Introduction to Statistics
Week 1 – Why do we need statistics? – Introduction to the course – frequency distributions
CH1 Introduction to statistics
Why do we even need statistics?
Maria is 26 years old, single, outspoken, and very bright. She majored in law. As a student, she was deeply
concerned with issues of discrimination and miscarriage of justice and participated in weekly animal-rights
demonstrations. Which is more probable?
A: Maria works in a law firm ←
B: Maria works in a law firm and does pro bono work for animal-rights activists
(Selective perception issue→relationship between 2 variables without taking all data)
- A variable is a characteristic or condition that changes or has different values for different individuals.
- Data (plural) are measurements or observations. A data set is a collection of measurements or observations. A
datum (singular) is a single measurement or observation and is commonly called a score or raw score.
Berkson’s paradox
Also holds for attractiveness and niceness in dating→the more attractive a person is the less nice they are
because of selective perception (tend to look at extremes)
Why should I care?
we are flooded with data + we want to make sense of the world around us (about human behaviour and society)
Statistics is the best way to do this.
Suppose you wanted to know..
-whether loneliness increased during lockdown?
-how much more dangerous COVID-19 is for people with cancer?
-how engagement in online communities relates to extremist world views?
-whether a curfew increases rioting?
Statistics is the ONLY way to meaningfully approach these questions!
What does it even mean? Statistics, the science of collecting, analyzing, presenting, and interpreting data.
-A branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of
numerical data.
→ a set of mathematical procedures for organizing, summarizing, and interpreting information.
Note: this is ≠ “statistics as a collection of data” Statistics is both a single number and the discipline
Synopsis of statistics
→ we work with data in a numerical sense, obtain information from these data, understand the uncertainty that
comes with data→this is one aspect where it differs from mathematical modelling
And lastly: the word data is the plural form of datum.
→not being able to interpret data can be dramatic+statistics is the discipline that is about data
deaths/cases vs deaths/population
The data never lie!? →people will use statistics to make their points, this can be used to mislead
- social + behavioural sciences have embraced quantitative methods, we seek to express processes/ attributes/
disorders as numbers so we also need methods to make sense of these numbers
The special role for Psychology→Human behaviour and social processes are very complex
We are often interested in the unobservables: intelligence, well-being, emotions (fear, sadness), loneliness
These are very hard to measure! And we need methods to learn about humans in general (=the population)
This is the essence of inferential statistics.
- A population is the set of all the individuals of interest in a particular study.
- A sample is a set of individuals selected from a population, intended to represent the population in a study
- A parameter is a value, usually a numerical value, that describes a population. A parameter is usually derived
from measurements of the individuals in the population→every population parameter has a corresponding sample
statistic
- A statistic is a value, usually a numerical value, that describes a sample. A statistic is usually derived from
measurements of the individuals in the sample.
1
,Two stances towards statistics:
→Statistics as a tool→you use it to serve your purpose (e.g. making an inference based on your data)
you have a pragmatic relationship with statistics (e.g. it’s needed to do research and to understand the world)
→Statistics as a discipline→about improving statistics, about better ways to model data, make inferences, quantify
uncertainty esp. now: making sense of massive volumes of data (never use the term Big Data)
The connection to AI
Basic statistics today is what reading was yesterday, If you invest the time to fully understand the content in this
module (always ask if things are unclear), Every more advanced approach builds on these basic ideas
Basic ideas in statistics→The idea of “data” + Types of statistical thinking + First look at distributions
Approaches of statistics
- Descriptive statistics→statistical procedures used to summarize, organize, and simplify data (about describing the
data, often through summary statistics)
e.g. on average a Spanish women is 1.63m tall / The wealthiest 1% own 50% of the equity/shares in companies
- Inferential statistics→techniques that allow us to study samples and then make generalizations about the
populations from which they were selected→we want to make an inference from something to something else
here: we want to make an inference from the sample to the population (sampling as the process of selecting a few
candidates from the population, test the sample, generalize it to the population)
Sampling error is the naturally occurring discrepancy, or error, that exists between a sample statistic and the
corresponding population parameter (even when the sample is representative)
→data≠data (there are different kinds of data) eg. Height (in cm), Annual income (in EUR)
Smoker vs. non-smoker (don’t get a value), Pet (dog, cat, bunny), Support for Trump (scale from -5 to +5)
Statistics in experimental research:
Step 1 Experiment: Compare two studying methods, Data=Test scores for the students in each sample
Step 2 Descriptive statistics: Organize and simplify
Step 3 Inferential statistics: Interpret results
Data structures, Research methods and statistics
- Individual Variables: Descriptive Research→to describe individual variables as they exist naturally (eating)
- Relationships Between Variables→to examine relationships between two or more variable (breakfast-grades)
→Correlational method→two variables are observed to determine whether there is a relationship between them
no cause-effect relationship
→Experimental method→one variable is manipulated while another variable is observed and measured. To
establish a cause-and-effect relationship between the two variables, an experiment attempts to control all other
variables to prevent them from influencing the results (manipulation+control) Rule out alternative explanations:
1. Participant Variables→characteristics as age, gender, intelligence that vary from one individual to another
2. Environmental Variables→characteristics of the environment as lighting, time of day, and weather conditions
+random assignment, matching (equivalent groups), holding them constant, control condition (no treatment)
→Non-experimental method→the “independent variable” that is used to create the different groups of scores is
often called the quasi-independent variable.
