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Statistics and probability

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#### Course content

#### Analyzing categorical data

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#### Displaying and comparing quantitative data

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#### Summarizing quantitative data

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#### Modeling data distributions

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#### Exploring bivariate numerical data

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#### Study design

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#### Probability

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#### Counting, permutations, and combinations

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#### Random variables

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#### Sampling distributions

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#### Confidence intervals

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389 Lessons

Analyzing one categorical variable

Identifying individuals, variables and categorical variables in a data set

Reading pictographs

Reading bar graphs

Reading bar graphs: Harry Potter

Creating a bar graph

Reading bar charts: comparing two sets of data

Reading bar charts: putting it together with central tendency

Reading pie graphs (circle graphs)

Two-way relative frequency tables

Interpreting two-way tables

Categorical data example

Analyzing trends in categorical data

Two-way tables

Two-way frequency tables and Venn diagrams

Distributions in two-way tables

Marginal and conditional distributions

Displaying quantitative data with graphs

Representing data

Frequency tables & dot plots

Creating a histogram

Interpreting a histogram

Stem-and-leaf plots

Reading stem and leaf plots

Describing and comparing distributions

Shapes of distributions

Clusters, gaps, peaks & outliers

Comparing distributions with dot plots (example problem)

Comparing dot plots, histograms, and box plots

Example: Comparing distributions

More on data displays

Reading line graphs

Misleading line graphs

Measuring center in quantitative data

Statistics intro: Mean, median, & mode

Mean, median, & mode example

More on mean and median

Comparing means of distributions

Means and medians of different distributions

Impact on median & mean: removing an outlier

Impact on median & mean: increasing an outlier

Missing value given the mean

Median & range puzzlers

Interquartile range (IQR)

Interquartile range (IQR)

Variance and standard deviation of a population

Measures of spread: range, variance & standard deviation

Variance of a population

Population standard deviation

Mean and standard deviation versus median and IQR

Statistics: Alternate variance formulas

Variance and standard deviation of a sample

Sample variance

Sample standard deviation and bias

More on standard deviation

Why we divide by n – 1 in variance

Simulation showing bias in sample variance

Simulation providing evidence that (n-1) gives us unbiased estimate

Review and intuition why we divide by n-1 for the unbiased sample variance

Box and whisker plots

Worked example: Creating a box plot (odd number of data points)

Worked example: Creating a box plot (even number of data points)

Constructing a box plot

Reading box plots

Interpreting box plots

Judging outliers in a dataset

Other measures of spread

Range and mid-range

Mean absolute deviation (MAD)

Mean absolute deviation example

Percentiles

Calculating percentile

Analyzing a cumulative relative frequency graph

Z-scores

Normal distribution problem: z-scores (from ck12.org

Comparing with z-scores

Effects of linear transformations

How parameters change as data is shifted and scaled

Density curves

Density Curves

Median, mean and skew from density curves

Density curve worked example

Worked example finding area under density curves

Normal distributions and the empirical rule

Qualitative sense of normal distributions (from ck12.org)

Normal distribution problems: Empirical rule (from ck12.org)

Standard normal distribution and the empirical rule (from ck12.org)

More empirical rule and z-score practice (from ck12.org)

Normal distribution calculations

Standard normal table for proportion below

Standard normal table for proportion above

Standard normal table for proportion between values

Finding z-score for a percentile

Threshold for low percentile

Normal distribution excel exercise

More on normal distributions

Deep definition of the normal distribution

Introduction to scatterplots

Constructing a scatter plot

Example of direction in scatterplots

Scatter plot: smokers

Bivariate relationship linearity, strength and direction

Correlation coefficients

Example: Correlation coefficient intuition

Calculating correlation coefficient r

Introduction to trend lines

Fitting a line to data

Estimating the line of best fit exercise

Estimating with linear regression (linear models)

Line of best fit: smoking in 1945

Least-squares regression equations

Introduction to residuals and least squares regression

Calculating residual example

Calculating the equation of a regression line

Interpreting slope of regression line

Interpreting y-intercept in regression model

Interpreting a trend line

Interpreting computer regression data

Impact of removing outliers on regression lines

Assessing the fit in least-squares regression

Residual plots

R-squared or coefficient of determination

Standard deviation of residuals or Root-mean-square error (RMSD)

