1. Communicating Quantitative Information Normal distribution Values, changes, changes of changes Female/Male income. Questions on Midterm Homework: prepare for midterm. (Post project I proposal)

2. Normal distribution Aka bell shaped curve Distribution: horizontal axis different values, vertical axis, number of incidents of that value Mean = mode= median and specific falling away from mean/mode/median –Much less (fewer?) incidence of values away from the mean/mode/median. These are also called outliers –Exact description is given in terms of standard deviation. Square root of (1/N * Sum(x i -mean) 2 )

3. (continuous) normal curve 34.1% Red area = 68.2% of total Red+green = 95.4% Red + green + blue (6 sigma) = 99.6% 1 standard deviation

4. Why is normal important Empirically, many things in nature appear to exhibit this type of distribution Mathematically, binomial → normal What does this mean? Remember coin tossing: Probabilities of 0, 1, 2 heads in tossing coin 2 times is: 1/4, 2/4, 2/4, 1/4 Probabilities of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 heads in tossing coin 10 times is:

5. 012345678910 1E-030.010.0440.1170.2050.2460.2050.1170.0440.011E-03 Note: 1E-03 means 1 times 10 raised to the negative 3 power means.001 Notice: numbers falling off from center.

6. Normal So, if a condition/value is the result of many different more or less on and off effects (binomial = 2 things), then cumulative effect is to be normal….

7. Graphs What graph and/or diagram and/or picture reveals the important aspects of the story? Pie is for parts of a whole Bar is for comparing things or sets of things or … Line graph generally used when the underlying phenomenon has continuous nature For bar or for line, time is frequent variable, generally on the horizontal axis.

8. Change Dental expense and inflation showed change, NOT underlying values Can show change over time Can show change of change These ideas are familiar to you! Think of velocity of car change in position / change in time Acceleration change in velocity / change in time

9. Women wages vs Men wages Say, John Smith made $100,000 and Jane Doe made $117,000….then Jane made 17% more than John. –John's salary is the base But, they're talking about group of women and group of men They decided to use median as their 'measure of centrality'.

10. Women/Men wages, cont. Case 1: –5 men making 30000 30000 100000 100000 200000 MEDIAN is 100000 Mean (aka average) = 92000 –3 women making 30000 117000 200000 MEDIAN is 117000 Mean =115666.67 Using medians or means, these women making more than these men!!!

11. Women/Men wages, cont. Case 2: –5 men making 30000 30000 100000 100000 1000000 MEDIAN is 100000 Mean (aka average) = 252000 –3 women making 30000 117000 200000 MEDIAN is 117000 Mean =115666.67 In this case, women are ahead if using medians, men ahead if using means.

12. Mean vs Median issue? No, in conversations with the researcher, the mean vs median would not have changed the result –This population of women were making more than the men by either measure However, the real news was the composition of each group.

13. Women/Men, cont. Schematic of actual case –Men in two groups, MH (men high school) and MC (men with college degree). More MH then MC. –Women in two groups, WH and WC, more WC than WH. –Mean or Median WH < MH < WC < MC Because the median (or the mean) of the Men tends more to that measure of the MH and the median (or the mean) of the Women tends to that of the WC, the total has women outearning men! It actually is something like case 2 with the one very high earning man!

14. New York Times article Graph included ethnic break down that was confusing if you tried to 'do arithmetic' because you can't do this with medians. Look at our postings and re-read article!

15. Presentation: graphs and diagrams Communication issue Consider –Stock market –Polls –Health, development issues –? Recall 5 D's –Chart should show 'what is the difference' and distribution and dimension –May or may not show denominator –Need to check definition(s)

16. Midterm Your questions?