1. Statistical Reasoning for everyday life Intro to Probability and Statistics Mr. Spering – Room 113

2. 4.2 Shapes of Distribution How do we represent situations using graphs?  Study the entire situation…  Think about the problem logically…  Consider divergent possibilities…  Keep in mind most situations are open to different interpretations…  Remember a relationship does not imply causation…  Be cognizant of misleading possibilities (data)…

3. 4.2 Shapes of Distribution Let’s try some…..YEAH!

4. 4.2 Shapes of Distribution THE FAMOUS FOREST PROBLEM…… Stumping the top 10% of the nation for decades! HA!HA!

5. Timothy R. Lucas and Associates The Fifth Discipline Fieldbook Project - Schools That Learn LucasRPS@aol.comLucasRPS@aol.com 201-236-8696

6. The FOREST Problem Core Facts Our farmer runs a steady state system. She always has three types of trees: 1. Saplings 2. Trees Coming of Age 3. Mature Trees That Can Be Harvested She has made only one change in her system. Timothy R. Lucas and Associates The Fifth Discipline Fieldbook Project - Schools That Learn LucasRPS@aol.comLucasRPS@aol.com 201-236-8696

7. Timothy R. Lucas and Associates The Fifth Discipline Fieldbook Project - Schools That Learn LucasRPS@aol.comLucasRPS@aol.com 201-236-8696 SCARY PROBLEM

8. Timothy R. Lucas and Associates The Fifth Discipline Fieldbook Project - Schools That Learn LucasRPS@aol.comLucasRPS@aol.com 201-236-8696 **This static graph will hold true as long as the product of the number of years for maturity and 50 is not greater than the number of mature trees.**

9. 4.2 Shapes of Distribution Uniform Distribution:  All values have the same frequency.  i.e. A broken watch reads 10:26 am……………… No matter the time it reads 10:26 am 10:26 am

10. 4.2 Shapes of Distribution SINGLE-PEAKED:  Distribution with a single mode…(unimodal)

11. 4.2 Shapes of Distribution BIMODAL:  Distribution with two modes…  …etc……

12. 4.2 Shapes of Distribution How many modes?  Heights of 1000 randomly selected adult women  Unimodal  Heights of 1000 randomly selected adult Americans  Bimodal  Weekly sales throughout the year at a retail store  Multimodal, back-to-school, spring sales, holiday sales, etc.  The number of people with a particular last digit (0 through 9) in their social security number  Uniform, because essentially social security numbers are random therefore the last digit being any specific digit should be about 10% of the population for 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

13. 4.2 Shapes of Distribution SYMMETRIC vs. SKEWED: Symmetric identical about a mirror line. Skewed implies non-symmetric. Symmetric Butterfly Picture Non- Symmetric Butterfly Picture

14. 4.2 Shapes of Distribution SKEWED : Right-Skewed values are more spread out toward the right compared to a symmetric distribution. (POSITIVE SKEW) Left-Skewed values are more spread out toward the left compared to a symmetric distribution. (NEGATIVE SKEW)

15. 4.2 Shapes of Distribution  More SKEWED examples:  What type of skew is possible?  Family income in the United States… Positive/Right Skewed, United States mean income is about $ 60,528 (200,4 Census Bureau), however many people make much more.  Speeds of cars on a road where a visible patrol car is using radar… Negative/left Skewed, slow down when see PO’s!PO’s!  Heights of women… Symmetric, large population “normal” distribution

16. 4.2 Shapes of Distribution Variation:  Describes how widely data are spread out about the center of a distribution.  ????How would you expect the variation to differ between times in the Olympic marathon and times in the New York marathon???? EXPLAIN Olympic less variation, all elite runners New York more variation, runners of all abilities

17. 4.2 Shapes of Distribution SUMMARY: Figure 10. Frequencies of times between eruptions of the old faithful geyser. Notice the two distinct peaks: one at 1.85 and the other at 3.85. BIMODAL Remember: Inferential statistics are techniques that allow us to study samples and then make generalizations about the populations from which they were selected.

18. 4.2 Shapes of Distribution HOMEWORK # 15:  Pg 161 # 1-10 all  SELF-IMPROVEMENT starts with SELF-CONTROL  Foolish people believe in luck. Intelligent people believe in cause and effect.