AP Statistics Introduction and Descriptive Statistics

AP Statistics Study Session : AP Statistics Study Session Introduction Descriptive Statistics

Introduction to Sessions : Introduction to Sessions Purpose of these sessions is to help you to succeed in the AP statistics exam It is also to help you to learn and retain statistics for University studies Research Job/Career Civic duties Being an informed and knowledgeable citizen

Who uses statistics? : Who uses statistics? Actuaries Scientists Psychologists Economists Engineers Pollsters Businesses

Who uses statistics? : Who uses statistics? Physicians Medical researchers Education Administrators Teachers Many, many different kinds of peoples If you do a search on PLoS for “p-value” you get about 5,000 hits. http://plosjournal.deepdyve.com/search?query=%22p-value%22

Format of Sessions : Format of Sessions Goals of the session: 1. Present review of material 2. Do questions together for practice and to help speed and understanding. We’ll pause for between 1 and 3 minutes to let everybody answer the question. Total class performance can be reported . If answer is wrong possibly offer material for further study.

Format of Sessions : Format of Sessions 3. Offer suggestions on how to study for the exam. 4. Can also cover material that is related to statistics but will not be on the exam (i.e. how to use R to make graphs) 5. Have fun! Feel free to ask questions

Format of Sessions : Format of Sessions Sessions will be free for now Informal and experimental Time of sessions I’ve set up a moodle site for the class http://www.dkfriedman.name/moodle May or may not use moodle. No specific schedule at this point, but I have a tentative list of topics

AP Statistics Curriculum : AP Statistics Curriculum Graphs and descriptive statistics Introduction to probability (binomial distribution, geometric distribution, etc.) The normal distribution Inference, confidence intervals for single variables Inference, hypothesis testing for single variables Confidence intervals and hypothesis testing for two variables

AP Statistics Curriculum : AP Statistics Curriculum Survey design, bias, and sampling methods Test of independence of variables, and test of homogeneity of populations Regression analysis Inference for regression analysis AP grading specific material No calculatorspeak Write legibly Four step method for hypothesis tests

Study Materials : Study Materials Jay Devore’s: Probability and Statistics for Scientists and Engineers, 5th edition The book that I used in my college class Martin Sternstein’s: AP Statistics, Barron’s. TI-NSpire calculator with a TI-84 keypad The R software package Bamboo tablet, headset, and camera

Textbooks & Calculators : Textbooks & Calculators Required review book: Martin Sternstein’s: AP Statistics, Barron’s. Auxiliary review book: Duane Hinders’s: 5 Steps to a 5: AP Statistics, McGraw Hill Calculator TI-84 calculator will be used in examples

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My Background : My Background Attended University of Virginia and graduated with highest distinction (majoring in computer science) Attended Johns Hopkins University and graduated with an M.S.E in computer science Currently working as a tutor for tutor.com Enjoying working with students and teaching them statistics and math.

Audience : Audience All are welcome AP Statistics students AP Statistics teachers Students who are considering taking AP Statistics Students in general Statistics professionals Mathematicians, scientists, and engineers Online teachers General public Feedback and comments are welcome

First Session : First Session In this initial pilot session we will talk about: Summary statistics/descriptive statistics Different kinds of charts and graphs

Summary Statistics : Summary Statistics Measures of center (central tendency) Mean Median Measures of spread Sample standard deviation Sample variance Range

Population vs. Sample : Population vs. Sample Example of population: all registered voters in the United States (millions) Example of sample: 500 selected voters in the United States It is often not practical to examine the whole population so we examine a sample and then seek to make an inference from that sample.

Population standard deviation (for a finite population) : Population standard deviation (for a finite population) s denotes the population standard deviation N is the total size of the entire population.

Population standard deviation(as generated by a probability distribution) : Population standard deviation(as generated by a probability distribution) We can cover this in more detail when we talk about probability However, for a discrete or continuous random variable X: Where the variance is well defined mathematically.

