QUANTITATIVE METHODS FOR SOCIAL SCIENCES
• types and dimensions of data (i.e. nominal, ordinal and cardinal variables)
- different types of data (variables and observations)
- types of variables (quantitative and categorical)
• univariate statistics
▪ frequency tables + statistical indexes to summarise data for a variable (mean, median mode)
▪ measures of dispersion (i.e. variance and standard deviation)
▪ measures of distribution (e.g. percentiles)
▪ graphing (i.e. histograms, pie charts, trend lines)
▪ new trends in infographics (e.g. Statista)
• bivariate statistics
▪ contingency tables
▪ Pearson and Spearman correlations indexes
• simple regression analysis
▪ linear relationship between two variables
▪ OLS regression analysis
• basic concepts of sampling distribution and inference analysis
▪ samples and population
▪ probability and probability distribution
▪ sampling distribution
▪ inferential statistics
point estimate and confidence intervals
testing hypothesis
3/10/19
What is statistics
Statistics is the science of designing studies and analyzing the data that those studies produce. Its ultimate goal is
translating data into knowledge and understanding of the world around us
• uses math formulas to generate synthetic indexes
• uses graphs to visualize data in order to describe socio-economic facts using real data (= descriptive statistics)
to make decisions or predictions about “population” (= inferential statistics)
• Its goal is translating data into knowledge and understanding of the world around us
Why should we study statistics in IRGA program?
1st reason:
If we consider international “hot” topics (ex. migration) that are not just a matter of Italy there’s statistics: we try to
read numbers and create indexes / compare them to understand social-economic phenomena (=descriptive statistics)
→table representing the number of international migrants in millions in the world
The number of migrants increased in17 years by 1.5 (𝑟𝑎𝑡𝑖𝑜 258/172=1.5)
Destinations are different:
▪ in 2000 60% [(103/172)x100] of migrants preferred more developed areas and 40% less developed areas.
▪ In 2017 percentages changed: 57% in more developed areas and 43% in less developed areas.
In 2017 3.4% of total population is involved in migration but
the weight on the total population in different areas is
differentiated: 11.6% versus 1.8% = some areas are affected
more and some less by migration.
→Which kind of policies should we improve in a particular
country?
, 2nd reason:
In order to compute and “infer” the relationship between variables
→this graph shows there’s a negative relation between the
emigration rate and the GDP of each country: people migrate if the
country is poor, otherwise if the GDP per capita is quiet high, the
migration in low
In order to predict the results of the political elections, dealing with a “margin of error” (Ex. “The good wife”)
→do exit pools and try to understand who’ll win political elections using
probabilities and some margin of error.
In this graph we see the percentage of people who are going to vote for
republicans and democrats, what to worry about? White women, because
it’s not sure who are they going to vote for
4/10/19
Statistical method
It helps us to investigate on an objective manner and to figure out how to apply our indexes in order to summarise
information and have some hints on the answer to our research question
Statistical problem solving is an investigative process that involves 4 components:
1) Formulate a research question: arises not from statistics but from other disciplines (economics, political science,
sociology, history…)
Once you design your research question plan how to obtain the data in order to answer it
2) Collect data: they are available most of the time on the web, but in some cases data are fake and need to think how
to collect true data.
(In order to do this be careful to Wikipedia and try to use international reliable databases (IMF, WB, national
statistical offices – Istat-, Eurostat…) the problem with this data is that they’re organised in a different way and need
to learn to assess them; also surfing in google public data it is possible to get the data needed)
Otherwise you can formulate your own research question and gather your own data (questionnaire ex.)