Dimensions of the “data” idea
- Constructs vs operationalizations
Whether there is a relationship between construct A or B→need to operationalize them to actually measure them
Constructs→internal characteristics not directly observed but are useful for describing/explaining behavior
Operational definition→measurement procedure (a set of operations) for measuring an external behavior and uses
the resulting measurements as a definition and a measurement of a hypothetical construct.
Two components: 1.describes a set of operations for measuring a construct.
2. defines the construct in terms of resulting measurements.
- Discrete vs continuous variables
Discrete variable→ separate, indivisible categories. No values can exist between two neighboring categories.
→only consist of a limited number of categories: e.g. gender, eye color, native language
but also: no. of pets, no. of siblings, how often were on holiday, no value between 1 and 2 pets
2
, Continuous variable→infinite number of possible values that fall between any two observed values→divisible into
an infinite number of fractional parts e.g. income, height, weight, speed→your height can, in principle, be
expressed as 1.75123461 meters, thus a value of a continuous variable (1.75m) is an interval
Real limits→boundaries of intervals for scores that are represented on a continuous number line. The real limit
separating two adjacent scores is located exactly halfway between the scores. Each score has two real limits. The
upper real limit is at the top of the interval, and the lower real limit is at the bottom.
- Scales of measurements
The nominal scale→set of categories that have different names. Measurements label and categorize observations,
but do not make any quantitative distinctions between observations
→named categories (e.g., dog, cat, hamster), no quantitative(you cannot say a dog is more than a cat), no zero!
The ordinal scale→set of categories that are organized in an ordered sequence. Measurements on an ordinal scale
rank observations in terms of size or magnitude→ranked named categories (e.g., 1st, 2nd, 3rd) (small, medium,
large), no equal distance between ranks, no zero!
The interval scale→ordered categories that are all intervals of exactly the same size. Equal differences between
numbers on scale reflect equal differences in magnitude. However, the zero point on an interval scale is arbitrary
and does not indicate a zero amount of the variable being measured→consists of equally-sized intervals between
values, each unit has the same size e.g. temperature: going from to C21∘ C26∘, going from to C1∘ C6∘, both have
the same difference but: no real zero! (arbitrarily chosen)
The ratio scale→interval scale with the additional feature of an absolute zero point. With a ratio scale, ratios of
numbers do reflect ratios of magnitude→, each unit has the same size, but now we do have an absolute zero e.g.
distance: a distance of zero means your bike has not moved! (eg. height/weight)
Representing data raw data ( id | answers)
Raw scores→original, unchanged scores obtained in the study
N=number of scores in a population, n=number of scores in a sample
Summation Notation: Σ, Σx (add all scores for variable x), Σ(x-1) (first calculate x-1, then add the results)
se alla seconda→squaring before adding!!, no parenthesis Σx-1 (first summation, then -1)
How many pets do you have? we ask 10 people, they state n. of pets that currently lives in their household
the construct is “number of pets”, the operationalisation is “the number of pets…” (discrete variable)
We may want some more structure starting from the raw data to extract information
- maybe we can count how often each option occurs→from sample to a population
- i.e. how many people have 0, 1, 2, ... pets?
These are called the frequencies of values→how frequent each single value is
Frequencies of values
A structured table is then called a frequency distribution table (variable | frequency)
CH 2 Frequency Distributions
Frequency distribution→organized tabulation of the number of individuals located in each category on the scale of
measurement
→takes a disorganized set of scores and places them in order from highest to lowest, grouping together individuals
who all have the same score→shows whether the scores are generally high or low
Table/Graph→2 elements: set of categories + record of the frequency
Frequency Distribution tables (x | frequency)
→all of the possible values are listed in the table, wg no one had a score of X = 5→ (5 | 0)
ordinal, interval, ratio scale→listed highest to lowest (x represents the scale of measurement, not the actual set)
Obtaining ΣX from a Frequency Distribution Table→use all the information presented
eg. table shows the distribution has one 5, two 4s, three 3s, three 2s, one 1, for total of 10 score (add or multiply)
Grouped frequency distribution table
Because presenting groups of scores rather than individual values→groups=class intervals
1. Best=about 10 class intervals
2. The width of each interval should be a relatively simple number (2, 5, 10)
3
Week 1 – Why do we need statistics? – Introduction to the course – frequency distributions
CH1 Introduction to statistics
Why do we even need statistics?
