STA 291 Fall 2009 1 LECTURE 4 8 September 2009 Administrative 2 (Review) Chapter 3: Sampling Plans Chapter 4: Graphical and Tabular Techniques Suggested problems from the textbook (not graded, but good as exam preparation): 3.1, 3.2, 3.3, 3.4, 3.5 , 3.7 Where we left off: Types of Bias 3 Selection Bias

Selection of the sample systematically excludes some part of the population of interest Measurement/Response Bias Method of observation tends to produce values that systematically differ from the true value Nonresponse Bias Occurs when responses are not actually obtained from all individuals selected for inclusion in the sample Where we left off: Sampling Error 4 Assume you take a random sample of 100 UK students and ask them about their political affiliation (Democrat, Republican, Independent) Now take another random sample of 100 UK students

Will you get the same percentages? Sampling Error (contd) 5 No, because of sampling variability. Also, the result will not be exactly the same as the population percentage, unless you 1. take a sample consisting of the whole population of 30,000 students (this would be called a ? ) - or 2. you get very lucky Sampling Error 6 Sampling Error is the error that occurs when a statistic based on a sample estimates or predicts the value of a population

parameter. In random samples, the sampling error can usually be quantified. In nonrandom samples, there is also sampling variability, but its extent is not predictable. Nonsampling Error 7 Any error that could also happen in a census, that is, when you ask the whole population Examples: Bias due to question wording, question order, nonresponse (people refuse to answer), wrong answers (especially to delicate questions) Descriptive Statistics 8

Summarize data Use graphs, tables (and numbers, see Chapter 4) Condense the information from the dataset Interval data: Histogram Nominal/Ordinal data: Bar chart, Pie chart Data Table: Murder Rates 9 Alabama Arizona California 11.6 Alaska 8.6 Arkansas 13.1 Colorado

9 10.2 5.8 Connecticut D.C. Georgia 6.3 Delaware 78.5 Florida 11.4 Hawaii 5 8.9 3.8 Difficult to see the big picture from these numbers

Try to condense the data Frequency Distribution 10 A listing of intervals of possible values for a variable And a tabulation of the number of observations in Murder Rate Frequency each interval. 0 2.9 5 3 5.9 16 6 8.9

12 9 11.9 12 12 14.9 4 15 17.9 0 18 20.9 1 > 21 1

Total 51 Frequency Distribution 11 Use intervals of same length (wherever possible) Intervals must be mutually exclusive: Any observation must fall into one and only one interval Rule of thumb: n If you have n observations, the number of intervals should be about Frequency, Relative Frequency, and Percentage Distribution 12

Murder Rate Frequency Relative Frequency Percentage 0 2.9 5 .10 ( = 5 / 51 ) 10 ( = .10 * 100% ) 3 5.9 16

.31 ( = 16 / 51 ) 31 ( = .31 * 100% ) 6 8.9 12 .24 24 9 11.9 12 .24 24

12 14.9 4 .08 8 15 17.9 0 0 0 18 20.9 1

.02 2 > 21 1 .02 2 Total 51 1 100 Frequency Distributions

13 Notice that we had to group the observations into intervals because the variable is measured on a continuous scale For discrete data, grouping may not be necessary (except when there are many categories) Frequency and Cumulative Frequency 14 Class Cumulative Frequency: Number of observations that fall in the class and in smaller classes Class Relative Cumulative Frequency: Proportion of observations that fall in the class and in smaller classes Cumulative Frequencies & Relative

Frequencies 15 Murder Rate Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency 0 2.9 5

.10 5 .10 3 5.9 16 .31 21 ( = 16 + 5) .41 (=.31 +.10) 6 8.9

12 .24 33 (= 12 + 21) .65(=.24+.4 1) 9 11.9 12 .24 12 14.9 4 .08

15 17.9 0 0 18 20.9 1 .02 > 21 1 .02 Total

51 1 Histogram (Interval Data) 16 Use the numbers from the frequency distribution to create a graph Draw a bar over each interval, the height of the bar represents the relative frequency for that interval Bars should be touching; i.e., equally extend the width of the bar at the upper and lower limits so that the bars are touching. Histogram (version I) 17 Histogram (version II) 18

Bar Graph (Nominal/Ordinal Data) 19 Histogram: for interval (quantitative) data Bar graph is almost the same, but for qualitative data Difference: The bars are usually separated to emphasize that the variable is categorical rather than quantitative For nominal variables (no natural ordering), order the bars by frequency, except possibly for a category other that is always last Pie Chart (Nominal/Ordinal Data) 20 First Step: Create a Frequency Distribution Highest Degree

Frequency (Number of Responses) Grade School 15 High School 200 Bachelors 185 Masters 55 Doctorate

70 Other 25 Total 550 Relative Frequency We could display this data in a bar chart 21 Pie Chart 22

Pie Chart: Pie is divided into slices; The area of each slice is proportional to the frequency of each class. Highest Degree Relative Frequency Angle ( = Rel. Freq. * 360 ) Grade School .027 ( = 15/550 ) 9.72 ( = .027 * 360 ) High School .364

131.04 Bachelors .336 120.96 Masters .100 36.0 Doctorate .127 45.72

Other .045 16.2 Pie Chart 23 Stem and Leaf Plot 24 Write the observations ordered from smallest to largest Each observation is represented by a stem (leading digit(s)) and a leaf (final digit) Looks like a histogram sideways Contains more information than a histogram, because every single measurement can be recovered

Stem and Leaf Plot 25 Useful for small data sets (<100 observations) Example of an EDA Practical problem: What if the variable is measured on a continuous scale, with measurements like 1267.298, 1987.208, 2098.089, 1199.082, 1328.208, 1299.365, 1480.731, etc. Use common sense when choosing stem and leaf Stem-and-Leaf Example: Age at Death for Presidents 26 Attendance Survey Question #4 On an index card Please write down your name and section number

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