Transcript R Squared - Radical Math

R Squared
r = -.944
r = -.79
y = -0.9402x + 43.721
y = -0.8141x + 9.1332
if x = 15, y = ?
if x = 6, y = ?
y = -0.9402(15) + 43.721
y = -0.8141(6) + 9.1332
y = 29.6195
y = 4.2486
Which value for Y is a more accurate
prediction for the given X value?
To find how well the Line of
Best Fit actually fits the data,
we can find a number called
R-Squared by using the
following formula:
1-
Sum of squared distances between the
actual and predicted Y values
Sum of squared distances between the
actual Y values and their mean
For example, here’s how to find the R Squared
value for the data/graph below:
X
Y
3
40
10
35
11
30
15
32
22
19
22
26
23
24
28
22
28
18
35
6
Equation for Line of Best Fit:
Correlation = -.94
y = .94x + 43.7
Equation for Line of Best Fit:
X
Y
3
40
10
35
11
30
15
32
22
19
22
26
23
24
28
22
28
18
35
6
Mean:
Predicted
Y Value
Error
Sum:
Error
Squared
y = .94x + 43.7
Distance
between Y
values and
their mean
Sum:
Mean
distances
squared
Equation for Line of Best Fit:
y = .94x + 43.7
Error
Error
Squared
Distance
between Y
values and
their mean
Mean
distances
squared
X
Y
Predicted
Y Value
3
40
40.88
.88
.77
14.8
219.04
10
35
34.30
-.70
.49
9.8
96.04
11
30
33.36
3.36
11.29
4.8
23.04
15
32
29.60
-2.40
5.76
6.8
46.24
22
19
23.02
4.02
16.16
-6.2
38.44
22
26
23.02
-2.98
8.88
.8
.64
23
24
22.08
-1.92
3.69
-1.2
1.44
28
22
17.38
-4.62
21.34
-3.2
10.24
28
18
17.38
-.62
.38
-7.2
51.84
35
6
10.80
4.8
23.04
-19.2
368.65
Mean:
25.2
Sum:
91.81
Sum:
855.60
To calculate “R Squared”…
1-
Sum of squared distances between the
actual and predicted Y values
Sum of squared distances between the
actual Y values and their mean
11- 0.11
91.81
855.60
=.89
OK. Don’t kill me. Remember this was the
data/graph we were finding “R Squared” for?
The value we got for R Squared was .89
Here’s a short-cut. To find R Squared…
X
Y
3
40
10
35
11
30
15
32
22
19
22
26
23
24
28
22
28
18
35
6
…Square r
r = -.944
r2 = -.944 • -.944
r2 = .89
R Squared
• To determine how well the regression
line fits the data, we find a value
called R-Squared (r2)
• To find r2, simply square the
correlation
• The closer r2 is +1, the better the line
fits the data
• r2 will always be a positive number
r = -.944
r2
= .89
r = -.79
r2 = .62