Maria is 26 years old, single, outspoken, and very bright. She majored in law. As a student, she was deeply
concerned with issues of discrimination and miscarriage of justice and participated in weekly animal-rights
demonstrations. Which is more probable?
A: Maria works in a law firm ←
B: Maria works in a law firm and does pro bono work for animal-rights activists
(Selective perception issue→relationship between 2 variables without taking all data)
- A variable is a characteristic or condition that changes or has different values for different individuals.
- Data (plural) are measurements or observations. A data set is a collection of measurements or observations. A
datum (singular) is a single measurement or observation and is commonly called a score or raw score.
Berkson’s paradox
Also holds for attractiveness and niceness in dating→the more attractive a person is the less nice they are
because of selective perception (tend to look at extremes)
Why should I care?
we are flooded with data + we want to make sense of the world around us (about human behaviour and society)
Statistics is the best way to do this.
Suppose you wanted to know..
-whether loneliness increased during lockdown?
-how much more dangerous COVID-19 is for people with cancer?
-how engagement in online communities relates to extremist world views?
-whether a curfew increases rioting?
Statistics is the ONLY way to meaningfully approach these questions!
What does it even mean? Statistics, the science of collecting, analyzing, presenting, and interpreting data.
-A branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of
numerical data.
→ a set of mathematical procedures for organizing, summarizing, and interpreting information.
Note: this is ≠ “statistics as a collection of data” Statistics is both a single number and the discipline
Synopsis of statistics
→ we work with data in a numerical sense, obtain information from these data, understand the uncertainty that
comes with data→this is one aspect where it differs from mathematical modelling
And lastly: the word data is the plural form of datum.
→not being able to interpret data can be dramatic+statistics is the discipline that is about data
deaths/cases vs deaths/population
The data never lie!? →people will use statistics to make their points, this can be used to mislead
- social + behavioural sciences have embraced quantitative methods, we seek to express processes/ attributes/
disorders as numbers so we also need methods to make sense of these numbers
The special role for Psychology→Human behaviour and social processes are very complex
We are often interested in the unobservables: intelligence, well-being, emotions (fear, sadness), loneliness
These are very hard to measure! And we need methods to learn about humans in general (=the population)
This is the essence of inferential statistics.
- A population is the set of all the individuals of interest in a particular study.
- A sample is a set of individuals selected from a population, intended to represent the population in a study
- A parameter is a value, usually a numerical value, that describes a population. A parameter is usually derived
from measurements of the individuals in the population→every population parameter has a corresponding sample
statistic
- A statistic is a value, usually a numerical value, that describes a sample. A statistic is usually derived from
measurements of the individuals in the sample.
1
,Two stances towards statistics:
→Statistics as a tool→you use it to serve your purpose (e.g. making an inference based on your data)
you have a pragmatic relationship with statistics (e.g. it’s needed to do research and to understand the world)
→Statistics as a discipline→about improving statistics, about better ways to model data, make inferences, quantify
uncertainty esp. now: making sense of massive volumes of data (never use the term Big Data)
The connection to AI
Basic statistics today is what reading was yesterday, If you invest the time to fully understand the content in this
module (always ask if things are unclear), Every more advanced approach builds on these basic ideas
Basic ideas in statistics→The idea of “data” + Types of statistical thinking + First look at distributions
Approaches of statistics
- Descriptive statistics→statistical procedures used to summarize, organize, and simplify data (about describing the
data, often through summary statistics)
e.g. on average a Spanish women is 1.63m tall / The wealthiest 1% own 50% of the equity/shares in companies
- Inferential statistics→techniques that allow us to study samples and then make generalizations about the
populations from which they were selected→we want to make an inference from something to something else
here: we want to make an inference from the sample to the population (sampling as the process of selecting a few
candidates from the population, test the sample, generalize it to the population)
Sampling error is the naturally occurring discrepancy, or error, that exists between a sample statistic and the
corresponding population parameter (even when the sample is representative)
→data≠data (there are different kinds of data) eg. Height (in cm), Annual income (in EUR)
Smoker vs. non-smoker (don’t get a value), Pet (dog, cat, bunny), Support for Trump (scale from -5 to +5)
Statistics in experimental research:
Step 1 Experiment: Compare two studying methods, Data=Test scores for the students in each sample
Step 2 Descriptive statistics: Organize and simplify
Step 3 Inferential statistics: Interpret results
Data structures, Research methods and statistics
- Individual Variables: Descriptive Research→to describe individual variables as they exist naturally (eating)
- Relationships Between Variables→to examine relationships between two or more variable (breakfast-grades)
→Correlational method→two variables are observed to determine whether there is a relationship between them
no cause-effect relationship
→Experimental method→one variable is manipulated while another variable is observed and measured. To
establish a cause-and-effect relationship between the two variables, an experiment attempts to control all other
variables to prevent them from influencing the results (manipulation+control) Rule out alternative explanations:
1. Participant Variables→characteristics as age, gender, intelligence that vary from one individual to another
2. Environmental Variables→characteristics of the environment as lighting, time of day, and weather conditions
+random assignment, matching (equivalent groups), holding them constant, control condition (no treatment)
→Non-experimental method→the “independent variable” that is used to create the different groups of scores is
often called the quasi-independent variable.
