Head First Statistics. A Brain-Friendly Guide Dawn Griffiths

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

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