Quantitative Data Analysis: Statistics – Part 1

1. Quantitative Data Analysis: Statistics – Part 1

2. "... while man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician."

3. Overview • Part 1 • Picturing the Data • Pitfalls of Surveys • Averages • Variance and Standard Deviation • Part 2 • The Normal Distribution • Z-Tests • Confidence Intervals • T-Tests

4. ~ THE GOLDEN RULE ~ Statistics NEVER replace the judgment of the expert.

5. Approach to Statistical Research • Formulate a Hypothesis • State predictions of the hypothesis • Perform experiments or observations • Interpret experiments or observations • Evaluate results with respect to hypothesis • Refine hypothesis and start again (Basically the same as all other research)

6. Hypothesis Testing H0 :Null Hypothesis, status quo HA : Alternative Hypothesis, research question So, either : "The data does not support H0" or "We fail to reject H0"

7. Types of Data • Continuous • height, age, time • Discrete • # of days worked this week, # leaves on a tree • Ordinal • {Good, O.K., Bad} • Nominal • {Yes/No}, {Teacher/Chemist/Haberdasher}

8. Picturing The Data

10. Time-Series Plots • Time related Data • e.g. Stock Prices

12. Pie Charts • Nominal/Ordinal • Only suitable for data that adds up to 1 • Hard to compare values in the chart

14. Bar Charts • Nominal/Ordinal • Easier to compare values than pie chart • Suitable for a wider range of data

16. Histograms • Continuous Data • Divide Data into ranges

18. Dot Plots • Nominal/Ordinal • Represents all the data • Difficult to read

20. Scatter Plots • Excellent for examining association between two variables

22. Box Plots • Nominal/Ordinal • 1IQR, 3IQR - First interquartile range (IQR), third interquartile range (3QR) • Outliers

26. John Tukey • Born June 16, 1915 • Died July 26, 2000 • Born in New Bedford, Massachusetts • He introduced the box plot in his 1977 book,"Exploratory Data Analysis" • Also the Cooley–Tukey FFT algorithm and jackknife estimation

27. While working with John von Neumann on early computer designs, Tukey introduced the word "bit" as a contraction of "binary digit". The term "bit" was first used in an article by Claude Shannon in 1948. • The term "software", which Paul Niquette claims he coined in 1953, was first used in print by Tukey in a 1958 article in American Mathematical Monthly, and thus some people attribute the term to him. John Tukey Paul Niquette Claude Shannon John von Neumann

28. Question 1 In a telephone survey of 68 households, when asked do they have pets, the following were the responses : • 16 : No Pets • 28 : Dogs • 32 : Cats Draw the appropriate graphic to illustrate the results !!

29. Question 1 - Solution Total number surveyed = 68 Number with no pets = 16 =>Total with pets = (68 - 16) = 52 But total 28 dogs + 32 cats = 60 => So some people have both cats and dogs

31. Question 1 - Solution How many? It must be (60 - 52) = 8 people • No pets = 16 • Dogs = 20 • Cats = 24 • Both = 8 ------------------------- Total = 68

32. Question 1 - Solution Graphic: Pie Chart or Bar Chart

33. Question 1 - Solution Bar Chart is easier to read Graphic: Pie Chart or Bar Chart

34. Pitfalls of Surveys

35. The Literary Digest Poll • 1936 US Presidential Election • Alf Landon (R) vs. Franklin D. Roosevelt (D)

36. The Literary Digest Poll

37. Literary Digest had been conducting successful presidential election polls since 1916 They had correctly predicted the outcomes of the 1916, 1920, 1924, 1928, and 1932 elections by conducting polls. These polls were a lucrative venture for the magazine: readers liked them; newspapers played them up; and each “ballot” included a subscription blank. The Literary Digest Poll

38. The Literary Digest Poll • In 1936 they sent out 10 million ballots to two groups of people: • prospective subscribers, “who were chiefly upper- and middle-income people” • a list designed to "correct for bias" from the first list, consisting of names selected from telephone books and motor vehicle registries

40. The Literary Digest Poll • Response rate: approximately 25%, or 2,376,523 responses • Result: Landon in a landslide (predicted 57% of the vote, Roosevelt predicted 40%)

41. The Literary Digest Poll • Response rate: approximately 25%, or 2,376,523 responses • Result: Landon in a landslide (predicted 57% of the vote, Roosevelt predicted 40%) • Election result: Roosevelt received approximately 60% of the vote

42. The Literary Digest Poll • POSSIBLE CAUSES OF ERROR • Selection Bias: By taking names and addresses from telephone directories, survey systematically excluded poor voters. • Republicans were markedly overrepresented • in 1936, Democrats did not have as many phones,  not as likely to drive cars, and did not read the Literary Digest • “Sampling Frame” is the actual population of individuals from which a sample is drawn: Selection bias results when sampling frame is not representative of the population of interest

43. The Literary Digest Poll • POSSIBLE CAUSES OF ERROR • Non-response Bias: Because only 20% of 10 million people returned surveys, non-respondents may have different preferences from respondents • Indeed, respondents favored Landon • Greater response rates reduce the odds of biased samples

44. Definitions and Formula

45. Terminology • Population: is a set of entities concerning which statistical inferences are to be drawn. • Sample: a number of independent observations from the same probability distribution • Parameter: the distribution of a random variable as belonging to a family of probability distributions, distinguished from each other by the values of a finite number of parameters • Bias: a factor that causes a statistical sample of a population to have some examples of the population less represented than others.

46. Outliers (and their treatment)

47. Outliers (and their treatment) • An "outlier" is an observation that does not fit the pattern in the rest of the data • Check the data • Check with the measurer • If reason to believe it is NOT real, change it if possible, otherwise leave it out (but note). • If reason to believe it is real, leave it out and note.

48. The Mean • The Mean (Arithmetic) • The mean is defined as the sum of all the elements, divided by the number of elements. • The statistical mean of a set of observations is the average of the measurements in a set of data

49. The Mode • The mode is defined as the most frequently element in a set of elements. • For example [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] has a mode of 6. • Given the list of data [1, 1, 2, 4, 4] the mode is not unique - the dataset may be said to be bimodal, while a set with more than two modes may be described as multimodal.

50. The Median • The median is defined as the middle element, or the value separating the higher half of a sample from the lower half. • If there is an even number of elements, it is half the sum of the middle two elements. • Given the list of data [1, 1, 2, 4, 4] the median is 2.