OPIM 5103 Descriptive Statistics Random Sampling Intro to Probability and Discrete Distributions

1. OPIM 5103 Descriptive StatisticsRandom SamplingIntro to Probability and Discrete Distributions Jan Stallaert Professor of OPIM

2. Median

3. Measures of Central Tendency Central Tendency Average Median Mode Geometric Mean

4. Mean (Arithmetic Mean) • Mean (arithmetic mean) of data values • Sample mean • Population mean Sample Size Population Size

5. Mean (Arithmetic Mean) • The most common measure of central tendency • Affected by extreme values (outliers) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5 Mean = 6 Excel function: =average(range)

6. Median • Robust measure of central tendency • Not affected by extreme values • In an ordered array, the median is the “middle” number 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Median = 5 Excel function: =median(range)

7. Measures of Variation Variation Variance Standard Deviation Coefficient of Variation Range Population Variance Population Standard Deviation Sample Variance Sample Standard Deviation Interquartile Range

8. Example

9. Example

10. Range • Measure of variation • Difference between the largest and the smallest observations: • Ignores the way in which data are distributed Range = 12 - 7 = 5 Range = 12 - 7 = 5 7 8 9 10 11 12 7 8 9 10 11 12

11. Quartiles • Split Ordered Data into 4 Quarters • = Median, A Measure of Central Tendency 25% 25% 25% 25% Excel function: =quartile(range, number) =0: minimum value =1: Q1 … =4: maximum value

12. Interquartile Range • Measure of spread/dispersion • Also known as midspread • Spread in the middle 50% • Difference between the first and third quartiles • Not affected by extreme values

13. Variance • Important measure of variation • Shows variation about the mean • Sample variance: • “Average of squared deviations from the mean” • “Standard deviation” = square root of variance

14. Excel functions • Variance =VAR(range) • Standard Deviation =STDEV(range)

15. Comparing Standard Deviations Data A Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Data C Mean = 15.5 s = 4.57 11 12 13 14 15 16 17 18 19 20 21

16. Coefficient of Variation • Measures relative variation • Always in percentage (%) • Shows variation relative to mean • Is used to compare two or more sets of data measured in different units

17. Comparing Coefficient of Variation • Stock A: • Average price last year = $50 • Standard deviation = $5 • Stock B: • Average price last year = $100 • Standard deviation = $5 • Coefficient of variation: • Stock A: • Stock B:

18. Exploratory Data Analysis • Box-and-whisker plot • Graphical display of data using 5-number summary Median( ) X X largest smallest 12 4 6 8 10

19. Coefficient of Correlation • Measures the strength of the linear relationship between two quantitative variables

20. Features of Correlation Coefficient • Unit free • Ranges between –1 and 1 • The closer to –1, the stronger the negative linear relationship • The closer to 1, the stronger the positive linear relationship • The closer to 0, the weaker any positive linear relationship

21. Scatter Plots of Data with Various Correlation Coefficients Y Y Y X X X r = -1 r = -.6 r = 0 Y Y X X r = 1 r = .6

22. Producing Data • Sampling methods • Survey Errors

23. Probability Sampling • Subjects of the sample are chosen based on known probabilities Probability Samples Simple Random Systematic Stratified Cluster

24. Simple Random Samples • Every individual or item from the frame has an equal chance of being selected • Selection may be with replacement or without replacement • Samples obtained from table of random numbers or computer random number generators

25. Random Samples

26. Systematic Samples • Decide on sample size: n • Divide frame of N individuals into groups of k individuals: k=N/n • Randomly select one individual from the 1st group • Select every k-th individual thereafter N = 64 n = 8 k = 8 First Group

27. Stratified Samples • Population divided into two or more groups according to some common characteristic • Simple random sample selected from each group • The two or more samples are combined into one

28. Advantages and Disadvantages • Simple random sample and systematic sample • Simple to use • May not be a good representation of the population’s underlying characteristics • Stratified sample • Ensures representation of individuals across the entire population • Cluster sample • More cost effective • Less efficient (need larger sample to acquire the same level of precision)

29. Key Definitions • A population (universe) is the collection of things under consideration • A sample is a portion of the frame selected for analysis • A parameter is a summary measure computed to describe a characteristic of the population • A statistic is a summary measure computed to describe a characteristic of the sample

