Bivariate Data Pick up a formula sheet Notes

Bivariate Data Pick up a formula sheet, Notes for Bivariate Data – Day 1, and a calculator.

Archaeopteryx is an extinct beast having feathers like a bird but teeth and a long bony tail like a reptile. Only six fossil specimens are known. Because these specimens differ greatly in size, some scientists think they are different species rather than individuals from the same species. If the specimens belong to the same species and differ in size because some are younger than others, there should be a positive linear relationship between the bones from all individuals. An outlier from this relationship would suggest a different species. Here are data on the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five specimens that preserve both bones. femur humerus 38 41 56 59 63 70 64 74 72 84 Load data into list 1 and list 2 and make a scatterplot.

This is not enough. What do we need?

humerus length in cm 72 41 38 64 femur length in cm A “cheater” way to put scale on a scatterplot is to trace two points and label each axis with those two values.

humerus length in cm 72 41 64 38 femur length in cm explanatory variable? response variable? femur length in cm humerus length in cm But does it really matter here? No. But often it does.

Find the correlation coefficient and explain what it means.

Find the correlation coefficient and explain what it means. correlation coefficient Did you get it?

If you did not get the correlation coefficient, you must turn your diagnostics on. Push 2 nd then 0. Scroll down to diagnostics on. Push “enter” twice and little calculator guy will say “done”.

Find the correlation coefficient and explain what it means. correlation coefficient

r =. 994 The correlation coefficient is ALWAYS between -1 and 1. It does not change when the units or scale is transformed. Let’s check out the formula sheet.

r =. 994 What does it mean? The correlation coefficient describes the strength of the linear relationship. The closer it is to 1 or -1 the more the points line up. These points line up pretty well with a negative slope. The correlation coefficient would be close to -1. graph on the bottom of your notes

r =. 994 What does it mean? The correlation coefficient describes the strength of the linear relationship. The closer it is to 1 or -1 the more the points line up. These points line up pretty well with a positive slope. The correlation coefficient would be close to 0. 8 or 0. 9.

r =. 994 What does it mean? The correlation coefficient describes the strength of the linear relationship. The closer it is to 1 or -1 the more the points line up. These points don’t line up at all. The correlation coefficient would be nearly 0.

r =. 994 What does it mean? The correlation coefficient describes the strength of the linear relationship. The closer it is to 1 or -1 the more the points line up. These points line up sort of well with a negative slope. The correlation coefficient might be – 0. 6 or – 0. 7.

r =. 994 What does it mean? The correlation coefficient describes the strength of the linear relationship. The closer it is to 1 or -1 the more the points line up. These points don’t line up at all. The correlation coefficient would be fairly close to 0.

r =. 994 Here’s what you write: This suggests a strong, positive, linear relationship between femur length and humerus length.

So what’s the rest of this stuff?

slope y-intercept coefficient of determination equation: ŷ = 1. 197 x – 3. 660 This is hugely important! It means the predicted y.

equation: ŷ = 1. 197 x – 3. 660 where x = femur length and y = humerus length slope = 1. 197 ; For every 1 cm increase in femur length, the humerus length increases by 1. 197 cm, on average. y-intercept ; When the femur length is 0 cm, the humerus length is about -3. 660 cm. Of course, this is ridiculous and is an example of extrapolation.

Residuals A residual is the distance between the actual point and the predicted point. residual = y – ŷ since the point is above the line, the residual will be positive y ŷ ŷ y since the point is below the line, the residual will be negative

What is the residual for the point (38, 41)? residual = y – ŷ First find y. 41

What is the residual for the point (38, 41)? residual = 41 – ŷ Then find ŷ using the equation of the line. ŷ = 1. 197 x – 3. 660 ŷ = 1. 197(38) – 3. 660 ŷ = 41. 826 residual = 41 – 41. 826 = –. 826

What is the residual for the point (59, 70)? residual = 70 – 66. 963 = 3. 037

Residual Plot residuals The residual plot is a scatterplot of all the residuals paired with their corresponding x values. femur length

Residual Plot on the calculator Scroll down to L 2 to get to the y-list.

Push 2 nd – stat to get the list function. You should see resid. Enter resid. Remember: You need the equation of the line to find the residual. If you do not see “resid”, then you must first do the linear regression so the calculator knows which equation to use.

zoom 9 to display graph

residuals femur length This is not the greatest residual plot. We would like the points to be randomly scattered about the line. Notice that the sum of the residuals will always be 0. Could push trace to get the actual points

residuals swine population Here’s a better residual plot from a different data set. It’s scattered randomly above and below the line.

What does a residual plot tell us? If points are randomly scattered above and below the line, then our linear equation is good. If there is a definite pattern, then another type of regression equation is more appropriate. Could be quadratic or exponential or logarithmic or any of the infinite mathematical possibilities.

Coefficient of determination r 2 =. 988; 98. 8% of the variation in humerus length is related to the variation in femur length.

That’s it for dinosaurs today.