Head First Statistics

Table of contents

1. Dedication
2. A Note Regarding Supplemental Files
3. Advance Praise for Head First Statistics
4. Praise for other Head First books
5. Author of Head First Statistics
6. How to use this Book: Intro
   1. Who is this book for?
      1. Who should probably back away from this book?
   2. We know what you’re thinking
   3. We know what your brain is thinking
   4. Metacognition: thinking about thinking
   5. Here’s what WE did
      1. Here’s what YOU can do to bend your brain into submission
   6. Read Me
   7. The technical review team
   8. Acknowledgments
   9. Safari® Books Online
7. 1. Visualizing Information: First Impressions
   1. Statistics are everywhere
   2. But why learn statistics?
   3. A tale of two charts
   4. Manic Mango needs some charts
   5. The humble pie chart
      1. So when are pie charts useful?
   6. Chart failure
   7. Bar charts can allow for more accuracy
   8. Vertical bar charts
   9. Horizontal bar charts
   10. It’s a matter of scale
       1. Using percentage scales
   11. Using frequency scales
   12. Dealing with multiple sets of data
       1. The split-category bar chart
       2. The segmented bar chart
   13. Your bar charts rock
   14. Categories vs. numbers
       1. Categorical or qualitative data
       2. Numerical or quantitative data
   15. Dealing with grouped data
   16. To make a histogram, start by finding bar widths
   17. Manic Mango needs another chart
       1. A histogram’s bar area must be proportional to frequency
   18. Make the area of histogram bars proportional to frequency
   19. Step 1: Find the bar widths
   20. Step 2: Find the bar heights
   21. Step 3: Draw your chart—a histogram
   22. Histograms can’t do everything
   23. Introducing cumulative frequency
       1. So what are the cumulative frequencies?
   24. Drawing the cumulative frequency graph
   25. Choosing the right chart
   26. Manic Mango conquered the games market!
8. 2. Measuring Central Tendency: The Middle Way
   1. Welcome to the Health Club
   2. A common measure of average is the mean
   3. Mean math
      1. Letters and numbers
   4. Dealing with unknowns
   5. Back to the mean
      1. The mean has its own symbol
   6. Handling frequencies
   7. Back to the Health Club
   8. Everybody was Kung Fu fighting
   9. Our data has outliers
   10. The butler outliers did it
   11. Watercooler conversation
   12. Finding the median
   13. Business is booming
   14. The Little Ducklings swimming class
   15. Frequency Magnets
   16. Frequency Magnets
   17. What went wrong with the mean and median?
   18. Introducing the mode
       1. It even works with categorical data
   19. Congratulations!
9. 3. Measuring Variability and Spread: Power Ranges
   1. Wanted: one player
   2. We need to compare player scores
   3. Use the range to differentiate between data sets
      1. Measuring the range
   4. The problem with outliers
   5. We need to get away from outliers
   6. Quartiles come to the rescue
   7. The interquartile range excludes outliers
   8. Quartile anatomy
      1. Finding the position of the lower quartile
      2. Finding the position of the upper quartile
   9. We’re not just limited to quartiles
   10. So what are percentiles?
       1. Percentile uses
       2. Finding percentiles
   11. Box and whisker plots let you visualize ranges
   12. Variability is more than just spread
   13. Calculating average distances
   14. We can calculate variation with the variance...
   15. ...but standard deviation is a more intuitive measure
       1. Standard deviation know-how
   16. A quicker calculation for variance
   17. What if we need a baseline for comparison?
   18. Use standard scores to compare values across data sets
       1. Calculating standard scores
   19. Interpreting standard scores
       1. So what does this tell us about the players?
   20. Statsville All Stars win the league!
10. 4. Calculating Probabilities: Taking Chances
    1. Fat Dan’s Grand Slam
    2. Roll up for roulette!
    3. Your very own roulette board
    4. Place your bets now!
    5. What are the chances?
    6. Find roulette probabilities
    7. You can visualize probabilities with a Venn diagram
       1. Complementary events
    8. It’s time to play!
    9. And the winning number is...
    10. Let’s bet on an even more likely event
    11. You can also add probabilities
    12. You win!
    13. Time for another bet
    14. Exclusive events and intersecting events
    15. Problems at the intersection
    16. Some more notation
    17. Another unlucky spin...
    18. ...but it’s time for another bet
    19. Conditions apply
    20. Find conditional probabilities
    21. You can visualize conditional probabilities with a probability tree
    22. Trees also help you calculate conditional probabilities
    23. Bad luck!
    24. We can find P(Black l Even) using the probabilities we already have
    25. Step 1: Finding P(Black ∩ Even)
    26. So where does this get us?
    27. Step 2: Finding P(Even)
    28. Step 3: Finding P(Black l Even)
    29. These results can be generalized to other problems
    30. Use the Law of Total Probability to find P(B)
    31. Introducing Bayes’ Theorem
    32. We have a winner!
    33. It’s time for one last bet
    34. If events affect each other, they are dependent
    35. If events do not affect each other, they are independent
    36. More on calculating probability for independent events
    37. Winner! Winner!
11. 5. Using Discrete Probability Distributions: Manage Your Expectations
    1. Back at Fat Dan’s Casino
    2. We can compose a probability distribution for the slot machine
    3. Expectation gives you a prediction of the results...
    4. ... and variance tells you about the spread of the results
    5. Variances and probability distributions
       1. So how do we calculate E(X – μ)2?
    6. Let’s calculate the slot machine’s variance
    7. Fat Dan changed his prices
    8. There’s a linear relationship between E(X) and E(Y)
    9. Slot machine transformations
    10. General formulas for linear transforms
    11. Every pull of the lever is an independent observation
    12. Observation shortcuts
        1. Expectation
        2. Variance
    13. New slot machine on the block
    14. Add E(X) and E(Y) to get E(X + Y)...
    15. ... and subtract E(X) and E(Y) to get E(X – Y)
