1

The R.M.S. Error

Regression for Prediction

2

Prediction Errors

y

x

Regression Line

Error or “residual”

3

A Measure of the Size of the Errors: RMS Error

RMS Error =

(error)2 + (error)2 + ... + (error)2

number of errors

.

4

10 15 20 25 30 35 40 45 50 55 600.5

1

1.5

2

2.5

3

3.5

4

4.5

Quantitative GMAT

MB

A G

PA

Quantitative GMAT Predicts MBA GPA? (r = .34)

5

10 15 20 25 30 35 40 45 50 55 600.5

1

1.5

2

2.5

3

3.5

4

4.5

Quantitative GMAT

MB

A G

PA

RMS Error = .46 Regression line plus one RMS error

Regression line minus one RMS error

6

Interpretation of the RMS Error

• It can be shown that the residuals have average = 0. The RMS error is thus their SD.

• Rule of thumb: about 68% of the residuals are smaller in magnitude than one RMS error. About 95% are smaller in magnitude than two RMS errors

7

The RMS error is a measure of the error around the regression line, in the same sense that the SD is a measure of variability around the mean.

8

10 15 20 25 30 35 40 45 50 55 600.5

1

1.5

2

2.5

3

3.5

4

4.5

Quantitative GMAT

MB

A G

PA

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.50

50

100

150

200

250

300

residuals

RMS Error

RMS Error

Scatter Diagram Histogram of Residuals

9

Match the RMS error: .2 1 5

10

Predicting MBA GPA• Using the GMAT, the error measure would be

the RMS error = .46

• Without knowledge of GMAT, the average would be your best prediciton. A measure of the error would be the SD of MBA GPAs. SD = .49

So you don’t gain much by using the GMAT

11

RMS Error, Correlation, the SD of Y: The Picture

Y Y

X X

Low Correlation High Correlation

12

RMS Error, Correlation, and the SD of Y: The Formula

RMS Error = 1 - r2 x SD of Y

13

Residual Plot: Focus on Prediction Errors

0

Residual = Observed minus Predicted

Scatter Diagram Residual Plot

14

Example: Predicting MBA GPA from Quantitative GMAT

10 15 20 25 30 35 40 45 50 55 600.5

1

1.5

2

2.5

3

3.5

4

4.5

Quantitative GMAT

MB

A G

PA

10 15 20 25 30 35 40 45 50 55 60-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Quantitative GMAT

Res

idua

ls

Scatter Diagram Residual Plot

15

Predicting MBA GPA from Undergrad GPA

1 1.5 2 2.5 3 3.5 40.5

1

1.5

2

2.5

3

3.5

4

4.5

Undergrad GPA

MB

A G

PA

1 1.5 2 2.5 3 3.5 4-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Undergrad GPA

Resi

dual

Scatter Diagram Residual Plot

16

Stream Flow Rate versus Depth

Scatter Diagram Residual Plot

17

Inside Vertical Strips

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SDs in Vertical Strips

10 15 20 25 30 35 40 45 50 55 600.5

1

1.5

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2.5

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3.5

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4.5

Quantitative GMAT

MB

A G

PA

19

Terminology

• Homoscedastic: same RMS errors in each vertical strip.

• Heteroscedastic: different RMS errors in vertical strips

20

A Heteroscedastic Scatter Plot

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A Football-Shaped Scatter Diagram is Homoscedastic?

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The Normal Curve Approximation within a Vertical Strip: The Picture

Regression Lin

e

The data in this vertical strip have an average given by the regression line and an SD equal to the RMS error. The normal approximation can be used with this average and SD.

23

The Normal Curve Approximation within a Vertical Strip: The Algorithm

• Find the average in the strip from the regression line

• The SD within the strip is the RMS error

• Convert to standard units using this mean and SD

• Refer to table of normal curve

24

Example

E3: Average height of father = 68 inches; SD = 2.7

Average height of son = 69 inches; SD = 2.7

r = .50

Scatter diagram is football shaped.

Q: What percent of the sons were over 6 feet tall?

6 feet = 72 inches.

Standard Unit =72 - 69

2.7= 1.11

25

From the Table:

-1.10 1.1072.87 %

13.57 %

So about 14 % of the sons are taller than 72 inches

26

E3: Average height of father = 68 inches; SD = 2.7

Average height of son = 69 inches; SD = 2.7

r = .50

Scatter diagram is football shaped.

Q: What percent of the 6 foot fathers had sons over 6 feet tall?

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1. Find the average height of the sons from the regression line:

A 6 foot father is 72 - 68

2.7= 1.48 SDs

higher than the father average.

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Son SD

Father SD r x Son SD

Regression Line and SD Line

29

1.48 father SDs

r x 1

.48

son S

Ds

son average + r x 1.48 son SDs

69 + .5 x 1.48 x 2.7 = 71 inches

Point of Averages

30

So the average in this strip is 71 inches. What is the SD?

RMS Error = 1 - r2 x SD of Y

= 1- .52 x 2.7

= 2.33

31

So: In the 72 inch father strip the average son height is 71 inches and the SD is 2.33.

To answer question, “What percent in this strip are over 6 feet tall?” use the normal curve.

6 feet = 72 inches

= (72 - 71)/2.33 standard units

= .43 standard units

34.73 32.6 32.6

About 33% of the sons of 6 foot fathers are taller than 6 feet.