Statistics and probability | 1 Big School (By Blue Brain)

Course Content

Analyzing categorical data

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 and comparing quantitative data

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

Summarizing quantitative data

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

Modeling data distributions

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

Exploring bivariate numerical data

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

Study design

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

Probability

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

Counting, permutations, and combinations

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

Random variables

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

Sampling distributions

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

Confidence intervals

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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Course Content

Analyzing categorical data

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 and comparing quantitative data

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

Summarizing quantitative data

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