    16. You can also add and subtract linear transformations
        1. Adding aX and bY
        2. Subtracting aX and bY
    17. Jackpot!
12. 6. Permutations and Combinations: Making Arrangements
    1. The Statsville Derby
    2. It’s a three-horse race
    3. How many ways can they cross the finish line?
    4. Calculate the number of arrangements
       1. So what if there are n horses?
    5. Going round in circles
    6. It’s time for the novelty race
    7. Arranging by individuals is different than arranging by type
    8. We need to arrange animals by type
    9. Generalize a formula for arranging duplicates
    10. It’s time for the twenty-horse race
    11. How many ways can we fill the top three positions?
    12. Examining permutations
    13. What if horse order doesn’t matter
    14. Examining combinations
    15. It’s the end of the race
13. 7. Geometric, Binomial, and Poisson Distributions: Keeping Things Discrete
    1. Meet Chad, the hapless snowboarder
    2. We need to find Chad’s probability distribution
    3. There’s a pattern to this probability distribution
    4. The probability distribution can be represented algebraically
    5. The pattern of expectations for the geometric distribution
    6. Expectation is 1/p
    7. Finding the variance for our distribution
    8. You’ve mastered the geometric distribution
    9. Should you play, or walk away?
    10. Generalizing the probability for three questions
        1. What’s the missing number?
    11. Let’s generalize the probability further
    12. What’s the expectation and variance?
        1. Let’s look at one trial
    13. Binomial expectation and variance
    14. The Statsville Cinema has a problem
        1. It’s a different sort of distribution
        2. So how do we find probabilities?
    15. Expectation and variance for the Poisson distribution
        1. What does the Poisson distribution look like?
    16. So what’s the probability distribution?
    17. Combine Poisson variables
    18. The Poisson in disguise
    19. Anyone for popcorn?
14. 8. Using the Normal Distribution: Being Normal
    1. Discrete data takes exact values...
    2. ... but not all numeric data is discrete
    3. What’s the delay?
    4. We need a probability distribution for continuous data
    5. Probability density functions can be used for continuous data
    6. Probability = area
    7. To calculate probability, start by finding f(x)...
    8. ... then find probability by finding the area
    9. We’ve found the probability
    10. Searching for a soul sole mate
    11. Male modelling
    12. The normal distribution is an “ideal” model for continuous data
    13. So how do we find normal probabilities?
    14. Three steps to calculating normal probabilities
    15. Step 1: Determine your distribution
    16. Step 2: Standardize to N(0, 1)
    17. To standardize, first move the mean...
    18. ... then squash the width
    19. Now find Z for the specific value you want to find probability for
    20. Step 3: Look up the probability in your handy table
        1. So how do you use probability tables?
    21. Julie’s probability is in the table
    22. And they all lived happily ever after
        1. But it doesn’t stop there.
15. 9. Using the Normal Distribution ii: Beyond Normal
    1. Love is a roller coaster
    2. All aboard the Love Train
    3. Normal bride + normal groom
    4. It’s still just weight
    5. How’s the combined weight distributed?
    6. Finding probabilities
    7. More people want the Love Train
    8. Linear transforms describe underlying changes in values...
       1. So what’s the distribution of a linear transform?
    9. ...and independent observations describe how many values you have
    10. Expectation and variance for independent observations
    11. Should we play, or walk away?
    12. Normal distribution to the rescue
    13. When to approximate the binomial distribution with the normal
        1. Finding the mean and variance
    14. Revisiting the normal approximation
    15. The binomial is discrete, but the normal is continuous
    16. Apply a continuity correction before calculating the approximation
    17. All aboard the Love Train
    18. When to approximate the binomial distribution with the normal
        1. When λ is small...
        2. When λ is large...
        3. So how large is large enough?
    19. A runaway success!
16. 10. Using Statistical Sampling: Taking Samples
    1. The Mighty Gumball taste test
    2. They’re running out of gumballs
    3. Test a gumball sample, not the whole gumball population
       1. Gumball populations
       2. Gumball samples
    4. How sampling works
    5. When sampling goes wrong
    6. How to design a sample
       1. Define your target population
       2. Define your sampling units
    7. Define your sampling frame
    8. Sometimes samples can be biased
       1. Unbiased Samples
       2. Biased Samples
    9. Sources of bias
    10. How to choose your sample
    11. Simple random sampling
        1. Sampling with replacement
        2. Sampling without replacement
    12. How to choose a simple random sample
        1. Drawing lots
        2. Random number generators
    13. There are other types of sampling
    14. We can use stratified sampling...
    15. ...or we can use cluster sampling...