More on regression

Squared error of regression line

Proof (part 1) minimizing squared error to regression line

Proof (part 2) minimizing squared error to regression 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

Calculating R-squared

Covariance and the regression line

Statistical questions

Statistical questions

Statistical and non statistical questions

Sampling and observational studies

Reasonable samples

Identifying a sample and population

Examples of bias in surveys

Example of undercoverage introducing bias

Correlation and causality

Sampling methods

Picking fairly

Techniques for generating a simple random sample

Techniques for random sampling and avoiding bias

Types of studies (experimental vs. observational)

Types of statistical studies

Worked example identifying experiment

Worked example identifying observational study

Worked example identifying sample study

Appropriate statistical study example

Experiments

Introduction to experiment design

Matched pairs experiment design

Basic theoretical probability

Intro to theoretical probability

Simple probability: yellow marble

Simple probability: non-blue marble

Intuitive sense of probabilities

The Monty Hall problem

Probability using sample spaces

Probability with counting outcomes

Example: All the ways you can flip a coin

Die rolling probability

Subsets of sample spaces

Basic set operations

Intersection and union of sets

Relative complement or difference between sets

Universal set and absolute complement

Subset, strict subset, and superset

Bringing the set operations together

Experimental probability

Experimental probability

Theoretical and experimental probabilities

Making predictions with probability

Randomness, probability, and simulation

Experimental versus theoretical probability simulation

Random number list to run experiment

Random numbers for experimental probability

Statistical significance of experiment

Addition rule

Probability with Venn diagrams

Addition rule for probability

Free-throw probability

Three-pointer vs free-throw probability

Probability without equally likely events

Independent events example: test taking

Die rolling probability with independent events

Multiplication rule for independent events

Sample spaces for compound events

Compound probability of independent events

Probability of a compound event

Coin flipping probability

Multiplication rule for dependent events

Dependent probability introduction

Dependent probability: coins

Dependent probability example

Independent & dependent probability

Dependent probability

Conditional probability and independence

Calculating conditional probability

Conditional probability explained visually

Conditional probability tree diagram example

Conditional probability and independence

Analyzing event probability for independence

Counting principle and factorial

Count outcomes using tree diagram

Counting outcomes: flower pots

Permutations

Permutation formula

Zero factorial or 0!

Factorial and counting seat arrangements

Possible three letter words

Ways to arrange colors

Ways to pick officers

Combinations

Intro to combinations

Combination formula

Handshaking combinations

Combination example: 9 card hands

Combinatorics and probability

Probability using combinations

Probability & combinations (2 of 2)

Example: Different ways to pick officers

Example: Combinatorics and probability

Getting exactly two heads (combinatorics)

Exactly three heads in five flips

Generalizing with binomial coefficients (bit advanced)

Example: Lottery probability

Conditional probability and combinations

Mega millions jackpot probability

Birthday probability problem

Discrete random variables

Random variables

Discrete and continuous random variables

Discrete and continuous random variables

Constructing a probability distribution for random variable

Probability models example: frozen yogurt

Valid discrete probability distribution examples

Probability with discrete random variable example

Mean (expected value) of a discrete random variable

Variance and standard deviation of a discrete random variable

Continuous random variables

Probability density functions

Probabilities from density curves

Transforming random variables

Impact of transforming (scaling and shifting) random variables

Example: Transforming a discrete random variable

Combining random variables

Mean of sum and difference of random variables

Variance of sum and difference of random variables

Intuition for why independence matters for variance of sum

Deriving the variance of the difference of random variables

Example: Analyzing distribution of sum of two normally distributed random variables

