100+ Statistics Concepts You Should Know

Summary

This video introduces statistics as a means to control information and make sense of randomness in data. It covers key statistical concepts like types of data, probability distributions, measures of central tendency and variation, hypothesis testing, sampling distributions, confidence intervals, regression modeling, experimental design, parametric vs nonparametric models, prediction, machine learning models, causal inference, counterfactuals, robust statistics, and recommendations for learning statistics like R programming and simulation.

Takeaways

• 😀 Statistics helps us analyze data and make sense of randomness and uncertainty
• 📊 There are two main types of data: quantitative (numbers) and qualitative (words)
• 🎲 Probability theory describes the inherent randomness in data
• 📈 Descriptive statistics summarize and describe the properties of data
• 🌡️ Inferential statistics allow us to make conclusions about populations from samples
• 🧪 Experiment design and data collection methods impact the conclusions we can draw
• 😯 Machine learning uses statistics to make predictions from data
• 🤯 Modern statistics tackles very complex, high-dimensional data analysis problems
• 📐 Assumptions are crucial in statistics - models can fail if assumptions are violated
• 👩‍💻 Statistical programming languages like R help analyze data and test models

Q & A

What is the main purpose of statistics?

The main purpose of statistics is to make sense of data and information despite the presence of randomness. It aims to uncover patterns, relationships, and insights from data.

What are the two main types of data?

The two main types of data are quantitative data (numerical data) and qualitative data (categorical or text data).

What is a random variable and what does its probability distribution describe?

A random variable is a variable that can take on different values probabilistically. Its probability distribution describes the probabilities associated with each potential value.

What is the difference between descriptive and inferential statistics?

Descriptive statistics summarize and describe the actual data collected, while inferential statistics make inferences about an unobservable larger population based on a sample of data.

What is the purpose of hypothesis testing?

Hypothesis testing is used to make decisions about hypotheses/claims made about a population. It allows us to conclude whether we have enough evidence to reject the initial hypothesis.

What are Type I and Type II errors in hypothesis testing?

Type I error occurs when we reject a true null hypothesis. Type II error occurs when we fail to reject a false null hypothesis.

What is the difference between observational and experimental studies?

Experimental studies involve randomization and manipulation of conditions, allowing causality conclusions. Observational studies do not involve manipulation, only observation.

What is semi-parametric modeling?

Semi-parametric modeling involves using both parametric components (with a finite number of parameters) and non-parametric components (without a predefined structure) in a model.

Why is programming important for statisticians?

Programming allows statisticians to implement statistical techniques, analyze data, run simulations to test models, automate tasks, and more effectively work with data.

What advice is offered to those wanting to get started with statistics?

The video advises learning a statistical programming language like R, running simulations to test models, and recognizing that building statistical skills requires hard work over time.

Outlines

00:00

😊 Overview of data types and role of statistics

This paragraph provides an introduction to different types of data like quantitative, qualitative, discrete, continuous, binary, count, time-to-event, etc. It also introduces key concepts like randomness, probability theory, random variables, probability distributions that are commonly used in statistics to make sense of data and draw insights from it.

05:00

😀 Random variables, conditional probability and statistical modeling

This paragraph discusses additional probability concepts like conditional probability, independence of random variables, Bayes' theorem. It then explains core ideas in statistics - defining a population, drawing a sample from it, making assumptions/models about data generation, using sample to estimate unknown population parameters, distinction between descriptive and inferential statistics.

10:01

📈 Hypothesis testing, errors, test types and experimental design

This paragraph provides an overview of hypothesis testing, null and alternate hypotheses, p-values, confidence intervals, type 1 and type 2 errors, setting significance levels, different types of tests like one sample, two sample, ANOVA, chi-square for different data types and study designs. It emphasizes the importance of experimental design in causal inference.

Keywords

💡data

Data refers to quantitative information that is observed and collected, acting as the raw material for statistical analysis. As stated in the video, "In the beginning there was Data" - data provides the foundation for statistics. Different types of data include quantitative (numerical) and qualitative (categorical) data. Examples of data from the script include clinical biomarkers, age, and binary (0/1) data.

💡random variable

A random variable is a key concept used to represent data statistically. It can take on different values with different probabilities. As stated, random variables "have special functions that describe the probabilities associated with each value". Common examples are continuous or discrete random variables. Random variables are important for modeling randomness and variation in data.

💡probability distribution

The probability distribution of a random variable describes the probabilities associated with different values it can take. It captures the structure behind the randomness in data. Examples given include the normal, binomial and Poisson distributions. The shape of the distribution, described by the probability density function, allows statisticians to model and understand variation in data.

💡estimator

An estimator is a statistic used to estimate an unknown population parameter. As samples are collected from a population, estimators construct a best guess for parameters of interest. The sample mean is a commonly used estimator to infer information about the population mean. Estimators allow statisticians to make inferences about unobserved populations.

