S 519 Evaluation of Information Systems Social Statistics

S 519: Evaluation of Information Systems Social Statistics Chapter 7: Are your curves normal?

Last week

This week l l l Why understanding probability is important? What is normal curve How to compute and interpret z scores.

What is probability? l l l The chance of winning a lotter The chance to get a head on one flip of a coin Determine the degree of confidence to state a finding

Normal curve Symmetrical: (bellshaped) mean=median=mode Asymptotic: tail closer to the horizontal axis, but never touch.

Normal distribution l Figure 7. 4 – P 157 l l Almost 100% of the scores fall between (-3 SD, +3 SD) Around 34% of the scores fall between (0, 1 SD)

Normal distribution The distance between contains Range (if mean=100, SD=10) Mean and 1 SD 34. 13% of all cases 100 -110 1 SD and 2 SD 13. 59% of all cases 110 -120 2 SD and 3 SD 2. 15% of all cases 120 -130 >3 SD 0. 13% of all cases >130 Mean and -1 SD 34. 13% of all cases 90 -100 -1 SD and -2 SD 13. 59% of all cases 80 -90 -2 SD and -3 SD 2. 15% of all cases 70 -80 < -3 SD 0. 13% of all cases <70

Z score – standard score l l If you want to compare individuals in different distributions Z scores are comparable because they are standardized in units of standard deviations.

Z score l Standard score X: the individual score : the mean S: standard deviation

Z score l l Z scores across different distributions are comparable Z scores represent a distance of z score standard deviation from the mean Raw score 12. 8 (mean=12, SD=2) z=+0. 4 Raw score 64 (mean=58, SD=15) z=+0. 4 Equal distances from the mean

Excel for z score l l Standardize(x, mean, standard deviation) (a 2 -average(a 2: a 11))/STDEV(a 2: a 11)

What z scores represent? l l Raw scores below the mean has negative z scores Raw scores above the mean has positive z scores Representing the number of standard deviations from the mean The more extreme the z score, the further it is from the mean,

What z scores represent? l l 84% of all the scores fall below a z score of +1 (why? ) 16% of all the scores fall above a z score of +1 (why? ) This percentage represents the probability of a certain score occurring, or an event happening If less than 5%, then this event is unlikely to happen

Lab Exercise l In a normal distribution with a mean of 100 and a standard deviation of 10, what is the probability that any one score will be 110 or above? 16% Table B. 1 (s-p 357)

Lab If z is not integer l Table B. 1 (S-P 357 -358) l Exercise l The probability associated with z=1. 38 l l l 41. 62% of all the cases in the distribution fall between mean and 1. 38 standard deviation, About 92% falls below a 1. 38 standard deviation How and why?

Between two z scores l What is the probability to fall between z score of 1. 5 and 2. 5 l l l Z=1. 5, 43. 32% Z=2. 5, 49. 38% So around 6% of the all the cases of the distribution fall between 1. 5 and 2. 5 standard deviation.

Lab Exercise l What is the percentage for data to fall between 110 and 125 with the distribution of mean=100 and SD=10 l Answer: 15. 25%

Excel l NORMSDIST(z) l To compute the probability associated with a particular z score

Lab Exercise l The probability of a particular score occurring between a z score of +1 and a z score of +2. 5 15%

What can we do with z score? l l Research hypothesis presents a statement of the expected event We use statistics to evaluate how likely that event is. l l Z tests are reserved for populations T tests are reserved for samples

Lab Exercise l Compute the z scores where mean=50 and the standard deviation =5 l l l 55 50 60 57. 5 46

Lab Exercise l Based on a distribution of scores with mean=75 and the standard deviation=6. 38 l l What is the probability of a score falling between a raw score of 70 and 80? What is the probability of a score falling above a raw score of 80? What is the probability of a score falling between a raw score of 81 and 83? What is the probability of a score falling below a raw score of 63?