30. Population and Sample Population Sample Use statistics to summarize features Use parameters to summarize features Inference on the population from the sample

31. Reasons for Drawing a Sample • Less time consuming than a census • Less costly to administer than a census • Less cumbersome and more practical to administer than a census of the targeted population

32. Evaluating Survey Worthiness • What is the purpose of the survey? • Is the survey based on a probability sample? • Coverage error – appropriate frame • Nonresponse error – follow up • Measurement error – good questions elicit good responses • Sampling error – always exists whensample ≠ population

33. Types of Survey Errors • Coverage error • Non response error • Sampling error • Measurement error Excluded from frame. Follow up on non responses. Chance differences from sample to sample. Bad Question!

34. Measurement Errors • Question Phrasing Avoid negations • Telescoping Effect • “Halo” Effect • Overzealous/Underzealous

35. Probability

36. Probability • Probability is the numerical measure of the likelihood that an event will occur • Value is between 0 and 1 • Sum of the probabilities of all mutually exclusive and collective exhaustive events is 1 1 Certain .5 0 Impossible

37. Computing Probabilities • The probability of an event E: • Each of the outcomes in the sample space is equally likely to occur e.g. P() = 2/36 (There are 2 ways to get one 6 and the other 4)

38. Empirical Probability Example: Find the probability that a randomly selected person will be struck by lightning this year . The sample space consists of two simple events: the person is struck by lightning or is not. Because these simple events are not equally likely, we can use the relative frequency approximation (Rule 1) or subjectively estimate the probability (Rule 3). Using Rule 1, we can research past events to determine that in a recent year 377 people were struck by lightning in the US, which has a population of about 274,037,295. Therefore, P(struck by lightning in a year) = 377 / 274,037,295 = 1/727,000

39. Computing Joint Probability • The probability of a joint event, A and B:

40. Computing Compound Probability • Probability of a compound event, A or B:

41. Compound Probability (Addition Rule) P(A or B ) = P(A) + P(B) - P(A and B) P(A and B) P(B) P(A) For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

42. Computing Conditional Probability • The probability of event A given that event B has occurred:

43. Conditional Probability Q: What is the probability that a randomly selected student is American, knowing that the student is female?

44. Conditional Probability and Joint Probability • Conditional probability: • Multiplication rule for joint probability:

45. Conditional Probability and Statistical Independence (continued) • Events A and B are independent if • Events A and B are independent when the probability of one event, A, is not affected by another event, B

46. Example • A company has two suppliers A and B. Rush orders are placed to both. If no raw material arrives in 4 days, the process shuts down. • A can deliver within 4 days with 55% probability. • B can deliver within 4 days with 35% probability. • What is the probability that A and B deliver within 4 days? • What is the probability the process shuts down? • What is the probability at least one delivers in 4 days?

47. Stock Trader’s Almanac • 1998 stock trader’s almanac has 48 years of data (1950-1997) • Stocks up in January: 31 times • Stocks up in year: 36 times • Stocks up in January AND year: 29 times

48. Binomial Probability Distribution • ‘n’ identical trials • e.g.: 15 tosses of a coin; ten light bulbs taken from a warehouse • Two mutually exclusive outcomes on each trials • e.g.: Head or tail in each toss of a coin; defective or not defective light bulb • Trials are independent • The outcome of one trial does not affect the outcome of the other • Constant probability for each trial • e.g.: Probability of getting a tail is the same each time we toss the coin

49. Excel’s Binomial Function =BINOMDIST(no. of successes, no. of trials, prob. of success, cumulative?) Example =BINOMDIST(2,8,0.5, FALSE) (=0.11) “Probability of tossing (exactly) two heads within 8 trials” =BINOMDIST(2,8,0.5, TRUE) (=0.14) “Probability of tossing two heads or less within 8 trials”

50. Binomial Setting Examples • Number of times newspaper arrives on time (i.e., before 7:30 AM) in a week/month • Number of times I roll “5” on a die in 20 rolls • Number of times I toss heads within 20 trials • Students pick random number between 1 and 10. Number of students who picked “7” • Number of people who will vote “Republican” in a group of 20 • Number of left-handed people in a group of 40