💡hypothesis test

Hypothesis tests are procedures used to make decisions about population parameters based on sample data. They involve stating a null hypothesis and an alternative hypothesis. Based on the data, statisticians determine whether to reject the null in favor of the alternative or not. Examples in the video include one sample, two sample and ANOVA hypothesis tests.

💡p-value

The p-value is the probability of obtaining sample results at least as extreme as observed, assuming the null hypothesis is true. Small p-values suggest the null is unlikely and provide evidence to reject it in favor of the alternative hypothesis. P-values are central to most hypothesis testing procedures.

💡statistical model

A statistical model is an mathematical representation used to describe and understand the data generation process. Models make assumptions about the mechanisms behind the data. Examples given include regression models, generalized linear models, longitudinal models and more. Appropriate modeling is crucial for valid statistical inference.

💡experimental design

Experimental design refers to how data collection is planned and structured. Elements of design like randomization of treatment groups allow causal interpretations. The video emphasizes that understanding the experimental design is vital for knowing what conclusions can validly be drawn from data.

💡assumptions

Assumptions refer to the often unverified conditions and beliefs underlying statistical models and procedures. Common assumptions discussed include independent and identically distributed data, linear relationships, normality of data, etc. The video stresses assumptions must be kept in mind as model violations can render statistical analysis invalid.

💡exploratory data analysis

Exploratory data analysis involves preliminary investigation of data to understand key properties, patterns and relationships before formal modeling. The video recommends learning programming to facilitate such exploration through summary statistics, visualizations and more. This can guide choices involved in statistical analysis.

Highlights

Statistics is the study of data, which includes everything from data collection to data presentation.

The central object that we use to represent data is the random variable, which can take on different values with different probabilities.

The probability density function or probability mass function describes the probabilities associated with each value a random variable can take.

Measures of central tendency like the mean, median and mode describe typical values, while measures of scale like variance and standard deviation describe the spread of values.

Statisticians define a population, collect data from a sample, assume the data comes from a probability distribution, and try to infer the unknown population parameters.

Hypothesis tests allow us to make decisions by assuming a null hypothesis, seeing how likely our data would be under this assumption, and potentially rejecting this assumption.

There are many hypothesis tests for different types of data and questions, ranging from one sample tests to ANOVA and regression models.

Experimental design with randomization allows us to make causal claims, while observational data usually only provides correlational evidence.

Non-parametric models make fewer assumptions, while robust statistics enable proper inference even when assumptions fail.

Prediction tries to forecast future observations based on models fit to past data, connecting statistics to machine learning.

High dimensional statistics deals with more predictors than samples, while causal inference allows for making causal claims from observational studies.

Simulation studies test models before publication, so programming skills are important for statisticians to develop and evaluate methods.

R is a good programming language to learn for doing statistics since it specializes in data analysis and visualization.

Statistics requires hard work to master, but the growing need for data analysis means demand for statisticians will continue rising.

By learning statistics you'll be better equipped to uncover insights from data and trick people into thinking you're good at board games.

