2. Statistics

What is Statistics?

Statistics is the science based on recolecting, organizing, presenting, analyzing, and interpreting of $\color{cyan}DATA$.

This results in facilitating informed decisions.

Objectives

We need to determine clearly what is (or are) the $\color{GOLD}OBJECTIVE$(s) of our study.

Measurement

We have to stablish the $\color{tomato}features$ relevant to our objective.

Subject

The entity measured.

• Could be an individual, a company, a phenomenon... We may be interested in multiple attributes, although not all of them may be measured directly.

Variable

The characteristic measurable. This is, we can assign a value specifically.

Value Type.

The set of all possible results of measurement.

• Numeric
• Identificators

Data

Set of observed values on the variables.

PROPIEDADES

• Varianza: $V(X) = E[(X-E(X))^2]=E(X^2) - E(X)^2$
• $E(c) = c$
• $E(cX) = cE(X)$
• $E(X+Y) = E(X) + E(Y)$
• $V(cX) = c^2 V(X)$

A modo de ayuda, consulta la siguiente tabla para el cálculo de probabilidades.

Situación	Calculo
P(X<=a)	F(a)
P(X>=a)	1-F(a-1)
P(X<a)	F(a-1)
P(X>a)	1-F(a)
P(a<X<b)	F(b-1)-F(a)
P(a<=X<b)	F(b-1)-F(a-1)
P(a<X<=b)	F(b)-F(a)
P(a<=X<=b)	F(b)-F(a-1)

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Statistics begins where certainty ends, but not where thought fails.

Probability studies what may happen.
Statistics studies what can be learned from what has happened.

That difference is small in wording and enormous in meaning.

A probability model begins with a space of possibilities.
A statistical inquiry begins with traces: measurements, observations, counts, records, errors, signals, fragments of a process already gone.

The world acts once.
We arrive later, with data.

🏵 This is the central tension of statistics:

• reality is rich
• observation is limited
• variation is everywhere
• knowledge is still possible

Statistics is the disciplined attempt to read structure from incomplete contact, to build careful bridges from observation to understanding.

A list of numbers is not yet statistics.
A graph is not yet statistics.
A mean is not yet statistics. Statistics begins when we ask:

• What process could have produced this?
• What kind of variation is present?
• What is signal, and what is noise?
• What can be concluded, and what must remain uncertain?

🔰 A world without variation would need no statistics.
A world without pattern would permit none.

Statistics exists because both are true at once:

• things differ
• and those differences are not pure chaos

This Part is about that middle territory.

About variation, uncertainty, distributions, models, and methods.
About the forms in which data spreads, clusters, fluctuates, and surprises.
About how we move from raw observation to structured reasoning.

Statistics is often taught as a toolbox.
It is better understood as a way of seeing.

A statistical mind does not merely calculate.
It asks what generated the data, what assumptions are being made, what distinctions matter, and what kind of error one is willing to tolerate.

⚜️ The aim is not to remove uncertainty.

It is to give uncertainty shape.

And that is one of the deepest things mathematics can do.