- Independent and dependent events
- Probability and combinatorics
- Random variables and probability distributions
- Descriptive statistics
- Regression
- Inferential statistics
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Independent and dependent events
Introduction to probability. Independent
and dependent events. Compound events. Mutual exclusive events.
Addition rule for probability.
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Basic probability
Can I pick a red frog out of a bag that only contains marbles? Is
it smart to buy a lottery ticket?
Even if we are unsure about whether something will happen, can we start
to be mathematical about the "chances" of an event (essentially
realizing that some things are more likely than others). This tutorial
will introduce us to the tools that allow us to think about random
events.
Venn diagrams and adding probabilities
What is the probability of getting a diamond or an ace from a deck
of cards? Well I could get a diamond that is not an ace, an ace that
is not a diamond, or the ace of diamonds. This tutorial helps us think
these types of situations through a bit better (especially with the help
of our good friend, the Venn diagram).
Compound, independent events
What is the probability of making three free throws in a row
(LeBron literally asks this in this tutorial).
In this tutorial, we'll explore compound events happening where the
probability of one event is not dependent on the outcome of another
(compound, independent, events).
- Compound Probability of Independent Events
- Getting At Least One Heads
- Example: Probability of rolling doubles
- LeBron Asks: What are the chances of making 10 free throws in a row?
- LeBron Asks: What are the chances of three free throws versus one three pointer?
- Frequency Probability and Unfair Coins
- Example: Getting two questions right on an exam
- Example: Rolling even three times
- Independent probability
- Frequency Stability
Dependent probability
What's the probability of picking two "e" from the bag in scrabble
(assuming that I don't replace the tiles). Well, the probability of
picking an 'e' on your second try depends on what happened in the first
(if you picked an 'e' the first time around, then there is one less 'e'
in the bag). This is just one of many, many type of scenarios involving
dependent probability.
Basic set operations
Whether you are learning computer science, logic, or probability
(or a bunch of other things), it can be very, very useful to have this
"set" of skills. From what a set is to how we can operate on them, this
tutorial will have you familiar with the basics of sets!
Old school probability (very optional)
Sal's old videos on probability. Covered better in other tutorials but here because some people actually like these better.
- Probability (part 1)
- Probability (part 2)
- Probability (part 3)
- Probability (part 4)
- Probability (part 5)
- Probability (part 6)
- Probability (part 7)
- Probability (part 8)
- Introduction to Random Variables
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Probability and combinatorics
Permutations and combinations. Using combinatorics to solve questions in probability.
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Permutations and combinations
If want to display your Chuck Norris dolls on your desk at school
and there is only room for five of them. Unfortunately, you own 50.
How many ways can you pick the dolls and arrange them on your desk?
What if you don't care what order they are in or how they are posed
(okay, of course you care about their awesome poses)?
Probability using combinatorics
This tutorial will apply the permutation and combination tools you
learned in the last tutorial to problems of probability. You'll
finally learn that there may be better "investments" than poring all
your money into the Powerball Lottery.
- Example: Probability through counting outcomes
- Example: All the ways you can flip a coin
- Getting Exactly Two Heads (Combinatorics)
- Probability and Combinations (part 2)
- Probability using Combinations
- Exactly Three Heads in Five Flips
- Example: Different ways to pick officers
- Example: Combinatorics and probability
- Example: Lottery probability
- Mega Millions Jackpot Probability
- Generalizing with Binomial Coefficients (bit advanced)
- Conditional Probability and Combinations
- Birthday Probability Problem
- Probability with permutations and combinations
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Random variables and probability distributions
Random variables. Expected value.
Probability distributions (both discrete and continuous). Binomial
distribution. Poisson processes.
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Descriptive statistics
Measures of central tendency and dispersion. Mean, median, mode, variance, and standard deviation.
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Measures of central tendency
This is the foundational tutorial for the rest of statistics. We
start thinking about how you can represent a set of numbers with one
number that somehow represents the "center". We then talk about the
differences between populations, samples, parameters and statistics.
Box-and-whisker plots
Whether you're looking at scientific data or stock price charts,
box-and-whisker plots can show up in your life. This tutorial covers
what they are, how to read them and how to construct them. We'd
consider this tutorial very optional, but it is a good application of
dealing with medians and ranges.
Variance and standard deviation
We have tools (like the arithmetic mean) to measure central
tendency and are now curious about representing how much the data in a
set varies from that central tendency. In this tutorial we introduce
the variance and standard deviation (which is just the square root of
the variance) as two commonly used tools for doing this.
- Variance of a population
- Sample variance
- Review and intuition why we divide by n-1 for the unbiased sample variance
- Simulation showing bias in sample variance
- Unbiased Estimate of Population Variance
- Another simulation giving evidence that (n-1) gives us an unbiased estimate of variance
- Simulation providing evidence that (n-1) gives us unbiased estimate
- Will it converge towards -1?