Sample Standard Deviation : Sample Standard Deviation s denotes the sample standard deviation

Standard deviation of the mean : Standard deviation of the mean The standard deviation of the mean is Where S is the standard deviation of the sample n is the size of the sample

Summary Statistics : Summary Statistics Five number summary: Minimum, First quartile, Median, Third quartile, Maximum We use the five number summary when creating a box plot

Formulas for Descriptive Statistics : Formulas for Descriptive Statistics Formula for mean (xi is the ith observation) Formula for sample standard deviation:

Median : Median The median is always that value such that half the sample is smaller than the median and half the sample is larger than the median. The formula for the median sample position depends on whether the sample size is an even or odd number.

Median : Median For a sample size n which is odd the median will be the ((n+1)/2)th value when they are in order (i.e. the 4th value in a sample of size 7, 3 or the left and 3 on the right) X1 X2 X3 X4 X5 X6 X7 Median

Median : Median For a sample size n which is even the median will be the average of the (n/2)th value and ((n/2) + 1)th value when they are in order (i.e. the 4th value and the 5th value in a sample of size 8) X1 X2 X3 X4 X5 X6 X7 X8 Median is the average of X4 and X5

First Questionhttp://www.surveymonkey.com/s/DZWPRZ6 : First Questionhttp://www.surveymonkey.com/s/DZWPRZ6

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Answers : Answer: A The median of (4,6,8,9,10,17,20) is 9 Answers

Quartiles : Quartiles According to McGraw Hill’s 5 Step to a 5: AP Statistics by Duane Hinders: First quartile is the median of the lower half (not including the median itself) Third quartile is the median of the upper half (not including the median itself) Other sources may give slightly different definitions. Roughly speaking the first quartile includes one fourth of the data and the third quartile includes three fourths of the data.

Quartiles : Quartiles For example, Excel uses a slightly different definition (i.e. it includes the median) Definitions may be similar but not exactly the same across different software or different scientific communities

Outliers : Outliers Definition of outliers is in terms of the quartiles: Interquartile range = third quartile – first quartile IQR = 3Q – 1Q An outlier is any point greater than 3Q + 1.5*IQR Or any point less than 1Q – 1.5*IQR

Descriptive Statistics : Descriptive Statistics Mean, median, minimum, maximum, quartiles, sample standard deviation, etc. On the AP exam you can use your calculator to compute these. On the TI-84: Put the values into a list (let’s say L1) Go to STAT->CALC->1-Var Stats 1-Var Stats L1 (can get L1 from the LIST menu) Hit ENTER

Standard Deviation Questions http://www.surveymonkey.com/s/D58NVZ7 : Standard Deviation Questions http://www.surveymonkey.com/s/D58NVZ7

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Answers : Answers

Answers : Answers 1. D. Would not change. This is consistent with the idea that we are measuring the spread—how far apart on average are the deviations from the mean. We can also see algebraically that the 3 cancels inside the formula (the mean will increase by 3 under this transformation).

Slide 42 : 2. B. Standard deviation would increase by a factor of 5. Under this transformation the mean increases by a factor of 5.In the equation for standard deviation we can factor out the 5 from within the summation. Then we can bring it out from the square root sign.

Slide 43 : 3. No. The standard deviation can never be negative. This is because it is the square root of a sum of squares and a square must be greater than or equal to 0.

Slide 44 : 4. Yes. Mathematically the standard deviation can be 0 if all data values are the same. In this case the mean will be the same as all the data values. In statistics though we usually deal with real data sets and not “pathological cases”. http://dictionary.reference.com/browse/pathological (Computing Dictionary)

Slide 45 : 5. No. The standard deviation is not resistant to outliers. If we added a large outlier this would result in a large difference from the new mean which would greatly increase the standard deviation.