    16. ...or even systematic sampling
    17. Mighty Gumball has a sample
        1. So what’s next?
17. 11. Estimating Populations and Samples: Making Predictions
    1. So how long does flavor really last for?
    2. Let’s start by estimating the population mean
    3. Point estimators can approximate population parameters
    4. Let’s estimate the population variance
    5. We need a different point estimator than sample variance
       1. So what is the estimator?
    6. Which formula’s which?
    7. Mighty Gumball has done more sampling
    8. It’s a question of proportion
       1. Predicting population proportion
    9. Buy your gumballs here!
       1. Introducing new jumbo boxes
    10. So how does this relate to sampling?
    11. The sampling distribution of proportions
    12. So what’s the expectation of Ps?
    13. And what’s the variance of Ps?
    14. Find the distribution of Ps
    15. Ps follows a normal distribution
        1. Ps—continuity correction required
    16. How many gumballs?
        1. There’s just one more problem...
    17. We need probabilities for the sample mean
    18. The sampling distribution of the mean
    19. Find the expectation for X̄
    20. What about the the variance of X̄?
    21. So how is X̄ distributed?
    22. If n is large, X̄ can still be approximated by the normal distribution
        1. Introducing the Central Limit Theorem
    23. Using the central limit theorem
        1. The binomial distribution
        2. The Poisson distribution
        3. Finding probabilities
    24. Sampling saves the day!
        1. You’ve made a lot of progress
18. 12. Constructing Confidence Intervals: Guessing with Confidence
    1. Mighty Gumball is in trouble
       1. They need you to save them
    2. The problem with precision
    3. Introducing confidence intervals
    4. Four steps for finding confidence intervals
    5. Step 1: Choose your population statistic
    6. Step 2: Find its sampling distribution
    7. Point estimators to the rescue
    8. We’ve found the distribution for X̄
    9. Step 3: Decide on the level of confidence
    10. How to select an appropriate confidence level
    11. Step 4: Find the confidence limits
    12. Start by finding Z
    13. Rewrite the inequality in terms of μ
    14. Finally, find the value of X̄
    15. You’ve found the confidence interval
    16. Let’s summarize the steps
    17. Handy shortcuts for confidence intervals
        1. What’s the interval in general?
    18. Just one more problem...
    19. Step 1: Choose your population statistic
    20. Step 2: Find its sampling distribution
    21. X̄ follows the t-distribution when the sample is small
    22. Find the standard score for the t-distribution
    23. Step 3: Decide on the level of confidence
    24. Step 4: Find the confidence limits
    25. Using t-distribution probability tables
    26. The t-distribution vs. the normal distribution
    27. You’ve found the confidence intervals!
19. 13. Using Hypothesis Tests: Look At The Evidence
    1. Statsville’s new miracle drug
    2. So what’s the problem?
    3. Resolving the conflict from 50,000 feet
    4. The six steps for hypothesis testing
    5. Step 1: Decide on the hypothesis
       1. The drug company’s claim
       2. So what’s the null hypothesis for SnoreCull?
    6. So what’s the alternative?
       1. The doctor’s perspective
       2. The alternate hypothesis for SnoreCull
    7. Step 2: Choose your test statistic
       1. What’s the test statistic for SnoreCull?
    8. Step 3: Determine the critical region
       1. At what point can we reject the drug company claims?
    9. To find the critical region, first decide on the significance level
       1. So what significance level should we use?
    10. Step 4: Find the p-value
        1. How do we find the p-value?
    11. We’ve found the p-value
    12. Step 5: Is the sample result in the critical region?
    13. Step 6: Make your decision
    14. So what did we just do?
    15. What if the sample size is larger?
    16. Let’s conduct another hypothesis test
    17. Step 1: Decide on the hypotheses
        1. It’s still the same problem
    18. Step 2: Choose the test statistic
