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OPIM 5103 Descriptive Statistics Random Sampling Intro to Probability and Discrete Distributions Jan Stallaert Professor of OPIM Median Measures of Central Tendency Central Tendency Average Median Mode n X X i 1 N i 1 N Geometric Mean X G X1 X 2 n X i i Xn 1/ n Mean (Arithmetic Mean) • Mean (arithmetic mean) of data values – Sample mean Sample Size n X X1 X 2 X n – Population n mean i 1 i Population Size N X i 1 N Xn i X1 X 2 N XN 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 Excel function: =average(range) Mean = 6 Median • Robust measure of central tendency • Not affected by extreme values 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 • In an ordered array, the median is the “middle” number Excel function: =median(range) Measures of Variation Variation Variance Range Population Variance Sample Variance Interquartile Range Standard Deviation Population Standard Deviation Sample Standard Deviation Coefficient of Variation Example 9 8 7 6 5 4 3 2 1 0 120.00% 100.00% 80.00% 60.00% 40.00% 20.00% 25 5 4. 3. 75 2 2. -0 1 .2 5 0. 5 1. 25 .00% -3 4 .2 5 -2 . -1 5 .7 5 Frequency Histogram Bins 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 120.00% 100.00% 80.00% 60.00% 40.00% 20.00% Bins 5 3. 5 4. 25 2. 7 2 .00% -3 4 .2 5 -2 . -1 5 .7 5 -0 1 .2 5 0. 5 1. 25 Frequency Histogram Example 9 8 7 6 5 4 3 2 1 0 120.00% 100.00% 80.00% 60.00% 40.00% 20.00% 25 5 4. 3. 75 2 2. -0 1 .2 5 0. 5 1. 25 .00% -3 4 .2 5 -2 . -1 5 .7 5 Frequency Histogram Bins 9 8 7 6 5 4 3 2 1 0 120.00% 100.00% 80.00% 60.00% 40.00% 20.00% Bins 4 1 1. 75 2. 5 3. 25 .00% -1 2 .2 5 -0 .5 0. 25 -4 .2 5 -3 .5 -2 .7 5 Frequency Histogram Range • Measure of variation • Difference between the largest and the smallest observations: Range X Largest X Smallest • 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 Quartiles • Split Ordered Data into 4 Quarters 25% 25% Q1 25% Q2 25% Q3 • Q2= Median, A Measure of Central Tendency Excel function: =quartile(range, number) =0: minimum value =1: Q1 … =4: maximum value 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 Variance • Important measure of variation • Shows variation about the mean – Sample variance: n S2 X i 1 i X 2 n 1 • “Average of squared deviations from the mean” • “Standard deviation” = square root of variance Excel functions • Variance =VAR(range) • Standard Deviation =STDEV(range) Comparing Standard Deviations Data A 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 3.338 Data B 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = .9258 Data C 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 4.57 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 • S CV X 100% 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: S CV X $5 100% 100% 10% $50 S CV X $5 100% 100% 5% $100 Exploratory Data Analysis • Box-and-whisker plot – Graphical display of data using 5-number summary X smallest Q 1 4 6 Median( Q2) 8 Q3 10 Xlargest 12 Coefficient of Correlation • Measures the strength of the linear relationship between two quantitative variables n r X i 1 n X i 1 i i X Yi Y X 2 n Y Y i 1 i 2 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 Scatter Plots of Data with Various Correlation Coefficients Y Y Y X r = -1 X r = -.6 Y X r=0 Y r = .6 X r=1 X Producing Data • Sampling methods • Survey Errors Probability Sampling • Subjects of the sample are chosen based on known probabilities Probability Samples Simple Random Systematic Stratified Cluster 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 Random Samples 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 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 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) 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 Population and Sample Population Sample Use statistics to summarize features Use parameters to summarize features Inference on the population from the sample 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 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 when sample ≠ population Types of Survey Errors • Coverage error Excluded from frame. • Non response error Follow up on non responses. • Sampling error • Measurement error Chance differences from sample to sample. Bad Question! Measurement Errors • Question Phrasing Avoid negations • Telescoping Effect • “Halo” Effect • Overzealous/Underzealous Probability Probability • Probability is the numerical measure of the likelihood that an event will occur 1 Certain • Value is between 0 and 1 • Sum of the probabilities of all mutually exclusive and collective exhaustive events is 1 .5 0 Impossible