- Variance
- Statistics: Standard Deviation
- Exploring Standard Deviation 1 Module
- Exploring standard deviation 1
- Standard deviation
- Statistics: Alternate Variance Formulas
Sal's old statistics videos
This tutorial covers central tendency and dispersion. It is
redundant with the other tutorials on this topic, but it has the benefit
of messy handwriting and a cheap microphone. This is Sal circa 2007 so
take it all with a grain of salt (or just skip it altogether).
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Regression
Fitting a line to points. Linear regression. R-squared.
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Linear regression and correlation
Even when there might be a rough linear relationship between two
variables, the data in the real-world is never as clean as you want it
to be. This tutorial helps you think about how you can best fit a line
to the relationship between two variables.
- Estimating the line of best fit
- Correlation and Causality
- Squared Error of Regression Line
- Proof (Part 1) Minimizing Squared Error to Regression Line
- Proof Part 2 Minimizing Squared Error to Line
- Proof (Part 3) Minimizing Squared Error to Regression Line
- Proof (Part 4) Minimizing Squared Error to Regression Line
- Regression Line Example
- Second Regression Example
- R-Squared or Coefficient of Determination
- Calculating R-Squared
- Covariance and the Regression Line
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Inferential statistics
Making inferences based on sample data. Confidence intervals. Margin of error. Hypothesis testing.
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Normal distribution
The normal distribution (often referred to as the "bell curve" is
at the core of most of inferential statistics. By assuming that most
complex processes result in a normal distribution (we'll see why this is
reasonable), we can gauge the probability of it happening by chance.
To best enjoy this tutorial, it is good to come to it understanding what
probability distributions and random variables are. You should also be
very familiar with the notions of population and sample mean and
standard deviation.
- Introduction to the Normal Distribution
- Normal Distribution Excel Exercise
- ck12.org Normal Distribution Problems: Qualitative sense of normal distributions
- ck12.org Normal Distribution Problems: Empirical Rule
- ck12.org Normal Distribution Problems: z-score
- ck12.org Exercise: Standard Normal Distribution and the Empirical Rule
- Empirical rule
- ck12.org: More Empirical Rule and Z-score practice
- Z scores 1
- Z scores 2
- Z scores 3
Sampling distribution
In this tutorial, we experience one of the most exciting ideas in
statistics--the central limit theorem. Without it, it would be a lot
harder to make any inferences about population parameters given sample
statistics. It tells us that, regardless of what the population
distribution looks like, the distribution of the sample means (you'll
learn what that is) can be normal.
Good idea to understand a bit about normal distributions before diving
into this tutorial.
Confidence intervals
We all have confidence intervals ("I'm the king of the world!!!!")
and non-confidence intervals ("No one loves me"). That is not what
this tutorial is about.
This tutorial takes what you already know about the central limit
theorem, sampling distributions, and z-scores and uses these tools to
dive into the world of inferential statistics. It may seem magical at
first, but from our sample, we can now make inferences about the
probability of our population mean actually being in an interval.
Bernoulli distributions and margin of error
Ever wondered what pollsters are talking about when they said that
there is a 3% "margin of error" in their results. Well, this tutorial
will not only explain what it means, but give you the tools and
understanding to be a pollster yourself!
Hypothesis testing with one sample
This tutorial helps us answer one of the most important questions
not only in statistics, but all of science: how confident are we that a
result from a new drug or process is not due to random chance but due
to an actual impact.
If you are familiar with sampling distributions and confidence
intervals, you're ready for this adventure!
Hypothesis testing with two samples
You're already familiar with hypothesis testing with one sample.
In this tutorial, we'll go further by testing whether the difference
between the means of two samples seems to be unlikely purely due to
chance.
- Variance of Differences of Random Variables
- Difference of Sample Means Distribution
- Confidence Interval of Difference of Means
- Clarification of Confidence Interval of Difference of Means
- Hypothesis Test for Difference of Means
- Comparing Population Proportions 1
- Comparing Population Proportions 2
- Hypothesis Test Comparing Population Proportions
Chi-square probability distribution
You've gotten good at hypothesis testing when you can make
assumptions about the underlying distributions. In this tutorial, we'll
learn about a new distribution (the chi-square one) and how it can help
you (yes, you) infer what an underlying distribution even is!
Analysis of variance
You already know a good bit about hypothesis testing with one or
two samples. Now we take things further by making inferences based on
three or more samples. We'll use the very special F-distribution to do
it (F stands for "fabulous").
- ANOVA 1 - Calculating SST (Total Sum of Squares)
- ANOVA 2 - Calculating SSW and SSB (Total Sum of Squares Within and Between).avi
- ANOVA 3 -Hypothesis Test with F-Statistic
