Relevant Chapter Review Exercise(s) Intro 3 R1.1. Related Example on Page(s) Intro 3 R R1.2, R R R R1.

Transcription

1 Name Chapter 1 Learning Objectives Identify the individuals and variables in a set of data. Classify variables as categorical or quantitative. Display categorical data with a bar graph. Decide whether it would be appropriate to make a pie chart. Identify what makes some graphs of categorical data deceptive. Calculate and display the marginal distribution of a categorical variable from a two-way table. Calculate and display the conditional distribution of a categorical variable for a particular value of the other categorical variable in a two-way table. Describe the association between two categorical variables by comparing appropriate conditional distributions. Make and interpret dotplots and stemplots of quantitative data. Describe the overall pattern (shape, center, and spread) of a distribution and identify any major departures from the pattern (outliers). Identify the shape of a distribution from a graph as roughly symmetric or skewed. Make and interpret histograms of quantitative data. Compare distributions of quantitative data using dotplots, stemplots, or histograms. Calculate measures of center (mean, median). Calculate and interpret measures of spread (range, IQR, standard deviation). Choose the most appropriate measure of center and spread in a given setting. Identify outliers using the 1.5 IQR rule. Make and interpret boxplots of quantitative data. Use appropriate graphs and numerical summaries to compare distributions of quantitative variables. Section Related Example on Page(s) Relevant Chapter Review Exercise(s) Intro 3 R1.1 Intro 3 R R1.2, R R R R R Dotplots: 25 Stemplots: 31 R Dotplots: 26 R1.6, R R1.6, R1.7, R1.8, R R1.7, R R1.8, R Mean: 49 Median: 52 IQR: 55 Std. dev: 60 R1.6 R R R1.6, R1.7, R R R1.8, R1.10 Can I do this? 1

2 1.1 Analyzing Categorical Data Read 2 4 Fr/Soph/Jr/Sr Exterior color address Name Bus route Phone number Days absent Address Credits earned Allergies Current on immunizations Total car length Number of cylinders Cost Model VIN Type of sound system Size of fuel tank g.p.a mileage What do we call these two kinds of variables? What s the difference? Why do people sometimes confuse the two kinds of variables? What is a distribution? It s all the values that a variable can take on and how often. 2

3 Alternate Example: Willott s music Here is information about 12 randomly selected songs in Willott s music library. Song Title Artist Album Track Tracks on Genre year Length the album Double Dare Bauhaus :54 Gothic 9 1 Carpe Noctum Tiesto :03 Dance/Electronic 12 4 She Wolf Shakira :10 Latin 12 1 Come as You Are Nirvana :39 Alternative 12 3 The Heinrich Maneuver Shake It Out My Songs Know What You Did in the Dark (Light Em Up) Track Number Interpol :35 Alternative 11 4 Florence + The Machine :38 Alternative 12 2 Fall Out Boy :07 Alternative 11 2 Locked Out of Heaven Bruno Mars :53 Pop 10 2 Womanizer Britney Spears :44 Pop 13 1 Iceolate Front Line Assembly :13 Industrial 10 7 I Bet You Look Good Arctic On The Dancefloor Monkeys :54 Indie 13 2 Meat is Murder The Smiths :06 Alternative 9 9 (a) Who are the individuals in this data set? (b) What variables are measured? Identify each as categorical or quantitative. In what units were the quantitative variables measured? (c) Describe the individual in the first row. Read 7 11 What's the difference between a data table, a frequency table, and a relative frequency table? Data table Frequency table Relative frequency table tells values of variables for individuals tells distribution of 1 variable in table form tells distribution of 1 variable as a %, decimal, or fraction Which one was the previous example? When making pie charts and bar graphs, what do people often mess up? 3

4 Bar Graphs Pie Charts Pros Quick & easy Show part-whole relationships well Cons part-whole relationships are hard to see They re hard to make by hand. Don't use when percents don't add up to 100%. Let's search "misleading graph" and see some examples. Identify some particular problems many of these graphs share. HW #11: page 7 (1, 3, 5, 7, 8), page 22 (11, 13, 15, 17, 18) Read Examples of: two-way table (2 variables are shown with counts or frequencies) Senior Non-senior Boy 8 3 Girl 15 4 marginal distribution (totals for rows & columns; the distribution for each variable) Senior Non-senior Totals Boy Girl Totals conditional distribution (distribution of one variable as a % of the other variable) Senior Non-senior Boy 35% 43% Girl 65% 57% Totals 100% 100% Senior Non-senior Totals Boy 73% 27% 100% Girl 79% 21% 100% How do we know which variable to condition on? Divide by the explanatory variable totals. Died Survived Hospital A Hospital B 4