Characterizing a distribution : Characterizing a distribution Center Spread Shape

Center : Center The mean and median are measures of center The median is resistant to outliers while the mean is not

Spread : Spread Range The range is the difference between the largest value and the smallest value Sample standard deviation Formula for standard deviation Sample variance Square of the sample standard deviation

Shape : Shape Bell shaped, normal Skewed right Skewed left

Shape: Bell-shaped, symmetric, approximately normal : Shape: Bell-shaped, symmetric, approximately normal

Shape: Skewed right : Shape: Skewed right

Shape: Skewed right : Shape: Skewed right Income levels are generally skewed right. Most people make an average income but there are a few people who make very large incomes Therefore the mean income level is higher than the median income level http://www.wolframalpha.com/input/?i=median+salary+vs+mean+salary

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Shape: Skewed left : Shape: Skewed left

Shape: Skewed left : Shape: Skewed left Weights of rowers on a rowing team Most rowers are probably fairly big and strong However, the coxswains are very light Life expectancy in a developing country could be skewed left. As soon as people reach adulthood their life expectancy increases because they can fend for themselves, but many die young because of malnutrition, disease, etc.

Different Kinds of Charts and Graphs : Different Kinds of Charts and Graphs Bar graph Stem-and-leaf plot Dot plot Histogram Box plot How to make plots with different software packages will not be on the AP exam but can be useful for projects.

Grade Dataset : Grade Dataset 50 students took an exam, and here are their grades: 70, 71, 72, 73, 74, 74, 75, 77, 77, 78, 81, 81, 81, 81, 82, 82, 82, 82, 84, 84, 84, 84, 84, 84, 85, 86, 86, 87, 87, 87, 88, 88, 89, 89, 90, 90, 90, 91, 91, 91, 91, 91, 91, 91, 92, 93, 94, 95, 95, 97

A histogram created with R : A histogram created with R Note: Bins exclude the lower bound but include the upper bound (a,b]. This is also the default in Excel. In R the first bin includes both end points, [a,b]

R commands used to create the histogram : R commands used to create the histogram grades <- c(70,71,72,73,74,74,75,77,77,78,81,81,81,81,82,82,82,82,84,84,84,84,84,84,85,86,86,87,87,87,88,88,89,89,90,90,90,91,91,91,91,91,91,91,92,93,94,95,95,97) myhist <- hist( grades, labels=TRUE, ylim = c(0,16), main="Histogram for Student Grades", xlab="Grades", ylab="Frequency" )

R doc pages : R doc pages Check doc page for function hist and for function plot

R investigation : R investigation Activity for somebody who would be interested and might have the time How to get a nice histogram with specified bucket ranges? This is what you get by just passing in a different bucket range

Bad Histogram : Bad Histogram The tick marks on the x axis aren’t in the right place when you put in your own breaks

R book on graphics : R book on graphics Murrell, P. (2005) R Graphics. Chapman & Hall/CRC Press.

R Resources : R Resources http://riki.wikidot.com/ wiki.r-project.org/ - No longer functioning Explore the program and report on your results R could also be used for an end of semester project.

Excel Histograms : Excel Histograms Excel can also do histograms. However, the bounds aren’t as nicely specified Appears to use bar graph code Means that labels are not to the right and left of the bar but below it However, it produces a table to compare the histogram with and you can also put data labels on top of the bars

Excel Histograms : Excel Histograms

Histogram Bucket Size : Histogram Bucket Size People have explored different ways to automate the choice of bucket size in a histogram Sturgis’s Rule: Set the number of intervals as close as possible to 1+Log2(N) This will not be on the AP exam.

