| Fifty Fathoms Contents |
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Fifty Fathoms is a set of fifty statistics demonstrations. This is a list of them, with a description of the topics each demo addresses. You can also see the names of the files each demo uses. (The files are on the CD that comes with the book.)
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| Measures of Center and Spread | ||
| 1 | The Meaning of Mean | The mean • How individual values affect the mean |
| Meaning of Mean.ftm | ||
| 2 | Mean and Median | Measures of center: mean, median, and midrange • Resistance: what happens to the measures when you move one point |
| Mean and Median.ftm | ||
| 3 | What Do Normal Data Look Like? | Normally-distributed data • The effect of changing the mean and standard deviation |
| Normal Data.ftm | ||
| 4 | Transforming the Mean and Standard Deviation | What happens to mean and standard deviation when you add a constant to every value or multiply every value by a constant |
| Transform Mean and SD.ftm | ||
| 5 | The Mean is Least Squares, Too | Defining the mean as the place where the sum of squares is a minimum (just like the least-squares line) • The median, and what it minimizes |
| Mean is Least Squares.ftm Mean is Least Squares 2.ftm | ||
| Regression and Correlation | ||
| 6 | Least Squares Linear Regression | Explore the squares in least squares • Minimizing the areas of the squares built on residuals |
| Least Squares.ftm | ||
| 7 | Standard Scores | Using standard scores to compare unlike scales • Making a scale in terms of standard deviations |
| Standard Scores.ftm | ||
| 8 | Devising the Correlation Coefficient | How the correlation coefficient measures what it does |
| Correlation.ftm | ||
| 9 | Correlation Coefficients of Samples | How samples from a correlated population yield different values for the correlation • How sample size affects that sampling distribution |
| Correlation in Samples.ftm Correlation in Samples 2.ftm | ||
| 10 | Regression Towards the Mean | Regression towards the mean • The meaning—and asymmetry—of the least-squares line |
| Regression Towards the Mean.ftm | ||
| Random Walks and the Binomial Distribution | ||
| 11 | Flipping Coins—the Law of Large Numbers | How the proportion of heads approaches 0.5 as sample size increases • How the number of heads does not approach half the sample size |
| Law of Large Numbers.ftm Law of Large Numbers 2.ftm | ||
| 12 | How Random Walks Go as Root N | How the distance from the origin increases with the number of steps |
| Random Walk Root n.ftm Random Walk Root n part 2.ftm | ||
| 13 | Building the Binomial Distribution | Constructing the binomial distribution by resampling • How the distribution depends on the population proportion |
| Building Binomial.ftm Building Binomial part 2.ftm Building Binomial part 3.ftm | ||
| 14 | More Binomial | How the binomial distribution depends on sample size for small N • The relationship between the distribution of sample proportions and the distribution of sample counts |
| Binomial Small N.ftm | ||
| 15 | Two-Dimensional Random Walks | Unexpected behavior in 2D random walks • How the 2D walk eventually looks like a 2D normal distribution |
| 2D Random Walk.ftm | ||
| Standard Deviation, Standard Error, and Student's t | ||
| 16 | Standard Error and Standard Deviation | Getting a feel for the difference between standard deviation and standard error |
| SD and SE.ftm | ||
| 17 | What Is Standard Error, Really? | The connection between standard error and the sampling distribution of the mean • How the sample size connects standard deviation and standard error |
| What is SE.ftm | ||
| 18 | The Road to Student's t | Using standard error as the scale for measuring how far a sample mean is from the true mean • How these quantities are not normally distributed; in fact they follow a t distribution |
| Road to t.ftm | ||
| 19 | A Close Look at the t Statistic | How sample mean, standard deviation, t, and P interrelate • How they depend on the values of individual points in a sample |
| Close Look at Student's t.ftm | ||
| Sampling Distributions | ||
| 20 | The Distribution of Sample Proportions | How sample size and population proportion affect the distribution |
| Dist of Sample Props.ftm | ||
| 21 | Adding Uniform Random Variables | What happens when you add two uniform random variables • How that corresponds to adding two dice |
| Adding Uniform.ftm | ||
| 22 | How Errors Add | Basic error analysis • If two quantities each have some measurement error, finding the error in their sum |
| Adding Errors.ftm | ||
| 23 | Sampling Distributions and Sample Size | How sampling distributions (of the mean) get narrower as you increase sample size |
| Sampling Dists Sample Size.ftm | ||