5 What is a segmented (or stacked) bar graph? Use a segmented bar graph to compare conditional distributions, to look for differences, and to look for patterns. When knowing the value of one variable helps predict the value of the other, we say that the variables are associated. Association appears in a segmented bar graph when we see big differences in the proportions. The proportions may be flipped or reversed. Careful! An association does NOT automatically mean that there is a cause-and-effect relationship. The boy/girl senior/non-senior graphs did not show much association. Alternate Example: Horseshoe Crabs Two members of the University of Florida at Gainesville Department of Zoology collected data on Horseshoe Crabs on a Delaware beach during 4 days in the late spring of Based on the color of the shells, they classified each crab as Young, Intermediate, or Old and whether the crabs could right themselves when flipped on their backs or whether they were stranded for at least a certain period of time. Here are the results. Young Intermediate Old Total Stranded Not Stranded Total (a) Explain what it would mean if there was no association between age and strandedness. (b) Does there appear to be an association between age and strandedness in this sample? Justify. 5

6 HW #12: page 22 (19, 21, 23, 25, 27 34) And now, we change from categorical data to quantitative data 1.2 Displaying Quantitative Data with Graphs Elmer and Ethel have retired and want to move someplace warm. The couple is considering nine different cities. The dotplots below show the distribution of average daily high temperatures in December, January, and February for each of these cities. Help them pick a city by answering the questions below, based on the data shown in the graph. Average High Temperatures Dot Plot palmspring... atlantah phoenixh sandiegoh orlandoh miamih keywesth honoluluh sanjuanh What is the typical high temperature for these months in Phoenix, Orlando, and San Juan? Which of those 3 cities is most similar in this respect to Palm Springs? (Look for the center: the average, median, or typical value.) 2. Are daily high temperatures for these months more predictable in Palm Springs or in Orlando? (Look at the spread: the variation, including the range.) 3. What might be unique to Atlanta, San Diego, and Honolulu? (Look for outliers: unusual values.) 4. What makes San Juan and San Diego somewhat similar to one another? Likewise, Palm Springs, Phoenix, and Orlando are similar to one another in this way, but different from the first group. (Look at the shape: symmetry vs. asymmetry.) 6

7 Read Notice that we are now looking at quantitative data! How should we describe the distribution of a quantitative variable? Use SOCS Center- Typical value, such as the mean or the median Spread- Range for now (we'll also use standard deviation and interquartile range "IQR") Outliers- Unusual values for now (we'll eventually use the "1.5IQR Rule") Shape- Address the graph's # of peaks and its symmetry (unimodal, bimodal, multimodal, uniform, symmetric, asymmetric, skewed left, skewed right) Read Examples and descriptions of various shapes of distributions: Unimodal Symmetric Curve Dotplot Histogram Heights on adult women Expected sums on 36 rolls of two 6-sided dice Length of growing seasons in St. Louis Bimodal Curve Dotplot Histogram Heights of men and women Observed sums on 35 rolls of a 4-sided die and an 8-sided die Maximum angle of a sample of roller coasters Unimodal Skewed Left Curve Dotplot Histogram Heights of kids at a middle school dance Time to finish a difficult test Heights in my extended family Unimodal Skewed Right Curve Dotplot Histogram Salaries of MLB players Selling prices of homes in a new subdivision Uniform Curve Dotplot Histogram Scores on a multiple choice pre-test over completely new material Expected outcomes of spins of a spinner with equally-sized spaces Outcomes of 36 rolls Ages of students numbered 1-10 of a 6-sided die in a school district 7

8 Here are the number of calories per item for 16 convenience store sandwiches, along with a dotplot of the data Describe the shape, center, and spread of the distribution. Are there any outliers? Read When asked to compare two distributions, be sure that you compare and don t just describe! Be sure that you use less, more, and -er words. How does the annual energy consumption (kwh/year) compare for top-loading washing machines and frontloading washers? The data below is from the Home Depot website. There are 26 front-loaders and 32 toploaders included. Read Caution! Remember to include a key when making a stemplot (stem-and-leaf-plot). If you write "19 7", is that 197, 19.7, 1970,...? 8

