1. Body Parts – Lab Write-up A hiker was walking in the woods and found a 54 cm fibula bone buried in the ground Forensic investigators were called to the scene and also found an old size 12 (12-inch) men’s shoe It is known that there were 12 missing persons in the area ranging from 5’3” to 6’8” Six of these were men and six were women Using this information and the data from your class, what, if anything, can the investigators conclude?

2. Making Charts What data would you plot? If you would like to plot this data what kind of chart would you make?

3. Chart of Radius Length vs. Height

4. Radius Length vs. Height – Best Fit Line

5. Radius Length vs. Height (Chart) Note the following key characteristics of the chart: 1.The chart has a title 2.The x-axis and y-axis are labeled 3.The labels have units (cm in this case) 4.The chart uses space well (not a lot of empty space (compare with the chart on the next page)

6. What’s Wrong with this Chart? (3 things)

7. Radius Length vs. Height (“Best-Fit” Line) Using Excel, I created a “best-fit” line for the data An estimated “best-fit” line is drawn so that approximately half of the data points are above it and half are below it (drawing a real best-fit line requires statistics, but Excel does this for you) This “best-fit” line is called the regression line Any line has a slope and an intercept: y = mx + b o m = slope = ∆y/∆x (“rise/run”) o b = intercept = y (when x = 0)

8. Radius Length vs. Height (Equation) Equation for this line: y = 2.4x +114 How could we find this equation from a paper graph (do exercise on board) If a person’s radius is 27cm, how tall would we predict him or her to be? Is this reasonable based on our chart? What if we know a person is 184 cm, how long would we expect his radius to be? CAUTION: You cannot use the equation to extrapolate values too far out of the range of points you have.

9. Radius Length vs. Height (Correlation) Correlation refers to how strongly two variables are related to each other Visually, the closer the data points are to a “best-fit” line the better the correlation The statistical term that measures how strongly two variables are correlated is R 2 (“R-squared”), which is expressed as a number between 0 and 1 The closer R 2 is to 1, the better the correlation

10. Radius Length vs. Height (R-Squared) Degree of correlation: o R 2 =1.0  Perfect correlation o 0.8<R 2 <1.0  Very strong correlation o 0.6<R 2 <0.8  Strong correlation o 0.4<R 2 <0.6  Moderate correlation o 0.2<R 2 <0.4  Weak correlation o 0.0<R 2 <0.2  Very weak correlation o R 2 =0.0  No correlation The R 2 for radius length vs. height is 0.6 The more data points, the better the R 2 – why?

11. Perfect Correlation (R 2 = 1)

12. No Correlation (R 2 = 0)

13. Perfect Correlation What do you think the equation is for the line on the previous slide? y = mx + b o m = ? o b = ? If a person is 185 cm, how many inches is he? If a person is 67 in, how many centimeters is she?

14. Some terminalogy Remember: y = mx + b When we say “x predicts y” or “y depends on x” o Independent variable: x o Dependent variable: y In the previous case, we can say “radius length predicts height,” in which case: o Independent variable: radius length o Dependent variable: height Some examples of dependent and independent variables are…

15. Outliers Outliers are points that are numerically distant from the rest of the data They are usually due to measurement error (either human error or instrument error) If you get data that look like outliers as you are doing the lab, the best thing to do is to take the measurement again o DO NOT erase the outliers from your original data o In your lab write-up explain that you are going to disregard outliers and proceed with the data analysis (see next page):

16. Chart with Outliers NY Yankees

17. Chart (Adjusted for Outliers)

18. MLB Payroll Question If a team spends $75 million on payroll, how many wins on average can that team expect to achieve? Why do you think the R-squared is so low for payroll vs. wins?