S519: Evaluation of Information Systems Social Statistics Chapter 7: Are your curves normal?

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

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

Normal distribution Figure 7.4 – P157 Almost 100% of the scores fall between (-3SD, +3SD) Around 34% of the scores fall between (0, 1SD) Are all distributions normal?

Normal distribution The distance betweencontainsRange (if mean=100, SD=10) Mean and 1SD34.13% of all cases SD and 2SD13.59% of all cases SD and 3SD2.15% of all cases >3SD0.13% of all cases>130 Mean and -1SD34.13% of all cases SD and -2SD13.59% of all cases SD and -3SD2.15% of all cases70-80 < -3SD0.13% of all cases<70

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

Z score Standard score X: the individual score : the mean : standard deviation Sample or population?

Z score Mean and SD for Z distribution? Mean=25, SD=2, what is the z score for 23, 27, 30?

Z score 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

Comparing apples and oranges: Eric competes in two track events: standing long jump and javelin. His long jump is 49 inches, and his javelin throw was 92 ft. He then measures all the other competitors in both events and calculates the mean and standard deviation: Javelin: M = 86ft, s = 10ft Long Jump: M = 44, s = 4 Which event did Eric do best in?

Excel for z score Standardize(x, mean, standard deviation) (x-average(array))/STDEV(array)

What z scores represent? 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? 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

Exercise 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? Lab What about 6σ

If z is not integer Table B.1 (S-P ) NORMSDIST(z) To compute the probability associated with a particular z score

Exercise The probability associated with z= % 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? Lab

Between two z scores What is the probability to fall between z score of 1.5 and 2.5 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.

Exercise What is the percentage for data to fall between 110 and 125 with the distribution of mean=100 and SD=10 Lab

Exercise The probability of a particular score occurring between a z score of +1 and a z score of +2.5 Lab

Exercise Compute the z scores where mean=50 and the standard deviation = Lab

Exercise The math section of the SAT has a μ = 500 and σ = 100. If you selected a person at random: a) What is the probability he would have a score greater than 650? b) What is the probability he would have a score between 400 and 500? c) What is the probability he would have a score between 630 and 700? Lab

Determine sample size Expected response rate: obtain based on historical data Number of responses needed: use formula to calculate

Number of responses needed n=number of responses needed (sample size) Z=the number of standard deviations that describe the precision of the results e=accuracy or the error of the results =variance of the data for large population size

Deciding from previous surveys intentionally use a large number conservative estimation e.g. a 10-point scale; assume that responses will be found across the entire 10-point scale 3 to the left/right of the mean describe virtually the entire area of the normal distribution curve =10/6=1.67; =2.78

Example Z=1.96 (usually rounded as 2) =2.78 e=0.2 n=278 (responses needed) assume response rate is 0.4 Sample size=278/0.4=695

Exercise Z=1.96 (usually rounded as 2) 5-point scale (suppose most of the responses are distributed from 1-4) error tolerance=0.4 assume response rate is 0.6 What is sample size?