9 How do gas prices in St. Charles County compare to those in Madison County, where Alton, Illinois is located? A sample of gas prices was taken on several days in July Make a back-to-back stemplot and compare the distributions. St. Charles Co.: 2.56, 2.56, 2.57, 2.57, 2.58, 2.58, 2.58, 2.58, 2.59, 2.59, 2.59, 2.59, 2.60, 2.60, 2.61 Madison Co.: 2.67, 2.68, 2.69, 2.69, 2.70, 2.70, 2.70, 2.71, 2.71, 2.71, 2.71, 2.72, 2.72, 2.73, 2.74 HW #13: page 41 (37, 39, 43, 45, 47) 1.2 Histograms The following table presents the total number of triples (3B) for the 30 MLB teams in the 2014 regular season. Make a dotplot to display the distribution of triples for the season. Then, use your dotplot to make a histogram of the distribution. Team 3B Team 3B Team 3B Arizona 47 Pittsburgh 30 Toronto 24 San Francisco 42 San Diego 30 Tampa Bay 24 Colorado 41 Kansas City 29 Cleveland 23 LA Dodgers 38 Milwaukee 28 Atlanta 22 Miami 36 Texas 28 St. Louis 21 Oakland 33 Minnesota 27 Boston 20 Chicago Sox 32 Washington 27 Cincinnati 20 Seattle 32 Philadelphia 27 Houston 19 LA Angels 31 Detroit 26 NY Mets 19 Chicago Cubs 31 NY Yankees 26 Baltimore 16 Read When you make a histogram......you can turn a dotplot into a histogram.... be consistent with "fence sitters".... be consistent with spacing and bin width. 9

10 Read When might we want a relative frequency histogram rather than a frequency histogram? to see part-whole relationships or to compare 2 groups HW #14: page 45 (51, 53, 55, 59 62) 1.3 Describing Quantitative Data with Numbers Read x is is a statistic; "x bar" is the sample mean. is a parameter; "mu" is the population mean. When adding a very large or very small data value to a data set (or changing a data value to something very large or very small) does not change the value of a statistic very much, or at all, we say that the statistic is resistant. The mean is not a resistant measure of center. Adding an extreme value, or altering a value to make it extreme, will change the value of the mean quite a bit. Think about what happens to the average age of people in the classroom when Mr. Willott walks in. The mean is the balancing point. Approximately where will the mean be located, when looking at a histogram or dotplot? Average High Temperatures Dot Plot StL_winter_Avg_High_Temps Read The median is a resistant measure of center. Adding an extreme value, or altering a value to make it extreme, will not change the value of the median much, if at all. Think about what happens to the median age of people in the classroom when Mr. Willott walks in. If we know the shape of a distribution, as shown below, then where are the mean and the median located in relation to one another? roughly symmetric exactly symmetric skewed 10

11 Read The range = highest data value minus lowest data value. The range is a single number and it is not a resistant measure of spread. An extreme value will affect the value of the range. Think about what happens to the range of ages of people in the classroom when Mr. Willott walks in. The median divides an ordered list of data into two equal groups. The quartiles divide an ordered list of data into four equal groups. The interquartile range (IQR) is the spread of the middle 50% of the data. The IQR is a resistant measure of spread. Think about what happens to the range of the middle 50% of ages of people in the classroom when Mr. Willott walks in. Here are data on the amount of fat (in grams) in 9 different Taco Bell menu items. Calculate the median, quartiles, and IQR. Read What is the 1.5 IQR Rule for identifying outliers? Item Crunchy Taco 10 Nachos Supreme 24 Cheese Quesadilla 26 Chicken Quesadilla 27 Mexican Pizza 31 Taco Salad (steak) 37 Nachos BellGrande 39 XXL Grilled Stuft Burrito Beef 41 Taco Salad (original) 42 Fat (g) Illustration by Kelly Boles 11