3) Analyse data: summarise and interpret them according to the research question (describe what’s going on)
4) Interpret your results: Do an inference analysis = try to understand what happens if something is changing
Complementary ways of analysing data
Three main components of statistics for answering a statistical question:
- Design: planning how to obtain data
- Description: means summarizing the data, files of raw data are often huge and not easy to assess SOO it is
more informative to use a few numbers or a graph to summarize data
- Inference: means making decisions or predictions based on the data; usually the decision or prediction refers to
a larger group of people, not merely those in the study
“infer” = to arrive at a decision or prediction by reasoning from known evidence
Complementary ways of analysing data: can use both to investigate questions which are important for society
,Statistical description (descriptive statistics)= a picture of what is going on 1st semester
▪ refers to methods for summarizing the collected data using graphs and numbers such as averages and
percentages
▪ simplifies data without distorting or loosing information
▪ can be done for sample and for population (w/census)
It’s much easier to get a sense of the data by looking at a graph than by reading through the questionnaires filled out by the
50,000 sampled households.
Inference analysis (inferential statistics) = a guessing 2nd semester
- making decisions or predictions about a population, based on data obtained from a sample of that population
- deals with probability= quantifying how likely various possible outcomes are
- important aspect of statistical inference involves reporting the likely precision of a prediction within a range
(how close is the sample value of % likely to be to the true percentage of the population: inferential
statistical analyses can predict characteristics of entire populations quite well by selecting samples that are
small relative to the population size.
For example, the population of China is about four times that of the United States, but a random sample of 1000 people from the
Chinese population and a random sample of 1000 people from the U.S. population would achieve similar levels of accuracy)
-> statistics consists of methods for designing investigative studies, describing (summarizing) data obtained for those
studies, and making inferences (decisions and predictions) based on those data to answer a statistical question
of interest.
What we can investigate
The entities we measure in a study are called subjects
• Individuals (want to see the average weight of this class)
• Schools (average performance of schools to see what’s the best school to go)
• Countries (evaluate GDP)
• Days: quiet recent topic, evaluate the probability of raining (ex.) and guess if in a specific day in a specific area is going to rain
•whatever we have in our mind
To our subjects (our unitof analysis) we must attach data (which are different features of our unit of analysis)
Population = total set of subjects in which we are interested, once you try to summarise this information you are
computing the “parameter” (statistical index that summarises what’s going on to the whole population)
usually too costly and time-consuming to obtain data related to all population
Sample = the subset of the population for whom we have data, the idea is that the subsample should be randomly
selected. Then statistics (number) tells you what’s going on with your sample (=only related to the subset)
ex. want to see average age of people in Milan, in order to get the parameter of this population, need to investigate the age
of every single person who’s living in Milan (quite costly).
BUUT there’s also the opportunity to have a sample by randomly select some subjects which represent a part of the milanese
population and once you get this information you come up with your statistics (= a number that’s summarising what’s going
on to the average age of the population that’s living in Milan)
Randomness and variability
Randomness is an extremely powerful tool for obtaining good samples and conducting experiments. A sample tends to
be a good reflection of a population when each subject in the population has the same chance of being included in
that sample. That’s the basis of random sampling, which is designed to make the sample representative of the
population.
Random sampling:
- allows us to make powerful inferences about populations
- is crucial to performing experiments well
10/10/19
Definitions
A variable is any characteristic observed/being measured in a study.
, The data are values that we observe for a variable, they are called observations; each observation can be a number or
may belong to a category.
A variable can be:
- Categorical, if each observation belongs to one of a set of categories (we count and we order data)
A key feature to describe categorical variables: the relative number of observations in the various categories
Ex. frequency table of shark attacks in various regions: each observation (shark attack) identifies a “region” where
the attack occurred -> region is the variable and it is categorical with categories representing regions of the world
- Quantitative/numerical, if observations take numerical values that represent different magnitudes of the
variable they measure “how much” of something”
Do you like
Final
Surname Vegetarian sports? (0=no,
ID Gender Weight (kg) mark in
name (yes=1) 1=quite, 2=very
QM
much)
1 Blue John male 1 1 88.7 30
2 Green Dana female 0 2 81.2 26
3 Yellow Zeb male 1 1 86.8 24
White
4 Robert male 0 2 73.21 19
5 Black Susan female 0 0 60.9 27
->table with columns and rows which identify different themes
Rows: represent single observation - Columns: represent a particular value
▪ ID: identifies each observation of my «sample»
ID is unique and allows us to manage possible homonymies if our samples include people’s names
▪ Surname and Name: identifies each student
▪ for each observation Gender identifies if the student is “male” or “female”.