Computing Probabilities • The probability of an event E: number of event outcomes P( E ) total number of possible outcomes in the sample space X T e.g. P( ) = 2/36 (There are 2 ways to get one 6 and the other 4) • Each of the outcomes in the sample space is equally likely to occur 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 Computing Joint Probability • The probability of a joint event, A and B: P(A and B) = P(A B) number of outcomes from both A and B total number of possible outcomes in sample space E.g. P(Red Card and Ace) 2 Red Aces 1 52 Total Number of Cards 26 Computing Compound Probability • Probability of a compound event, A or B: P( A or B) P( A B) number of outcomes from either A or B or both total number of outcomes in sample space E.g. P(Red Card or Ace) 4 Aces + 26 Red Cards - 2 Red Aces 52 total number of cards 28 7 52 13 Compound Probability (Addition Rule) P(A or B ) = P(A) + P(B) - P(A and B) P(A) P(A and B) P(B) For Mutually Exclusive Events: P(A or B) = P(A) + P(B) Computing Conditional Probability • The probability of event A given that event B has occurred: P( A and B) P( A | B) P( B) E.g. P(Red Card given that it is an Ace) 2 Red Aces 1 4 Aces 2 Conditional Probability American Int’l Total Men 0.25 0.15 0.40 Women 0.45 0.15 0.60 Total 0.70 0.30 Q: What is the probability that a randomly selected student is American, knowing that the student is female? Conditional Probability and Joint Probability • Conditional probability: P( A and B) P( A | B) P( B) • Multiplication rule for joint probability: P( A and B) P( A | B) P( B) P( B | A) P( A) Conditional Probability and Statistical Independence (continued) • Events A and B are independent if P( A | B) P ( A) or P( B | A) P( B) or P( A and B) P ( A) P ( B ) • Events A and B are independent when the probability of one event, A, is not affected by another event, B 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. 1. What is the probability that A and B deliver within 4 days? 2. What is the probability the process shuts down? 3. What is the probability at least one delivers in 4 days? 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 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 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” 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 Service Center Staffing 0.36417 0.371602 0.185801 0.06067 0.014548 0.002732 0.000418 5.36E-05 5.88E-06 5.6E-07 4.69E-08 3.48E-09 2.31E-10 1.38E-11 7.42E-13 3.64E-14 1.62E-15 6.63E-17 2.48E-18 8.52E-20 2.7E-21 7.86E-23 2.11E-24 5.25E-26 1.21E-27 2.56E-29 5.02E-31 9.11E-33 0.36417 0.735771 0.921572 0.982242 0.99679 0.999522 0.99994 0.999994 0.999999 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Assumptions - 50 computers sold - Prob. customer calls for service = 0.02 - Want < 5% that there is no engineer 1 0.9 0.8 0.7 Probability 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 Cumul. Prob. Probability 10 15 20 25 30 35 Number of Service Calls 40 45 50 Poisson Distribution • Poisson Process: P( X x | - x – Discrete events in an “interval” e • The probability of One Success x! in an interval is stable • The probability of More than One Success in this interval is 0 – The probability of success is independent from interval to interval – e.g.: number of customers arriving in 15 minutes – e.g.: number of defects per case of light bulbs Excel’s Poisson Function =POISSON(no. of occurences, mean, cumulative?) Example =POISSON(5,2,FALSE) (=0.036) “Probability that (exactly) five customers arrive wihtin an hour when the overall average is two” =POISSON(5,2,TRUE) (=0.983) “Probability that five or less customers arrive wihtin an hour when the overall average is two” Poisson Setting Examples • Number of accidents at an intersection in 6 months • Number of people entering a bank in a 30minute interval • Number of kids ringing the doorbell in 30 minutes for Halloween • Number of times a Microsoft machine crashes within 24 hours • Number of sewing flaws per (100) garment(s) Halloween 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.000335 0.002684 0.010735 0.028626 0.057252 0.091604 0.122138 0.139587 0.139587 0.124077 0.099262 0.07219 0.048127 0.029616 0.016924 0.009026 0.004513 0.002124 0.000944 0.000397 0.000159 0.000335 0.003019 0.013754 0.04238 0.099632 0.191236 0.313374 0.452961 0.592547 0.716624 0.815886 0.888076 0.936203 0.965819 0.982743 0.991769 0.996282 0.998406 0.99935 0.999747 0.999906 Assume: on average 4 kids /hour (=lambda) A Poisson Distribution 1 0.9 0.8 0.7 0.6 0.5 0.4 Probability Cum. Prob. 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9101112131415161718192021