Quantitative Data Analysis: Statistics – Part 2

1. Quantitative Data Analysis: Statistics – Part 2

2. 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

3. The Normal Distribution

4. The Normal Distribution • Imagine we asked 100 employees of an organisation to rate their satisfaction with their job on a scale of 1 to 10, and we plotted it: Number Of people 1 2 3 4 5 6 7 8 9 10 Scale

5. The Normal Distribution • Is it likely we’d get an even distribution across all 10 scale points? 10 Number Of people 1 2 3 4 5 6 7 8 9 10 Scale

6. The Normal Distribution Not really!

7. The Normal Distribution • Let’s imagine it’s a crappy organisation and no one likes working there, then we’d get this sort of distribution: 40 Number Of people 1 2 3 4 5 6 7 8 9 10 Scale

8. The Normal Distribution • Or let’s imagine it’s a great organisation, then we’d get this sort of distribution: 40 Number Of people 1 2 3 4 5 6 7 8 9 10 Scale

9. The Normal Distribution • But what if it’s a middling sort of organisation that just average to work for?

10. The Normal Distribution • Then we’ll get this: 40 Number Of people 1 2 3 4 5 6 7 8 9 10 Scale

11. The Normal Distribution • Which looks like this 40 Number Of people 1 2 3 4 5 6 7 8 9 10 Scale

12. The Normal Distribution • As does this: 40 Number Of people 1 2 3 4 5 6 7 8 9 10 Scale

13. The Normal Distribution • As does this: 40 Number Of people 1 2 3 4 5 6 7 8 9 10 Scale

14. The Normal Distribution • Abraham de Moivre, the 18th century statistician and consultant to gamblers was often called upon to make lengthy computations about coin flips. de Moivre noted that when the number of events (coin flips) increased, the shape of the binomial distribution approached a very smooth curve. • In 1809 Carl Gauss developed the formula for the normal distribution and showed that the distribution of many natural phenomena are at least approximately normally distributed.

15. Abraham de Moivre • Born 26 May 1667 • Died 27 November 1754 • Born in Champagne, France • wrote a textbook on probability theory, "The Doctrine of Chances: a method of calculating the probabilities of events in play". This book came out in four editions, 1711 in Latin, and 1718, 1738 and 1756 in English. • In the later editions of his book, de Moivre gives the first statement of the formula for the normal distribution curve.

16. Carl Friedrich Gauss • Born 30 April 1777 • Died 23 February 1855 • Born in Lower Saxony, Germany • In 1809 Gauss published the monograph “Theoriamotuscorporumcoelestium in sectionibusconicissolemambientium” where among other things he introduces and describes several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution.

19. The Normal Distribution

20. The Normal Distribution • Age of students in a class • Body temperature • Pulse rate • Shoe size • IQ score • Diameter of trees • Height?

21. The Normal Distribution

22. The Normal Distribution

24. Density Curves: Properties

25. The Normal Distribution • The graph has a single peak at the center, this peak occurs at the mean • The graph is symmetrical about the mean • The graph never touches the horizontal axis • The area under the graph is equal to 1

26. Characterization • A normal distribution is bell-shaped and symmetric. • The distribution is determined by the mean mu, m, and the standard deviation sigma, s. • The mean mu controls the center and sigma controls the spread.

27. Same Mean, Different Standard Deviation 1 10

28. Different Mean, Different Standard Deviation 1 10

29. Different Mean, Same Standard Deviation 1 10

39. The Normal Distribution • If a variable is normally distributed, then: • within one standard deviation of the mean there will be approximately 68% of the data • within two standard deviations of the mean there will be approximately 95% of the data • within three standard deviations of the mean there will be approximately 99.7% of the data

40. The Normal Distribution

41. Why? • One reason the normal distribution is important is that many psychological and organsational variables are distributed approximately normally. Measures of reading ability, introversion, job satisfaction, and memory are among the many psychological variables approximately normally distributed. Although the distributions are only approximately normal, they are usually quite close.

42. Why? • A second reason the normal distribution is so important is that it is easy for mathematical statisticians to work with. This means that many kinds of statistical tests can be derived for normal distributions. Almost all statistical tests discussed in this text assume normal distributions. Fortunately, these tests work very well even if the distribution is only approximately normally distributed. Some tests work well even with very wide deviations from normality.

43. So what? • Imagine we undertook an experiment where we measured staff productivity before and after we introduced a computer system to help record solutions to common issues of work • Average productivity before = 6.4 • Average productivity after = 9.2

44. So what? 0 After = 9.2 10 Before = 6.4

45. So what? Is this a significant difference? 0 After = 9.2 10 Before = 6.4

46. So what? or is it more likely a sampling variation? 0 After = 9.2 10 Before = 6.4

47. So what? 0 After = 9.2 10 Before = 6.4

48. So what? 0 After = 9.2 10 Before = 6.4

49. So what? How many standard devaitions from the mean is this? 0 After = 9.2 10 Before = 6.4

50. So what? How many standard devaitions from the mean is this? and is it statistically significant? 0 After = 9.2 10 Before = 6.4