Example: Analyzing the difference in distributions

Graphing basketball binomial distribution

Binompdf and binomcdf functions

Binomial random variables

Binomial variables

Recognizing binomial variables

10% Rule of assuming “independence” between trials

Binomial distribution

Visualizing a binomial distribution

Binomial probability example

Generalizing k scores in n attempts

Free throw binomial probability distribution

Binomial mean and standard deviation formulas

Mean and variance of Bernoulli distribution example

Bernoulli distribution mean and variance formulas

Expected value of a binomial variable

Variance of a binomial variable

Finding the mean and standard deviation of a binomial random variable

Geometric random variables

Geometric random variables introduction

Probability for a geometric random variable

Cumulative geometric probability (greater than a value)

Cumulative geometric probability (less than a value)

TI-84 geometpdf and geometcdf functions

Proof of expected value of geometric random variable

More on expected value

Term life insurance and death probability

Getting data from expected value

Expected profit from lottery ticket

Expected value while fishing

Comparing insurance with expected value

Law of large numbers

Poisson distribution

Poisson process 1

Poisson process 2

What is a sampling distribution?

Introduction to sampling distributions

Sample statistic bias worked example

Sampling distribution of a sample proportion

Sampling distribution of sample proportion part 1

Sampling distribution of sample proportion part 2

Normal conditions for sampling distributions of sample proportions

Probability of sample proportions example

Sampling distribution of a sample mean

Inferring population mean from sample mean

Central limit theorem

Sampling distribution of the sample mean

Sampling distribution of the sample mean 2

Standard error of the mean

Example: Probability of sample mean exceeding a value

Introduction to confidence intervals

Confidence intervals and margin of error

Confidence interval simulation

Interpreting confidence level example

Estimating a population proportion

Confidence interval example

Margin of error 1

Margin of error 2

Conditions for valid confidence intervals

Conditions for confidence intervals worked examples

Critical value (z*) for a given confidence level

Example constructing and interpreting a confidence interval for p

Determining sample size based on confidence and margin of error

Estimating a population mean

Introduction to t statistics

Simulation showing value of t statistic

Conditions for valid t intervals

Example finding critical t value

Example constructing a t interval for a mean

Confidence interval for a mean with paired data

Sample size for a given margin of error for a mean

More confidence interval videos

T-statistic confidence interval

Small sample size confidence intervals

The idea of significance tests

Simple hypothesis testing

Idea behind hypothesis testing

Examples of null and alternative hypotheses

P-values and significance tests

Comparing P-values to different significance levels

Estimating a P-value from a simulation

Error probabilities and power

Introduction to Type I and Type II errors

Type 1 errors

Examples identifying Type I and Type II errors

Introduction to power in significance tests

Examples thinking about power in significance tests

Calculating a z statistic in a test about a proportion

Calculating a P-value given a z statistic

Making conclusions in a test about a proportion

Tests about a population proportion

Constructing hypotheses for a significance test about a proportion

Conditions for a z test about a proportion

Tests about a population mean

Writing hypotheses for a significance test about a mean

Conditions for a t test about a mean

When to use z or t statistics in significance tests

Example calculating t statistic for a test about a mean

Using TI calculator for P-value from t statistic

Using a table to estimate P-value from t statistic

Comparing P-value from t statistic to significance level

Free response example: Significance test for a mean

More significance testing videos

Hypothesis testing and p-values

One-tailed and two-tailed tests

Z-statistics vs. T-statistics

Small sample hypothesis test

Large sample proportion hypothesis testing

Comparing two proportions

Comparing population proportions 1

Comparing population proportions 2

Hypothesis test comparing population proportions

Comparing two means

Statistical significance of experiment

Statistical significance on bus speeds

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

Inference about slope

Introduction to inference about slope in linear regression

Conditions for inference on slope

Confidence interval for the slope of a regression line

Calculating t statistic for slope of regression line

Using a P-value to make conclusions in a test about slope

Using a confidence interval to test slope

Nonlinear regression

Comparing models to fit data example

Transforming nonlinear data

Worked example of linear regression using transformed data

Analysis of variance (ANOVA)

ANOVA 1: Calculating SST (total sum of squares)

ANOVA 2: Calculating SSW and SSB (total sum of squares within and between)

ANOVA 3: Hypothesis test with F-statistic

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