Joint Frequency Distributions. Lecture 6--Monday, Sept 27, 2010 Announcements. Lecture 6--Monday, Sept 27, 2010 Announcements 2 HOURS ONLY

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1 Lecture 6--Monday, Sept 27, 2010 Announcements Join PRS depending on the side of the room you are sitting on! Right side (as you look at the screens) use: < x >, < y > Left side (as you look at the screens) use: < x >, < y > Where x can be a letter or a number And y will be some other letter or number Lecture 6--Monday, Sept 27, 2010 Announcements PreLec 7 by Wednesday 4 pm Note SHORT deadlines. PreDS4 by Friday 10 am. Quiz 1a & 1b in OWL go up Wed. Expire next Wed before class. Recall Quiz Rules: no feedback, only 3 tries in same time period, including errors, goofs,, and not always the same question do you get 3 tries. Must be on a team or told TA you opt out Today s lecture is heavily visual. Do NOT try to copy slides! 2 HOURS ONLY More/124495/?sid=at&utm_source=at&utm_medium=en Joint Frequency Distributions bivariate set of data joint characteristics of two vars e.g., Folks on the Titanic by Fate and Class Also called: Contingency Table Cross-Tabulation Table Drinking and Gender of 212 Students a Joint Frequency Distribution Drinks on Weekend Gender None Some Lots Totals Female Male Totals See SPARK for a Camtasia audio/inking version of this lecture. Page 1 of 11

2 Drinking and Gender of 212 Students a Joint Relative Frequency Distribution Drinks on Weekend Gender None Some Lots Totals What proportion of Males from this population drink lots on the weekend? Report answer to three decimal places Gender None Some Lots Totals Gender None Some Lots Totals Female Male Female Male Female Male Totals Totals Totals Scatter Diagrams a two-dimensional graph of two variables an ordered pair of observations One var on the vertical axis One var and the horizontal axis Dependent and Independent A dependent variable depends on the values of another variable. (y-axis) An independent variable impacts the values of the dependent variable. (xaxis). Changes in affect y y = f() Price and Size of Houses Two types of studies: Observational Studies show association, but not causation Designed Experiments designed to show cause and effect See SPARK for a Camtasia audio/inking version of this lecture. Page 2 of 11

3 Scatter Diagrams Depicting Linearity Figure 2.15b Scatter Diagrams Depicting Linearity Scatter Diagrams Depicting Linearity What kind of chart is this? Select best answer. What kind of chart is this? Select best answer. 1.Bar chart 2.Historgram 3.Frequency Distribution 4.Ojive 5.Pareto chart See SPARK for a Camtasia audio/inking version of this lecture. Page 3 of 11

4 Pictures are Wonderful, but Speed Informs quickly But also Speed Misinforms the unwary viewer Must be careful in viewing i pictures Number of Bins (class width of bars) Watch scales Watch for missing intervening years Cute is Fine as long as: Emphasis on Reveal Not Conceal Make it easy to grasp the key information. Area Principle Violates Area Principle The area corresponds to the magnitude of the value it represents. Area = Importance Big Area = Big Amount of the Data Small Area = Small Amount of the Data Frequency People on the Titanic by Class First Second Third Crew Ticket Class allons Millions of Ga Year See SPARK for a Camtasia audio/inking version of this lecture. Page 4 of 11

5 s Lecture 6. Fall 2010 Professor Rogers--Intro Stats 09/27/2010, Yes but Dramatically? Millions of Gallons Millions of Gallo ons Year Millions of Gallons Year 0 Year Freshmen earn higher GPAs GPA Last 5 Years at UMass GPA Last 5 Years at UMass USA Snapshots Love Clever Charts GPA GPA Fall Fall Source: Daily Collegian, 9/3/08 p. 1 USA Today Loves Clever Charts USA TODAY Snapshots - USATODAY.com Screen clipping taken: 9/28/2009, 8:52 AM See SPARK for a Camtasia audio/inking version of this lecture. Page 5 of 11

6 The average amount of beer consumed per person per year in the U.S. is 22 gallons, or 235 bottles. Source: The Beer Institute LO Deceptive Graphs Descriptive Stats Objective LO9: Recognize deceptive graphing techniques Error 1: Nonzero Origin Error 2: Elastic Graph Proportions Error 3: Dramatic Title Error 4: Distracting Pictures Error 5: Authority Figures Error 6: 3-D and Rotated Graphs Error 7: Missing Axis Demarcations Error 8: Missing Measurement Units or Definitions Error 9: Vague Source Error 10: Complex Graphs Error 11: Gratuitous Effects Error 12: Estimated Data Error 13: Area Trick Summarize pile of numbers [population or sample] The pile is the Raw data Trade off complete detail and accuracy for speed in conveying key points Summarize Exam Scores What do you want to know? Will you Trade off complete accuracy for speed? Do you want a table or picture? Some of the Raw Data See SPARK for a Camtasia audio/inking version of this lecture. Page 6 of 11

