three conditional relative distribution
of “survived (X)” given “class (Y)”,
normally written as X|Y
Two conditional relative distribution
of “class (Y)” given “survived (X)” ,
normally written as Y|X
27

The case of Titanic ---- contingency table ---conditional relative distribution
28

Why not just the frequencies?
Assume that
29

The Maths and Notations – Boring but Necessary
30

31

32

The case of Titanic
33

The case of Titanic
34

Make sure that you can
• Calculate the covariance and correlation coefficient;
• Understand their differences and properties;
• Conduct a comprehensive association analysis on quantitative variables
by adopting scatter plot, covariance, correlation with corresponding
interpretations;
• Understand the use of contingency table on dependency/independency
analysis for qualitative variables.
35

correlation.pptx

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