Transcripts

the world is full of information and

information is power as Tom Clancy once

said if you can control information you

can control the people but he never told

us what to do with that information once

we got it this video won't teach you how

to control people but it'll at least

tell you how to control information and

that's what statistics is for for many

people statistics was just one course we

took and forgot about but for those that

take it further they see the

possibilities that strong statistical

skills unlock you're not controlling

people but you're paid nicely to handle

data and information if you need more

stats in your life but don't know where

to start learning then welcome to the

statistical 100 if you can Master these

100 topics then you'll be one step

closer to tricking people into thinking

you're good at board games in the

beginning there was Data data is a

general catch-all term for information

that we observe this information can be

in number form also known as

quantitative data or can be in word form

known as qualitative data for this video

we focus on old part numbers data can

come in different flavors one of those

flavors is discrete data represented by

the integers one important example of

discrete data is binary data which can

only take two values zero and one zero

and one are useful because they can

represent on and off states such as true

and false we can represent group

membership with binary data using one to

indicate that someone is a part of one

group while zero can represent the other

group this logic can be taken further

with categorical data where there can be

more than two groups another important

example of discrete data is Count data

represented by the positive integers the

other flavor of data is continuous data

represented by the entire number line

continuous data is useful for things

that fall in the Continuum like a

clinical biomarker or age one important

example of continuous data is time to

event data which represents the amount

of time until some event such as debt or

remission there are many types of data

but they're all haunted by the same

demon Randomness Randomness prevents us

from making perfect predictions with

data and winning big at the casino 100

of the time this raises an important

question how can we still learn from

data despite this Randomness statistics

can be viewed as the study of data which

includes everything from data collection

to data presentation statisticians are

in the business of looking past the

randomness and data when we think of

Randomness we usually think of it as

chaotic and uncontrolled but if we

assume that this Randomness has some

kind of structure behind it then we can

use this to our advantage this is where

probability Theory enters statistics the

central object that we use to represent

data is the random variable a random

variable can take on different values

with different probabilities random

variables are usually represented by

capital letters while actual values

taken from a random variable usually

lowercase letter random variables have

special functions that describe the

probabilities associated with each value

this function is called a probability

density function or probability Mass

function depending on the nature of the

data the random variable needs to match

the data so it can be either discrete or

continuous the shape of a PDF describes

the structure behind the randomness in

the data you may also hear it described

as the law of the data technically the

distribution can take any shape but

there are a few common ones you need to

know about the uniform distribution

tells us that all values are equally

likely the Bernoulli distribution

describes the distribution for a binary

data and tells us How likely we'll

observe a 1 which we call a success the

binomial distribution is similar but

tells us the probability for multiple

coin flips or successes the poisson

distribution describes counts and the

most famous distribution of all is the

normal distribution which has a

characteristic Bell shape the normal

distribution is used everywhere in

statistics and it's the namesake of the

channel the PDF is not the only way to

describe Randomness in the data there's

also the cumulative distribution

function or CDF and the CDF is useful

for defining quantiles and percentiles

of a random variable there are other

useful values that we use to

characterize random data we might want

to know what a typical value with

typicalness is captured in the measures

of central tendency the one most people

know is the mean known in technical

terms as the expectation or expected

value another measure of typicalness is

the median which marks the middle of a

data set in terms of the CDF finally

there's the mode which describes the

most common value defined by the peak of

the PDF we might also be interested in

the range of values a random variable

might take this can be described using

the measures of scale the variance gives

us a sense of how far values can be from

the expected value and the standard

deviation tells us the spread of data in

terms of the original data units the

measures of shape tell us more specific

details about a shape of a probability

distribution the skewness tells us the

imbalance in the distribution towards

one side or the other skewness implies

the presence about liars or extreme

values in a distribution kurtosis tells

us about the pointiness of a

distribution often we have to deal with

functions or transformations of a random

variable most commonly we deal with

functions of normal random variables

such as the T and case Squared

Distribution other times we want to see

how the probability of one random

variable is influenced by another in

this case as we want to know about the

conditional probability of that random

variable when random variables don't

influence each other we think of them as

independent another important concept

using conditional probability is Bayes

rule the Israel tells us that we should

change our beliefs Based on data that we

observe humans are natural bayesians but

maybe not so much in this polarized

World pager will gave rise to a

framework of Statistics called

bayesianism but most students are

trained in frequent statistics instead

these two schools have a lot of beef but

that's a subject for another video now

we'll use these probability tools to our

advantage statisticians start by

defining a population population is a

group that we're interested in but often

don't have the resources to fully

observe instead we're forced to collect

data from a small subset which we call a

sample next we assume that the data was

generated by a probability distribution

or mathematical formula this assumption

is our statistical model statisticians

translate an aspect of this population

into a parameter within this model

because the population is unobservable

this parameter is also unknown our goal

is to use the data to construct the

guests for the parameter which we call

an estimator this process is called

inferential statistics because we're

trying to infer about an unknown

population based on collected data this

is distinct from descriptive statistics

which are used to describe the data we

collect the first estimator that people

learn about is the sample mean we learn

about the sample mean because it has

many good qualities as an estimator the

law of large numbers tells us that when

we collect large amounts of data the

sample mean will get very close to the

population mean we call this consistency

because samples are random by extension

the sample mean is also random this

means that estimators are also random

variables and it's crucial that we

understand the distribution of the

estimator this distribution is so

special we give it a name the sampling

distribution if we know what it is we

can tell if observing a single sample

mean is likely or rare depending on the