7 To summarize, show: Middle or central tendency Spread or dispersion Shape of fthe Distribution tib ti Which best describes the shape of this distribution? Last year s Exam scores: uniform 2 - normal 3 - skewed left 4 - skewed right 5 - J shaped 6 - Reversed J shaped 7 - bimodal Which is the best description of this visual display of Exam 1 scores? a frequency histogram 2 - a frequency table 3 - a single-value grouping histogram 4 - an unordered stem and leaf diagram 5 - an ordered stem and leaf diagram 6 - an unordered stem and leaf diagram with two lines per leaf 7 - an ordered stem and leaf diagram with two lines per stem In Chapter 3 we used tables: Age (years) Frequency, f Class mark or midpoint f = n= 15 Frequency Or a Picture: Frequency Distribution Histogram Age of UMass Undergrads Age (years) Quickly see: Middle Dispersion Shape Now in Chapter 4 Change to Numerical Numbers to Show 1. Middle or central tendency 2. Spread or dispersion 3. Shape of the Distribution See SPARK for a Camtasia audio/inking version of this lecture. Page 7 of 11

8 Measures of Central Tendency Middle of the data the "M" words. "Average" is often used to refer to the mean (arithmetic) but there are many averages or measures of the middle; we will cover four of them today hence, be clear which "measure or even average" you are referring to. Notation signals Population Parameter or Sample Statistic μ x is population mean (a parameter) is the sample mean (a statistic) Formulas for each: Population Mean: Sample Mean: x μ = x = N n Population Mean Formula Complete with all notation where: N i= 1 μ = N i μ = population mean (mu) = response variable of interest [omit if clear which variable is being used] N = number of data values in pop i = ith individual value of variable Population Mean: Mean Age of current U.S. Supreme Court Justices Justice Age Kagan 50 Roberts 55 Scalia 74 Kennedy 74 Sotomayor 56 Thomas 62 Ginsburg 77 Breyer 72 Alito μ Population Mean The 9 Justices The population mean for the mean age of our current 9 Justices is computed as follows: = = ( )/9 N = 580 / 9 μ = 64.4 years See SPARK for a Camtasia audio/inking version of this lecture. Page 8 of 11

9 From our Age of UMass Students Example The Sample Mean is: The 15 ages (rounded) in our sample: The Raw Data: unorganized, hard to grasp key points: middle, spread, shape. x x 323 = = = 21.5years n 15 Formula for the Mean from a Frequency Distribution Table xf n Note: You need a xf column to sum; So add it to your table as a new column Calculating the Mean from a Frequency Distribution Table Age (years) Frequency, f Class mark midpoint (x) x*f Leave blank! f = n= 15 Leave blank! Formula for the Mean from a Frequency Distribution Table xf n = = 21.4years 15 Use the new xf column Loss some accuracy but gained speed Our Second Measure of the Middle Median: x% Read as x tilda. That x value that is physically in the middle of the sorted raw data, sorted from low to high. See SPARK for a Camtasia audio/inking version of this lecture. Page 9 of 11

10 Let x be the number of ounces of meat in a serving, n= 9 [odd number] Sort the raw data from low to high: The median is 10 ounces Remove one observation (the big one) so that n= 8 [now an even number] Sort the raw data from low to high: ? Take the midpoint from the two adjacent values: (8+10)/2 = 9 ounces The Third Measure of the Middle Mode (no symbol is used) Fourth Measure of the Middle Midrange (again no symbol is used) The most frequently occurring value That value that repeats the most If f = 1 for each value, then no mode If more than one value repeats as often, then you have multiple modes (some say no mode) midrange = Min + Max 2 Four Measures of the Middle Mean Median Mode Midrange The M-words [measures of the center] Will they be the same number? Most likely not, but how they relate to each other tells us something about the shape of the underlying distribution ib ti of observations. When they all are the same then it implies we have a Symmetric Distribution. See SPARK for a Camtasia audio/inking version of this lecture. Page 10 of 11

11 What about the influence of one extreme value? find the average UMass student s summer income. Take a sample of 5 students: 2k 1k 1k 3k 100k (drug dealer?; AIG Intern?) =107 Calculate the 4 Measures of the Middle Mean = 107/5 = $21.4k Median = $2k Sort data: 1, 1, 2, 3, 100 Mode = $1k Midrange = (1+100)/2 = $50.5k Which is the most likely? Which would an anti-umass story use? Which measure of location is the best? Mean is generally used, unless extreme values (outliers) exist Then median is often used, since the median is not sensitive to extreme values. Example: Median home prices may be reported for a region less sensitive to outliers 4 middle measures of a rightskewed distribution Reverse the relative positions for Left Skew. Right-Skewed Mode < Median < Mean< MR Exaggerated for emphasis! Big values really pull the mean toward them. See SPARK for a Camtasia audio/inking version of this lecture. Page 11 of 11