1. Plan for today: Chapter 13: Normal distribution

2. Normal Distribution

3. Normal Distribution: Two variables we usually use to describe a sample. (Although five-number summary is also a good choice.) Measure of central tendency: sample mean: Measure of variability or “spread” : sample std dev:

4. Normal Distribution: So we want to find a distribution which is fully determined by those two numbers. For a general distribution, it’s not true.

5. Normal Distribution: Normal density curve is symmetric, bell-shaped.

6. Normal Distribution: A specific Normal curve is completely described by giving its mean and its standard deviation. (pic form: Wikipedia)

7. Normal Distribution: The mean determines the center of the distribution. It is located at the center of symmetry of the curve. (pic form: Wikipedia) Thus mean = median for Normal Distribution

8. Normal Distribution: The standard deviation determines the shape of the curve. (pic form: Wikipedia)

9. Why Normal Distribution: You may wonder: Why we call the distribution normal distribution? For example, flip a fair coin 100 times, the proportion of getting head is. What if we repeat the procedure m times. how those distributed.

10. Day 1Day 2Day 3Day 4Day 5… 0.490.430.560.550.57… Each day, I flipped a fair coin 100 times and got the proportion of getting a head.

11. 68-95-99.7 Rule (or Three-sigma Rule): 68% of the observations fall within one standard deviation of the mean. 95% of the observations fall within two standard deviations of the mean. 99.7% of the observations fall within three standard deviations of the mean.

12. 68-95-99.7 Rule (or Three-sigma Rule): A generalization of 68-95-99.7 rule is standard scores.

13. Standard Scores: A standard score of 1 says that the observation in question lies one standard deviation above the mean. Standard scores can be used to compare values in different distributions.

14. Example in the Textbook: Jennie scored 600 on the SAT and her friend Gerald scored 21 on the math part of ACT. Assuming that both tests measure the same kind of ability, who has the higher score? The performance depends on where those two scores lie in their distribution (percentile). Because the percentage makes more sense. SAT: mean = 500 standard deviation = 100 ACT: mean = 18 standard deviation = 6

16. Normal Distribution: The area under any density curve is always 1 which corresponding to 100 percent.

17. Calculation the Chance (a.k.a. Probability): So The area is designed to express the corresponding chance/probability a observation could be.

18. Calculation the Chance (a.k.a. Probability): 93.32 % of the area is shaded. So the chance that an observation is less than 1.5 is 93.32%.

19. Calculation the Chance (a.k.a. Probability): 86.64% area is shaded. So the chance that an observation is between -1.5 and 1.5 is 86.64%.

20. Calculation the Chance (a.k.a. Probability): 6.68% area is shaded. So the chance that an observation is larger than 1.5 is 6.68%.

21. Percentiles: The c th percentile is a value such that c percent of the observation lie below it. 93.32 % of the area is shaded. 1.5 is the 93.32 th percentile of this particular distribution.

22. Forward/Backward Problems: How can we find the corresponding area at test and HW? Answer: using Table B on the textbook. Caution: Table B only provides percentiles for the standard scores. Observation Standard Score Percentile Forward Problem Backward Problem

23. Normal Table (Forward): 65.54 % of the area is under 0.4. Once you obtained the standard score and know the area you should work on. Standard score Percentile 0.050.00 0.153.98 0.257.93 0.361.79 0.465.54 0.569.15 0.672.58 0.775.80 0.878.81 0.981.59 1.084.13 1.186.43 1.288.49 1.390.32 1.491.92 1.593.32

24. Normal Table (Forward): The area above 0.4 is 100-65.54. Standard score Percentile 0.050.00 0.153.98 0.257.93 0.361.79 0.465.54 0.569.15 0.672.58 0.775.80 0.878.81 0.981.59 1.084.13 1.186.43 1.288.49 1.390.32 1.491.92 1.593.32 Once you obtained the standard score and know the area you should work on.

25. The area between 0.2 and 1.5 is 93.32-57.93. Normal Table (Forward): Standard score Percentile 0.050.00 0.153.98 0.257.93 0.361.79 0.465.54 0.569.15 0.672.58 0.775.80 0.878.81 0.981.59 1.084.13 1.186.43 1.288.49 1.390.32 1.491.92 1.593.32 Once you obtained the standard score and know the area you should work on.