Correlation Value 4 Section 4 1 The Correlation

Correlation Value, 4 Section 4. 1

The Correlation Value, r The 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-values 2) For each data point, multiply the z-score of the xvalue times the z-score of the y-value 3) Add all the products together 4) Divide by "N - 1" where N = number of data points

Analyzing the meaning of r for linear 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. 98 Very 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. 93 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. 85 Moderate

Guess the r-value • r < 0 for a negative correlation • r > 0 for a positive correlation • | r | = 1 means perfectly linear correlation r = 0. 62 Weak

Guess the r-value • r < 0 for a negative correlation • r > 0 for a positive correlation • | r | = 1 means perfectly linear correlation r = – 0. 32 Weak

Guess the r-value r = – 0. 19 A 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-value r = – 1. 00 Perfectly Linear

Guess the r-value r = 0. 00 Why 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).

2 r It's more often more convenient to deal with r 2 instead of r (r 2 = r * r) • r 2 = 1 means perfectly linear for both positive & negative slopes (both 12 and -12 = 1) • r 2 = 0 means worst possible fit • The vale of r 2 = percent of the variation seen in variable y that is caused by variable x

– 61% of the variation can be accounted by this relationship. Number of Laps the Player Could Run Afterwards • A group of football players went out for pizza and ran laps afterwards. • What does r 2 mean? 12 R 2 = 0, 6062 10 8 6 4 2 0 0 2 4 6 8 10 Number of Slices of Pizza Eaten • 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 2 nd to last slice & last slice – Amount of Soda Consumed 12