Car Dealership Dataset : Car Dealership Dataset Over twenty months we count the number of cars sold by a car dealership

A stem-and-leaf plot created with R : A stem-and-leaf plot created with R

Do people use stem-and-leaf plots? : Do people use stem-and-leaf plots? Could only find one article in PLoS which uses a stem-and-leaf plot http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0005239 The idea was introduced by Tukey in his book Exploratory Data Analysis published in 1977 People could use it for back of the envelope type analysis/brainstorming which wouldn’t get into research papers Also, there are larger databases of research articles which could have more hits

Possible Project/Research Idea : Look at different statistical ideas and see how widely they are used Histogram has much more hits on PLoS Dotplot has got a few Boxplot and “box plot” get in total about 550 hits on PLoS Possible Project/Research Idea

Stem-and-leaf plots and the AP curriculum : Stem-and-leaf plots and the AP curriculum Are stem-and-leaf plots part of the AP curriculum? Yes.

Bar Graph Created Using R : Bar Graph Created Using R

R commands used to create the bar graph : R commands used to create the bar graph steel = c(4,2.5,3.5,4.5); barplot(steel, space=2, width=c(1,1,1,1), main="Tons of Alumnium Produced by Widget Inc.", xlab="Year", ylab="Aluminum Tons", names.arg = c("1950", "1951", "1952", "1953"), axes=FALSE); axis(1, labels=FALSE, lwd=1, lwd.ticks=0) ; axis(2, at=c(0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0,4.5,5.0,5.5)) ;

R shortcuts : R shortcuts Commands can be made simpler by using rep and seq rep(1,4) will produce a vector [1,1,1,1] seq(from=0,to=5.5,by=0.5) will produce [0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0,4.5,5.0,5.5]

R shortened command : R shortened command barplot(steel, space=2, width=rep(1,4), main="Tons of Alumnium Produced by Widget Inc.", xlab="Year", ylab="Aluminum Tons", names.arg = c("1950", "1951", "1952", "1953"), axes=FALSE); axis(1, labels=FALSE, lwd=1, lwd.ticks=0) ; axis(2, at=seq(from=0,to=5.5,by=0.5)) ;

Bar Graphs can also be done in Excel : Bar Graphs can also be done in Excel With Excel it may be easier to change style, color, labels, etc.

Lemonade Stand Data : Lemonade Stand Data Number of lemonades sold during a one week period

Dotplot Created Manually : Dotplot Created Manually Sunday Monday Tuesday Wednesday Thursday Friday Saturday Lemonade Stand Sales

R calls these stripcharts : R calls these stripcharts The dotplot function does something slightly different! Apparently the stripchart function can do this kind of chart http://en.wikipedia.org/wiki/Dot_plot_(statistics)

Chicago Bears Data : Chicago Bears Data Number of games won by the Chicago Bears (from wikipedia)

Boxplot Created Using R : Boxplot Created Using R Five number summary for Chicago Bears data Minimum: 4 First quartile: 5 Median: 7 Third quartile: 10 Maximum: 13

Boxplot Whiskers : Boxplot Whiskers The whiskers can either: Extend all the way out to the minimum and maximum Or only go up to the minimum and maximum values which are not outliers: An outlier is defined on the positive side to be anything larger than 3Q + 1.5*IQR And on the negative side to be anything smaller than 1Q – 1.5*IQR

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Graph Questionshttp://www.surveymonkey.com/s/CM8TSS7 : Graph Questionshttp://www.surveymonkey.com/s/CM8TSS7

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Answers : Answers

Slide 91 : 1. D. This is the agreed upon definition of outlier: Less than 1Q-1.5*IQR Greater than 3Q+1.5*IQR

Reviewer Comment : Reviewer Comment In 5 Steps to a 5 an older alternate definition of outlier is discussed:

Reviewer Comment : Reviewer Comment Is this a good definition? Under the empirical rule: If the distribution is normal and we sampled 1000 points, wouldn’t we expect to find about 40 points outside of 2 standard deviations and about 2 outside of 3 standard deviations? Shouldn’t our definition of outlier take into account the sample size? For larger sample size we would expect to find points outside of those bounds. Removing them would be removing valid data. Definition of Outlier:

Reviewer Comment: Response : Reviewer Comment: Response One could make the same argument about the quartile definition In the standard normal distribution the third quartile is at about +0.674 Meaning an outlier is anything above 0.674 + 1.5*(1.348) ˜ 2.70 The probability corresponding to 2.70 is about .997 so we could expect about 6 outliers in an n=1000 sample Also: under any definition there is no need to throw out outliers.