Dimensions of the “data” idea
- Constructs vs operationalizations
Whether there is a relationship between construct A or B→need to operationalize them to actually measure them
Constructs→internal characteristics not directly observed but are useful for describing/explaining behavior
Operational definition→measurement procedure (a set of operations) for measuring an external behavior and uses
the resulting measurements as a definition and a measurement of a hypothetical construct.
Two components: 1.describes a set of operations for measuring a construct.
2. defines the construct in terms of resulting measurements.
- Discrete vs continuous variables
Discrete variable→ separate, indivisible categories. No values can exist between two neighboring categories.
→only consist of a limited number of categories: e.g. gender, eye color, native language
but also: no. of pets, no. of siblings, how often were on holiday, no value between 1 and 2 pets
2
, Continuous variable→infinite number of possible values that fall between any two observed values→divisible into
an infinite number of fractional parts e.g. income, height, weight, speed→your height can, in principle, be
expressed as 1.75123461 meters, thus a value of a continuous variable (1.75m) is an interval
Real limits→boundaries of intervals for scores that are represented on a continuous number line. The real limit
separating two adjacent scores is located exactly halfway between the scores. Each score has two real limits. The
upper real limit is at the top of the interval, and the lower real limit is at the bottom.
- Scales of measurements
The nominal scale→set of categories that have different names. Measurements label and categorize observations,
but do not make any quantitative distinctions between observations
→named categories (e.g., dog, cat, hamster), no quantitative(you cannot say a dog is more than a cat), no zero!
The ordinal scale→set of categories that are organized in an ordered sequence. Measurements on an ordinal scale
rank observations in terms of size or magnitude→ranked named categories (e.g., 1st, 2nd, 3rd) (small, medium,
large), no equal distance between ranks, no zero!
The interval scale→ordered categories that are all intervals of exactly the same size. Equal differences between
numbers on scale reflect equal differences in magnitude. However, the zero point on an interval scale is arbitrary
and does not indicate a zero amount of the variable being measured→consists of equally-sized intervals between
values, each unit has the same size e.g. temperature: going from to C21∘ C26∘, going from to C1∘ C6∘, both have
the same difference but: no real zero! (arbitrarily chosen)
The ratio scale→interval scale with the additional feature of an absolute zero point. With a ratio scale, ratios of
numbers do reflect ratios of magnitude→, each unit has the same size, but now we do have an absolute zero e.g.
distance: a distance of zero means your bike has not moved! (eg. height/weight)
Representing data raw data ( id | answers)
Raw scores→original, unchanged scores obtained in the study
N=number of scores in a population, n=number of scores in a sample
Summation Notation: Σ, Σx (add all scores for variable x), Σ(x-1) (first calculate x-1, then add the results)
se alla seconda→squaring before adding!!, no parenthesis Σx-1 (first summation, then -1)
How many pets do you have? we ask 10 people, they state n. of pets that currently lives in their household
the construct is “number of pets”, the operationalisation is “the number of pets…” (discrete variable)
We may want some more structure starting from the raw data to extract information
- maybe we can count how often each option occurs→from sample to a population
- i.e. how many people have 0, 1, 2, ... pets?
These are called the frequencies of values→how frequent each single value is
Frequencies of values
A structured table is then called a frequency distribution table (variable | frequency)
CH 2 Frequency Distributions
Frequency distribution→organized tabulation of the number of individuals located in each category on the scale of
measurement
→takes a disorganized set of scores and places them in order from highest to lowest, grouping together individuals
who all have the same score→shows whether the scores are generally high or low
Table/Graph→2 elements: set of categories + record of the frequency
Frequency Distribution tables (x | frequency)
→all of the possible values are listed in the table, wg no one had a score of X = 5→ (5 | 0)
ordinal, interval, ratio scale→listed highest to lowest (x represents the scale of measurement, not the actual set)
Obtaining ΣX from a Frequency Distribution Table→use all the information presented
eg. table shows the distribution has one 5, two 4s, three 3s, three 2s, one 1, for total of 10 score (add or multiply)
Grouped frequency distribution table
Because presenting groups of scores rather than individual values→groups=class intervals
1. Best=about 10 class intervals
2. The width of each interval should be a relatively simple number (2, 5, 10)
3
