Correlation Value, 4

Section 4.1

The Correlation Value, rThe correlation value is a mathematical way of measuring how good a fit is.

1) Find the average and standard deviation for both the x-values and the y-values2) For each data point, multiply the z-score of the x- value times the z-score of the y-value3) Add all the products together4) Divide by "N - 1" where N = number of data points

Analyzing the meaning of r forlinear equations (straight lines)

• If r > 0 then the slope is positive• If r < 0 then the slope is negative

• If r = -1 or r = 1 then the correlation is perfectly linear• If r < -0.95 or r > 0.95 then the correlation is great• If r < -0.9 or r > 0.9 then the correlation is good• If r < -0.8 or r > 0.8 then the correlation is fair/ok• If -0.7 < r < 0.7 then the correlation is weak

• r is unaffected by units, r looks at the relative distances between points, when you change the units, it changes every point so the relative distances stay the same

Guess the r-value• r < 0 for a negative correlation• r > 0 for a positive correlation• | r | = 1 means perfectly linear correlation

r = 0.98Very Strong

Guess the r-value• r < 0 for a negative correlation• r > 0 for a positive correlation• | r | = 1 means perfectly linear correlation

r = 0.93Strong

Guess the r-value• r < 0 for a negative correlation• r > 0 for a positive correlation• | r | = 1 means perfectly linear correlation

r = 0.85Moderate

Guess the r-value• r < 0 for a negative correlation• r > 0 for a positive correlation• | r | = 1 means perfectly linear correlation

r = 0.62Weak

Guess the r-value• r < 0 for a negative correlation• r > 0 for a positive correlation• | r | = 1 means perfectly linear correlation

r = – 0.32Weak

Guess the r-valuer = – 0.19A strong correlation can get a bad r-value if you choose the wrong type of model. What type of model (equation) should we fit to this graph? quadratic

Guess the r-valuer = – 1.00Perfectly Linear

Guess the r-valuer = 0.00Why does y = 0 for a horizontal line no matter how well the data fits it?If the slope is 0 that means y doesn't change when x changes, y is completely unaffected by x, therefore there is no correlation,(no relation).

r2

It's more often more convenient to deal with r2 instead of r (r2 = r * r)

• r2 = 1 means perfectly linear for both positive & negative slopes (both 12 and -12 = 1)

• r2 = 0 means worst possible fit

• The vale of r2 = percent of the variation seen in variable y that is caused by variable x

• A group of footballplayers went out forpizza and ran lapsafterwards.

• What does r2 mean?– 61% of the variation can be

accounted by this relationship.

• What other variables causes the remaining 39% of the variation between data points (# of laps run)?– Athleticism of player– How much game time each player had earlier that day– Time between 2nd to last slice & last slice– Amount of Soda Consumed

0 2 4 6 8 10 120

2

4

6

8

10

12

f(x) = − 0.47326074774 x + 9.28488511728R² = 0.606247556260986

Number of Slices of Pizza Eaten

Num

ber o

f Lap

s th

e Pl

ayer

Cou

ld R

un A

f-te

rwar

ds