Slide 95 : 2. C. Since the buckets are smaller there will be less data points for that bucket and the height will be smaller.

Slide 96 : 3. B. It may or may not be an outlier. Keep in mind that when we add an additional point this changes the five number summary. If it is not that much larger it may not be an outlier in the new data set. 3. B. It may or may not be an outlier. Keep in mind that when we add an additional point this changes the five number summary. If it is not much larger it may not be an outlier in the new data set.

Slide 97 : 4. C. We need to know the five number summary: (minimum, first quartile, median, third quartile, and maximum) in order to make a boxplot.

A few brief additional comments(this will not be on the AP exam) : A few brief additional comments(this will not be on the AP exam)

Few Boxplot Comments : Few Boxplot Comments Boxplot is considered a misspelled word by Microsoft It is not included in Merriam-Webster’s dictionary nor is it included in dictionary.com’s dictionary (although “box plot” with a space is in dictionary.com’s dictionary)

Boxplot Comments : Boxplot Comments However, the statistics books I have all write boxplot and not “box plot”. Boxplot is not used in mainstream writing: (e.g. The Washington Post, The New York Times, etc.). Conjecture that it is considered too close to jargon by the dictionaries and not included. Eigenvalue (an older word) is included in both dictionaries.

Boxplot : Boxplot Who introduced the boxplot and when?

Boxplot : Boxplot John W. Tukey Famous statistician (classmate and friend of Richard Feynman – they were classmates at Princeton) First introduced in 1977 in Tukey’s book: Tukey, J. W. "Box-and-Whisker Plots." §2C in Exploratory Data Analysis. Reading, MA: Addison-Wesley, pp. 39-43, 1977.

Boxplot : Boxplot Wolfram MathWorld’s page also cites another book: Chambers, J.; Cleveland, W.; Kleiner, B.; and Tukey, P. Graphical Methods for Data Analysis. Belmont, CA: Wadsworth, 1983. http://mathworld.wolfram.com/Box-and-WhiskerPlot.html

Summary : Summary Descriptive statistics terms covered: Mean, median, range, sample variance, sample standard deviation, first quartile, third quartile, range Types of graphs covered: Histogram Stem-and-leaf plot Bar graph Dotplot Boxplot

Suggestions and Recommendations : Suggestions and Recommendations Purchase the review book if you haven’t already: Martin Sternstein, AP Statistics, Barron’s Going through the review book and doing the examples and problems can be a good way to prepare. Doing the examples and problems on your own will help you speed, accuracy, and increase your knowledge.

Suggestions and Recommendations : Suggestions and Recommendations The review book, textbook, and your teacher can be your first resources. However, if you get stuck on a problem or are having trouble understanding a concept I will be offering a tutoring service.

Tutoring Service : Tutoring Service The tutoring service will be through WiZiQ and will be at the following rates: $15 for 30 minutes $30 for 1 hour A $15 or $30 payment through PayPal is required in advance.

Suggestions and Recommendations : Suggestions and Recommendations Time for next study session If you are interested in the study session please fill out the survey so that we can determine a good time for the next one. http://www.surveymonkey.com/s/3VHQKWC

Next Session : Next Session The next topic: Regression Analysis Covered in Barron’s in topic 4. I’ll send out an e-mail announcement (about two or three days before) indicating when the next session is This can be a recurring time for the study session If you cannot make it to the session you can still see the recordings through WiZiQ.

Contact Information : Contact Information E-mail: David.Kit.Friedman@gmail.com