This is a “category” that could be converted into a number to be better treated (0 for male and 1 for female), but it
willalways remain a category and this codifying procedure doesn’t attach a particular value to numbers ( 0 & 1 in
this case)
->“male” = 0 and “female” = 1 doesn’t mean that we are evaluating male and female, the number attached to this
category means nothing
Gender = Categorical variable.
▪ for each observation Vegetarian identifies if the student has “vegetarian” tastes or “not”.
This variable is a “category” already converted into a number (0 for not veggie and 1 for veggie).
Vegetarian taste = Categorical variable
▪ This variable identifies how much the student appreciates football. Each student could evaluate his/her preferences
from 0 to 2.
Preferences on Football = Categorical variable already converted into a number
▪ Weight (kg) and Final Mark in QM are numerical/quantitative variables associated to each observation and we
can compare them
BUUT Weight are “continuous” numbers and Final Mark in QM is “discrete”
A quantitative variable is discrete if its possible values form a set of separate numbers, such as 0, 1, 2, 3, . . .
(e.g. number of books in a shelf)
A quantitative variable is continuous if its possible values form an interval, such as 1.1, 1.2, 1.205, 4.37, …
(e.g. share of people living below poverty line in African countries)
Which kind of analysis we can run if we know that we have variables that are very different
In fact, both categorical and quantitative variables can be translated into
• Nominal variables
Categorical variable that has two or more categories but there is no intrinsic ordering of them.
ex. gender (2 categories) :0=male; 1=female;
religion belief (more than 2 categories): 1 = Catholic; 2 = Protestant; 3 = Muslim
• types and dimensions of data (i.e. nominal, ordinal and cardinal variables)
- different types of data (variables and observations)
- types of variables (quantitative and categorical)
• univariate statistics
▪ frequency tables + statistical indexes to summarise data for a variable (mean, median mode)
▪ measures of dispersion (i.e. variance and standard deviation)
▪ measures of distribution (e.g. percentiles)
▪ graphing (i.e. histograms, pie charts, trend lines)
▪ new trends in infographics (e.g. Statista)
• bivariate statistics
▪ contingency tables
▪ Pearson and Spearman correlations indexes
• simple regression analysis
▪ linear relationship between two variables
▪ OLS regression analysis
• basic concepts of sampling distribution and inference analysis
▪ samples and population
▪ probability and probability distribution
▪ sampling distribution
▪ inferential statistics
point estimate and confidence intervals
testing hypothesis
3/10/19
What is statistics
Statistics is the science of designing studies and analyzing the data that those studies produce. Its ultimate goal is
translating data into knowledge and understanding of the world around us
• uses math formulas to generate synthetic indexes
• uses graphs to visualize data in order to describe socio-economic facts using real data (= descriptive statistics)
to make decisions or predictions about “population” (= inferential statistics)
• Its goal is translating data into knowledge and understanding of the world around us
Why should we study statistics in IRGA program?
1st reason:
If we consider international “hot” topics (ex. migration) that are not just a matter of Italy there’s statistics: we try to
read numbers and create indexes / compare them to understand social-economic phenomena (=descriptive statistics)
→table representing the number of international migrants in millions in the world
The number of migrants increased in17 years by 1.5 (𝑟𝑎𝑡𝑖𝑜 258/172=1.5)
Destinations are different:
▪ in 2000 60% [(103/172)x100] of migrants preferred more developed areas and 40% less developed areas.
▪ In 2017 percentages changed: 57% in more developed areas and 43% in less developed areas.
