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

2. Last week

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

4. What is probability? 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

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

6. Normal distribution Figure 7.4 – P157
Almost 100% of the scores fall between (-3SD, +3SD)
Around 34% of the scores fall between (0, 1SD)

7. Normal distribution The distance between contains
Range (if mean=100, SD=10)
Mean and 1SD
34.13% of all cases
1SD and 2SD
13.59% of all cases
2SD and 3SD
2.15% of all cases
>3SD
0.13% of all cases
>130
Mean and -1SD
90-100
-1SD and -2SD
80-90
-2SD and -3SD
70-80
< -3SD
<70

8. 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.

9. Z score Standard score X: the individual score : the mean
S: standard deviation

10. 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

11. Excel for z score Standardize(x, mean, standard deviation)
(a2-average(a2:a11))/STDEV(a2:a11)

12. 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,

13. 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

14. Exercise
Lab
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-p357)

15. If z is not integer Table B.1 (S-P357-358) Exercise
Lab
Table B.1 (S-P )
Exercise
The probability associated with z=1.38
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?

16. 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.

17. Exercise
Lab
What is the percentage for data to fall between 110 and 125 with the distribution of mean=100 and SD=10
Answer: 15.25%

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

19. Exercise
Lab
The probability of a particular score occurring between a z score of +1 and a z score of +2.5
15%

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

21. Exercise
Lab
Compute the z scores where mean=50 and the standard deviation =5 55 50 60
57.5 46

22. Exercise
Lab
Based on a distribution of scores with mean=75 and the standard deviation=6.38
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?