distribution we find the SIM Central

limit theorem tells us that a function

of the sample mean is a standard normal

distribution assuming that we have lots

of data the law of large numbers and the

central limit theorem are examples of

asymptotic theorems and they're the

reason we try to get the biggest sample

sizes we can when asymptotics don't

apply we can possibly turn to other

methods like the bootstrap statisticians

translate beliefs about a population in

two statements about the population

parameters sampling distributions are

crucial to understanding hypothesis

tests there are two hypotheses we create

in hypothesis test the first is the null

hypothesis which represents a belief

about the world that we want to disprove

the second hypothesis is the alternative

hypothesis which opposes the null as an

example our parameter of interest will

be the difference between treatment

groups our null hypothesis is that there

is no difference between the groups so

the parameter is equal to zero as we

collect data we have a decision to make

we assume the null hypothesis is correct

and extend our logic from there if we

assume the null hypothesis is true it

suggests that we have particular

sampling distribution then we see where

our estimator lies relative to this null

distribution we want to know the

probability of observing our sample mean

or more extreme value this probability

is known as the infamous p-value if this

probability is low enough it suggests

that it's unlikely that the world under

the null hypothesis would have produced

the sample mean that we got we can also

make a decision based on a confidence

interval a range of parameter values

that could have realistically produced

our sample mean if this interval doesn't

contain the null hypothesis value then

we can also reject there's a duality

between p-values and confidence

intervals so we know they'll lead to the

same decision after making a decision

there are two ways that we can be wrong

a type 1 error happens when the truth is

that the null hypothesis is actually

correct but we decide to reject it this

is like saying the treatment works when

it actually doesn't a type 2 error

happens when the null hypothesis is

actually false but we fail to reject it

this is like saying good medicine

doesn't work ideally we want to minimize

the probability that both of these

errors occur minimizing one increases

the chances of another instead we Define

a low enough probability that we can

tolerate for a type 1 error which we

call the significance level after

setting this we minimize the probability

of a type 2 error this is also known as

maximizing power

[Applause]

there are lots of hypothesis tests we

can conduct depending on the question we

want to answer if we want to

characterize the population we can

perform a one sample test but if we want

to compare two groups we can use a two

sample test the central limit theorem

tells us that the sampling distributions

for these will be normal and we can take

advantage of this to produce a z

statistic Z is usually used to denote a

standard normal variable with zero mean

and unit variance C statistics assume

that we know the population variance but

this is unrealistic in practice if we

have to estimate this too it converts

the Z statistic into a t statistic don't

be surprised but a t statistic comes

from a t distribution which has a

slightly wider shape than a normal if

we're comparing three groups we can use

an analysis of variance or Anova to

check if they'll have the same mean all

of these tests assume continuous data or

large sample sizes but if we're dealing

with binary data we can construct the

contingency table and perform a

chi-square test these hypotheses tests

are all types of univari analyzes since

they focused on single random variables

if we want to check relationships

between variables we need regression

linear regression lets us see how one

variable influences another if the

outcome is binary or a count then we can

use a generalized linear model instead

to estimate the parameters in these

regression models we need to turn the

maximum likelihood estimation or an

optimization algorithm like

newton-raphson you might suspect that

treatment effects will vary over time so

you can collect data from people across

multiple occasions if you do this you'll

no longer have independent and

identically distributed data so you'll

need to shift to longitudinal modeling

for this we can use a GE model or a

mixed effect model to account for the

clustering effects we may also want to

include multiple predictors in all of

these models and choosing this set is

called variable selection we can do this

by asking experts in the field or

checking past research another rule of

thumb is to include potential

confounders which can muddy the

predictor outcome relationship if you

don't account for them it's very

important to know how data has been

collected if you have data from a

randomized controlled trial the results

from your experiment could be considered

causal instead of correl relational

without this randomization our data

comes from an observational design and

we definitely can make causal statements

with it this is why statisticians care

heavily about experimental design so

that we can know precisely what we can

conclude from our data in statistics we

must always keep our assumptions in mind

models are assumptions themselves and we

may have more depending on the model the

models I've discussed so far are

examples of parametric statistics if we

don't want to use a parametric model we

can use a non-parametric model instead

for example the man Whitney test is a

non-parametric form of the two-sample

t-test then there's also semi-parametric

statistics the most famous

semi-parametric model is the Cox model

used in survival analysis one part of

the model is parametric which describes

how survival changes with the treatment

while the non-parametric part is an

entire function called the hazard

function inference and description are

not the only statistical goals there's

also prediction where we try to predict

the value of a future observation based

on a Model we've estimated this starts

to venture into the field of machine

learning where you'll start to see black

box models modern statisticians deal

with very specific but exciting modeling

problems it's very common to assume that

the number of predictors is much less

than the sample size but in feels like

genetics this is rarely the case so we

need High dimensional statistics and

other exciting areas causal inference

which encompasses a set of techniques

that allow us to make causal statements

from observational data one of the

pivotal ideas in causal inference is the

counterfactual framework as I've

mentioned before we need several

assumptions and statistics if these

assumptions are violated then our models

are useless some reachers develop robust

statistics that will enable proper

inference even if these assumptions are

wrong if you're excited to start doing

statistics how do you start I recommend

learning a statistical programming

language so you can start playing with

these ideas python is good but it's a

general use language I recommend picking

up R since it's dedicated to statistical

analysis R makes it easier to examine

data and perform exploratory data

analyzes that can help inform how we

model data statistician use programming

to do extensive simulation studies to

test new models before they can publish

their work so that's also another skill

to learn finally there's no free lunch

and statistics statistics is hard work

and because she jobs aren't easily

earned everyone has data so everyone

will eventually need a statistician so

hopefully by watching this video someone

will finally need you as well thanks for

watching and I'll see you in the next

one

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