12 How many fat grams would qualify as an outlier for the Taco Bell items? Are there outliers among the 9 taco bell items? Here are data for the calories for 12 McDonald s menu items. Are there any outliers? Sandwich Calorie 32 oz. Chocolate Shake 1160 Big Breakfast 740 Big Mac 540 Sausage Biscuit with Egg 510 McRib pc. McNuggets 460 Double Cheeseburger 440 Quarter Pounder 410 Filet-O-Fish 380 McChicken 360 Large Caramel Latte 330 Large Vanilla Iced Coffee 270 Read The five-number summary: Minimum, Q1, Median, Q3, Maximum A boxplot is a graph that is related to the five-number summary. Draw a boxplot for the Taco Bell data. Check yours against the one that the graphing calculator makes. Item Fat (g) Crunchy Taco 10 Nachos Supreme 24 Cheese Quesadilla 26 Chicken Quesadilla 27 Mexican Pizza 31 Taco Salad (steak) 37 Nachos BellGrande 39 XXL Grilled Stuft Burrito Beef 41 Taco Salad (original) 42 Here are parallel boxplots for the heights of baseball players for 5 of the 2005 MLB teams. Compare these distributions. 12

13 HW #15: page 47 (65, 69 74), page 69 (79, 81, 83, 85, 86, 87, 89, 91, 93) 1.3 Standard Deviation Arnold ran each afternoon for 5 days. His distances (in miles) were 10, 10, 10, 10, and 10. Find the mean (or average) number of miles that Arnold ran each day. Complete the table: Table for Arnold's distances Distances Difference from the mean Square of difference from the mean Sum of squared differences: Sum of squared differences divided by 4 (since there were 5 distances): Square root of the sum of squared differences divided by 4: That last value is the standard deviation for the distances Arnold ran. What are the units? The number above it is the variance for the distances. What are the units? Becky ran each afternoon for 5 days. Her distances (in miles) were 8, 9, 10, 11, and 12. Find the mean (or average) number of miles that Becky ran each day. Complete the table: Table for Becky's distances Distances Difference from the mean Square of difference from the mean Sum of squared differences: Sum of squared differences divided by 4 (since there were 5 distances): Square root of the sum of squared differences divided by 4: That last value is the standard deviation for the distances Becky ran. What are the units? 13

14 The number above it is the variance for the distances. What are the units? Caleb ran each afternoon for 5 days. His distances (in miles) were 7, 9, 10, 11, and 13. Find the mean (or average) number of miles that Caleb ran each day. Complete the table: Table for Caleb's distances Distances Difference from the mean Square of difference from the mean Sum of squared differences: Sum of squared differences divided by 4 (since there were 5 distances): Square root of the sum of squared differences divided by 4: That last value is the standard deviation for the distances Caleb ran. What are the units? The number above it is the variance for the distances. What are the units? Donna ran each afternoon for 5 days. Her distances (in miles) were 3, 3, 4, 5, and 35. Find the mean (or average) number of miles that Donna ran each day. Complete the table: Table for Donna's distances Distances Difference from the mean Square of difference from the mean Sum of squared differences: Sum of squared differences divided by 4 (since there were 5 distances): Square root of the sum of squared differences divided by 4: That last value is the standard deviation for the distances Donna ran. What are the units? The number above it is the variance for the distances. What are the units? 14

15 The standard deviation measures the typical distance the data are from the mean. The range, IQR, and standard deviation all measure variation or spread, but only the IQR is resistant. Read Standard deviation Variance Square root of variance Square of standard deviation s= sample standard deviation s 2 = sample variance σ= population standard deviation σ 2 = population variance If s =4, then s 2 =16. If s 2 =9, then s=3. If σ 2 =25, then σ =5. If σ =6, then σ 2 =36. Four important properties of the standard deviation: Standard deviation 0. (0 means no variation, a large number means lots of variation.) Standard deviation units are the same as the units for the data. Standard deviation is not resistant. Standard deviation measures spread around the mean. s=5 s=6.22 s=9.52 s=10.7 A random sample of 5 students was asked how many minutes they spent listening to music outside school hours the previous day. They responded: 20, 30, 60, 90, 120. Calculate and interpret the standard deviation. Read Of mean, median, IQR, and standard deviation, which summary statistics will we typically use for each situation? Symmetric Skewed Center Spread 15

16 HW #16: page 71 (95, 97, 99, , ) FRAPPY! page 74 HW #17: page 76 Chapter Review Exercises Review Chapter 1 HW #18: page 78 Chapter 1 AP Statistics Practice Test Chapter 1 Test 16