100+ Statistics Concepts You Should Know
Summary
TLDRThis 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
😊 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.
😀 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.
📈 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.
Mindmap
Keywords
💡data
💡random variable
💡probability distribution
💡estimator
💡hypothesis test
💡p-value
💡statistical model
💡experimental design
💡assumptions
💡exploratory data 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
foreign
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