In 2017 3.4% of total population is involved in migration but
the weight on the total population in different areas is
differentiated: 11.6% versus 1.8% = some areas are affected
more and some less by migration.
→Which kind of policies should we improve in a particular
country?
, 2nd reason:
In order to compute and “infer” the relationship between variables
→this graph shows there’s a negative relation between the
emigration rate and the GDP of each country: people migrate if the
country is poor, otherwise if the GDP per capita is quiet high, the
migration in low
In order to predict the results of the political elections, dealing with a “margin of error” (Ex. “The good wife”)
→do exit pools and try to understand who’ll win political elections using
probabilities and some margin of error.
In this graph we see the percentage of people who are going to vote for
republicans and democrats, what to worry about? White women, because
it’s not sure who are they going to vote for
4/10/19
Statistical method
It helps us to investigate on an objective manner and to figure out how to apply our indexes in order to summarise
information and have some hints on the answer to our research question
Statistical problem solving is an investigative process that involves 4 components:
1) Formulate a research question: arises not from statistics but from other disciplines (economics, political science,
sociology, history…)
Once you design your research question plan how to obtain the data in order to answer it
2) Collect data: they are available most of the time on the web, but in some cases data are fake and need to think how
to collect true data.
(In order to do this be careful to Wikipedia and try to use international reliable databases (IMF, WB, national
statistical offices – Istat-, Eurostat…) the problem with this data is that they’re organised in a different way and need
to learn to assess them; also surfing in google public data it is possible to get the data needed)
Otherwise you can formulate your own research question and gather your own data (questionnaire ex.)
3) Analyse data: summarise and interpret them according to the research question (describe what’s going on)
4) Interpret your results: Do an inference analysis = try to understand what happens if something is changing
Complementary ways of analysing data
Three main components of statistics for answering a statistical question:
- Design: planning how to obtain data
- Description: means summarizing the data, files of raw data are often huge and not easy to assess SOO it is
more informative to use a few numbers or a graph to summarize data
- Inference: means making decisions or predictions based on the data; usually the decision or prediction refers to
a larger group of people, not merely those in the study
“infer” = to arrive at a decision or prediction by reasoning from known evidence
Complementary ways of analysing data: can use both to investigate questions which are important for society
,Statistical description (descriptive statistics)= a picture of what is going on 1st semester
▪ refers to methods for summarizing the collected data using graphs and numbers such as averages and
percentages
▪ simplifies data without distorting or loosing information
▪ can be done for sample and for population (w/census)
It’s much easier to get a sense of the data by looking at a graph than by reading through the questionnaires filled out by the
50,000 sampled households.
Inference analysis (inferential statistics) = a guessing 2nd semester
- making decisions or predictions about a population, based on data obtained from a sample of that population
- deals with probability= quantifying how likely various possible outcomes are
- important aspect of statistical inference involves reporting the likely precision of a prediction within a range
(how close is the sample value of % likely to be to the true percentage of the population: inferential
statistical analyses can predict characteristics of entire populations quite well by selecting samples that are
small relative to the population size.
For example, the population of China is about four times that of the United States, but a random sample of 1000 people from the
Chinese population and a random sample of 1000 people from the U.S. population would achieve similar levels of accuracy)
-> statistics consists of methods for designing investigative studies, describing (summarizing) data obtained for those
studies, and making inferences (decisions and predictions) based on those data to answer a statistical question
of interest.
What we can investigate
The entities we measure in a study are called subjects
• Individuals (want to see the average weight of this class)
• Schools (average performance of schools to see what’s the best school to go)
• Countries (evaluate GDP)
• Days: quiet recent topic, evaluate the probability of raining (ex.) and guess if in a specific day in a specific area is going to rain
•whatever we have in our mind
To our subjects (our unitof analysis) we must attach data (which are different features of our unit of analysis)
Population = total set of subjects in which we are interested, once you try to summarise this information you are
computing the “parameter” (statistical index that summarises what’s going on to the whole population)
usually too costly and time-consuming to obtain data related to all population
Sample = the subset of the population for whom we have data, the idea is that the subsample should be randomly
selected. Then statistics (number) tells you what’s going on with your sample (=only related to the subset)
ex. want to see average age of people in Milan, in order to get the parameter of this population, need to investigate the age
of every single person who’s living in Milan (quite costly).
