STAT 100 Lecture 4:
Bivariate Data, Scatter Plots, Linear Regression,
and Correlations
Nate Strawn
http://www.math.umd.edu/∼nstrawn/
Nate Strawn STAT 100 Lecture 4: Bivariate Data, Scatter Plots, Linear Regression, and Correlations
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Class Note - Bivariate Data, Linear regression, and Correlation | STAT 100, Study notes of Probability and Statistics
Material Type: Notes; Class: ELEM STAT & PROB; Subject: Statistics and Probability; University: University of Maryland; Term: Unknown 1989;
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STAT 100 Lecture 4:
Bivariate Data, Scatter Plots, Linear Regression,
and Correlations
Nate Strawn
http://www.math.umd.edu/∼nstrawn/
Last Time...
(^1) Lots of definitions
The Variance: s^2 = 1 N − 1
∑^ N i=
(xi − x)^2
The Standard Deviation: s =
√√ √√ 1 N − 1
∑^ N i=
(xi − x)^2
The Range: Range = 1 max≤i≤N xi − (^1) ≤mini≤N xi The Interquartile Range: IQR = Q 3 − Q 1.
(^2) Boxplots and Normal Distributions
Bivariate Data
Bivariate Data is a fancy way to say, “Two-Variable Data.” Multivariate Data is a fancy way to say, “Many-Variable Data.” Bivariate Data is a list of numerical pairs
(x 1 , y 1 ), (x 2 , y 2 ),... , (xN , yN ).
The easiest way to visualize Bivariate Data is through a Scatter Plot.
Scatter Plots
(^014) 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16
5000 10000 15000 20000 25000 30000 35000 40000 45000
50000 Average Global Temperature vs Number of Pirates
Average Global Temperature
Number of Pirates
(^00 50 100 150 200 250 300 350 400 450 )
50
100
150
200
250 Degrees Fahrenheit vs Degrees Celsius
Degrees Fahrenheit
Degrees Celsius
(^015 10 5 0 5 10 )
20
40
60
80
100
120 Integers vs Squares
x
x^
(^00) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Stuff vs Junk
Stuff
Junk
Formula for the Correlation Coefficient
Formula
r = Sxy √ Sxx
Syy where Sxy =
(xi − x)(yi − y ), Sxx =
(xi − x)^2 , and Syy =
(yi − y )^2.
Mean Temp. Pirates xi − x yi − y (xi − x)^2 (yi − y )^2 (xi − x)(yi − y ) (x) (y ) 14.2 35000 -0.76 17797.57 0.57 316753548 - 14.3 45000 -0.66 27797.57 0.43 772704977 - 14.6 20000 -0.36 2797.57 0.13 7826405 - 14.9 15000 -0.06 -2202.43 0 4850691 125 15.2 5000 0.24 -12202.43 0.06 148899263 - 15.6 400 0.64 -16802.43 0.41 282321605 - 15.9 17 0.94 -17185.43 0.89 295338955 - Tot.=104.7 120417 0 0 2.5 1828695447 - x = 14. 96 y = 17202. 43 Sxx Syy Sxy
We then have that r = Sxy /(
Sxx
Syy ) ≈ − 0. 93
Scatter Plots and r-Values
For the following scatter plots, we have r = − 0 .93 and r = 1.0 for the first row, and r = − 0 .34 and r = 0.35 for the second row.
(^014) 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16
(^100005000) 1500020000
2500030000
3500040000
4500050000
Average Global Temperature vs Number of Pirates
Average Global Temperature
Number of Pirates (^00 50 100 150 200 250 300 350 400 450 )
50
100
150
200
250 Degrees Fahrenheit vs Degrees Celsius
Degrees Fahrenheit
Degrees Celsius
(^015 10 5 0 5 10 )
20
40
60
80
100
120 Integers vs Squares
x
x^
(^00) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
Stuff vs Junk
Stuff
Junk
Quick Review of Linear Equations
Remeber that a line is both an equation (y = β 0 + β 1 x) and a graph:
Here, β 0 is the y -intercept, and β 1 is the slope of the line.
Linear Regression
Definition The Least Squares Line fitting the data
(x 1 , y 1 ), (x 2 , y 2 ),... , (xN , yN )
is ˆy = ˆβ 0 + ˆβ 1 x, where β^ ˆ 1 = Sxy /Sxx =
(xi − x)(yi − y ) ∑ (xi − x)^2 and βˆ 0 = y − βˆ 1 x.
For Next Time
Quiz 01! Covers Sections 2.1, 2.2, 2.3, 2.4, and 2.5. Problems are taken directly from the book (with small changes!).
Read Section 3.4, 3.5, and 3.6 from Johnson and Bhattacharyya
Online homework 3.6: 3.35, 3.37, 3.39, 3.41, 3.
Group Problems: Group 1 2 3 4 5 6 7 8 9 10 Problem 3.38 3.40 3.42 3.46 3.49 3.51 3.52 3.53 3.54 3.