    19. Use the normal to approximate the binomial in our test statistic
    20. Step 3: Find the critical region
    21. SnoreCull failed the test
    22. Mistakes can happen
    23. Let’s start with Type I errors
        1. So what’s the probability of getting a Type I error?
    24. What about Type II errors?
        1. So how do we find β?
    25. Finding errors for SnoreCull
        1. Let’s start with the Type I error
        2. So what about the Type II error?
    26. We need to find the range of values
    27. Find P(Type II error)
    28. Introducing power
        1. So what’s the power of SnoreCull?
    29. The doctor’s happy
        1. But it doesn’t stop there
20. 14. The χ2 Distribution: There’s Something Going On...
    1. There may be trouble ahead at Fat Dan’s Casino
    2. Let’s start with the slot machines
    3. The χ2 test assesses difference
    4. So what does the test statistic represent?
    5. Two main uses of the χ2 distribution
       1. When v is 1 or 2
       2. When v is greater than 2
    6. v represents degrees of freedom
       1. So what’s v?
    7. What’s the significance?
       1. How to use χ2 probability tables
    8. Hypothesis testing with χ2
    9. You’ve solved the slot machine mystery
    10. Fat Dan has another problem
    11. the χ2 distribution can test for independence
    12. You can find the expected frequencies using probability
    13. So what are the frequencies?
        1. How do we find the frequencies in general?
    14. We still need to calculate degrees of freedom
    15. Generalizing the degrees of freedom
    16. And the formula is...
    17. You’ve saved the casino
21. 15. Correlation and Regression: What’s My Line?
    1. Never trust the weather
    2. Let’s analyze sunshine and attendance
    3. Exploring types of data
       1. All about bivariate data
    4. Visualizing bivariate data
    5. Scatter diagrams show you patterns
    6. Correlation vs. causation
       1. We need to predict the concert attendance
    7. Predict values with a line of best fit
    8. Your best guess is still a guess
       1. We need to find the equation of the line
    9. We need to minimize the errors
    10. Introducing the sum of squared errors
    11. Find the equation for the line of best fit
        1. Let’s start with b
    12. Finding the slope for the line of best fit
        1. We use x̄ and ȳ to help us find b
    13. Finding the slope for the line of best fit, part ii
    14. We’ve found b, but what about a?
    15. You’ve made the connection
    16. Let’s look at some correlations
        1. Accurate linear correlation
        2. No linear correlation
    17. The correlation coefficient measures how well the line fits the data
    18. There’s a formula for calculating the correlation coefficient, r
    19. Find r for the concert data
    20. Find r for the concert data, continued
    21. You’ve saved the day!
    22. Leaving town...
    23. It’s been great having you here in Statsville!
22. A. Leftovers: The Top Ten Things (we didn’t cover)
    1. #1. Other ways of presenting data
       1. Dotplots
       2. Stemplots
    2. #2. Distribution anatomy
       1. The empirical rule for normal distributions
       2. Chebyshev’s rule for any distribution
    3. #3. Experiments
       1. So what makes for a good experiment?
    4. Designing your experiment
       1. Completely randomized design
       2. Randomized block design
       3. Matched pairs design
    5. #4. Least square regression alternate notation
    6. #5. The coefficient of determination
       1. Calculating r2
    7. #6. Non-linear relationships
    8. #7. The confidence interval for the slope of a regression line
       1. The margin of error for b
    9. #8. Sampling distributions – the difference between two means
    10. #9. Sampling distributions – the difference between two proportions
    11. #10. E(X) and Var(X) for continuous probability distributions
    12. Finding E(X)
    13. Finding Var(X)
23. B. Statistics Tables: Looking Things Up
    1. #1. Standard normal probabilities
    2. #2. t-distribution critical values
    3. #3. X2 critical values
24. Index
25. About the Author
26. Copyright