BUUT there’s also the opportunity to have a sample by randomly select some subjects which represent a part of the milanese
population and once you get this information you come up with your statistics (= a number that’s summarising what’s going
on to the average age of the population that’s living in Milan)
Randomness and variability
Randomness is an extremely powerful tool for obtaining good samples and conducting experiments. A sample tends to
be a good reflection of a population when each subject in the population has the same chance of being included in
that sample. That’s the basis of random sampling, which is designed to make the sample representative of the
population.
Random sampling:
- allows us to make powerful inferences about populations
- is crucial to performing experiments well
10/10/19
Definitions
A variable is any characteristic observed/being measured in a study.
, The data are values that we observe for a variable, they are called observations; each observation can be a number or
may belong to a category.
A variable can be:
- Categorical, if each observation belongs to one of a set of categories (we count and we order data)
A key feature to describe categorical variables: the relative number of observations in the various categories
Ex. frequency table of shark attacks in various regions: each observation (shark attack) identifies a “region” where
the attack occurred -> region is the variable and it is categorical with categories representing regions of the world
- Quantitative/numerical, if observations take numerical values that represent different magnitudes of the
variable they measure “how much” of something”
Do you like
Final
Surname Vegetarian sports? (0=no,
ID Gender Weight (kg) mark in
name (yes=1) 1=quite, 2=very
QM
much)
1 Blue John male 1 1 88.7 30
2 Green Dana female 0 2 81.2 26
3 Yellow Zeb male 1 1 86.8 24
White
4 Robert male 0 2 73.21 19
5 Black Susan female 0 0 60.9 27
->table with columns and rows which identify different themes
Rows: represent single observation - Columns: represent a particular value
▪ ID: identifies each observation of my «sample»
ID is unique and allows us to manage possible homonymies if our samples include people’s names
▪ Surname and Name: identifies each student
▪ for each observation Gender identifies if the student is “male” or “female”.
This is a “category” that could be converted into a number to be better treated (0 for male and 1 for female), but it
willalways remain a category and this codifying procedure doesn’t attach a particular value to numbers ( 0 & 1 in
this case)
->“male” = 0 and “female” = 1 doesn’t mean that we are evaluating male and female, the number attached to this
category means nothing
Gender = Categorical variable.
▪ for each observation Vegetarian identifies if the student has “vegetarian” tastes or “not”.
This variable is a “category” already converted into a number (0 for not veggie and 1 for veggie).
Vegetarian taste = Categorical variable
▪ This variable identifies how much the student appreciates football. Each student could evaluate his/her preferences
from 0 to 2.
Preferences on Football = Categorical variable already converted into a number
▪ Weight (kg) and Final Mark in QM are numerical/quantitative variables associated to each observation and we
can compare them
BUUT Weight are “continuous” numbers and Final Mark in QM is “discrete”
A quantitative variable is discrete if its possible values form a set of separate numbers, such as 0, 1, 2, 3, . . .
(e.g. number of books in a shelf)
A quantitative variable is continuous if its possible values form an interval, such as 1.1, 1.2, 1.205, 4.37, …
(e.g. share of people living below poverty line in African countries)
Which kind of analysis we can run if we know that we have variables that are very different
In fact, both categorical and quantitative variables can be translated into
• Nominal variables
Categorical variable that has two or more categories but there is no intrinsic ordering of them.
ex. gender (2 categories) :0=male; 1=female;
religion belief (more than 2 categories): 1 = Catholic; 2 = Protestant; 3 = Muslim