| 24 | How the Width of the Sampling Distribution Depends on N | How the width (as measured by IQR) of a sampling distribution of the mean is inversely proportional to the square root of the sample size |
| Width of Dist Depends on N.ftm | ||
| 25 | Does n – 1 Really Work in the SD? | Unbiased estimators • How the familiar formula for sample standard deviation is not unbiased • Why we should care about variance |
| Estimating SD from a Sample.ftm | ||
| 26 | German Tanks | Unbiased estimators • Evaluating estimators from their sampling distributions • Even among unbiased estimators, some are better than others |
| Tanks.ftm Tanks2.ftm | ||
| 27 | The Central Limit Theorem | A demo of the CLT • How sampling distributions usually look normal • Cases where they do not |
| Central Limit Theorem.ftm | ||
| Confidence Intervals | ||
| 28 | The Confidence Interval of a Proportion | Defining the confidence interval • Looking at sample results in terms of plausibility |
| CI of a Proportion.ftm | ||
| 29 | Capturing with Confidence Intervals | How confidence intervals of a proportion do not always capture the population value |
| Capturing Props with CIs.ftm CI Stairway.ftm CI Stairway 2.ftm | ||
| 30 | Where Does That Root(p(1 – p)) Come From? | The standard deviation of a variable that's only zero or one • Connecting the “proportion” situation to the “mean” situation |
| SD of a Bernoulli Variable.ftm | ||
| 31 | Why np>10 is a Good Rule of Thumb | Explaining the np > 10 rule for using the normal approximation in the CI of a proportion |
| Why np is greater than 10.ftm Binomial v Normal.ftm | ||
| 32 | How the Width of the CI Depends on N | How the width of a confidence interval is inversely proportional to the square root of the sample size |
| Width of CI depends on N.ftm | ||
| 33 | Using the Bootstrap to Estimate a Parameter | The bootstrap • Using resampling (with replacement) to create an interval for a parameter |
| Bootstrap.ftm | ||
| 34 | Exploring the Confidence Interval of the Mean | How the CI depends on individual values |
| CI of the Mean.ftm | ||
| 35 | Capturing the Mean with Confidence Intervals | How confidence intervals of a mean do not always capture the population value • What repeated CIs look like |
| Capturing the Mean with CIs.ftm | ||
| Hypothesis Tests | ||
| 36 | Fair and Unfair Dice | Creating a measure of “fairness” • Sampling distributions • Testing hypotheses empirically • The chi-square statistic |
| Fair and Unfair Dice.ftm | ||
| 37 | Scrambling to Compare Means | Randomization test • Using scrambling to simulate the null hypothesis • Generating a sampling distribution |
| Scrambling.ftm Scrambling 2.ftm | ||
| 38 | Using a t-Test to Compare Means | Comparing means with Student's t |
| Compare Means Using t.ftm | ||
| 39 | Another Look at a t-Test | Repeated t-tests on samples from the same distribution • How t, P, mean, and standard deviation interrelate |
| Exploring t.ftm | ||
| 40 | On the Equivalence of Tests and Estimates | How a hypothesis test and a confidence interval are really the same |
| Tests and CIs.ftm | ||
| 41 | Paired Versus Unpaired | How a paired test gives a significant result more easily than its unpaired counterpart |
| Paired Versus Unpaired.ftm | ||
| 42 | Analysis of Variance | Assessing whether means are different in different groups • Introduction to ANOVA |
| Within and Between.ftm | ||
| Power in Tests | ||
| 43 | The Distribution of P-values | How the distribution of P is flat if the null hypothesis is true • How it changes if the null hypothesis is false |
| Distribution of P-values.ftm | ||
| 44 | Power | How power—the chance that you reject the null hypothesis— changes with the population parameters |
| Power.ftm | ||
| 45 | Power and Sample Size | How power—the chance that you reject the null hypothesis— changes with sample size |
| Power and Sample Size.ftm | ||
| 46 | Heteroscedasticity and its Consequences | Homoscedasticity is an assumption behind many statistical calculations. What happens when that assumption is not met? |
| Heteroscedasticity.ftm Heteroscedasticity 2.ftm | ||
| Distributions | ||
| 47 | Wait Time and the Geometric Distribution | The distribution of times until something happens • How this is the geometric distribution |
| Wait Time.ftm | ||
| 48 | The Exponential and Poisson Distributions | The continuous analog to the geometric distribution • How many events happen in a different time: a Poisson distribution |
| Exponential and Poisson.ftm | ||
| 49 | Sampling Without Replacement and the Hypergeometric Distribution | How distributions change when the sample is large compared to the population |
| Hypergeometric.ftm | ||
| 50 | The Bizarre Cauchy Distribution | The Cauchy distribution • The meaning of mean and standard deviation; how it's possible for a distribution to have neither |
| Cauchy.ftm | ||
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