Download Scatter Plots and Best Fitting Lines

Scatter Plots
and
Best Fitting Lines
MM2D2. Students will determine an algebraic model to quantify the association
between two quantitative variables.
a. Gather and plot data that can be modeled with linear and quadratic functions.
b. Examine the issues of curve fitting by finding good linear fits to data using simple
methods such as the median-median line and “eyeballing.”
c. Understand and apply the processes of linear and quadratic regression for curve fitting
using appropriate technology.
d. Investigate issues that arise when using data to explore the relationship between two
variables, including confusion between correlation and causation.
Scatter Plot
A graph of a set of data points (x, y)
Correlation
(connection between the different values in one set of data
and in a second set of data)
If y tends to increase as
x increases, then the
data has a
POSITIVE
CORRELATION
If y tends to decrease
as x increases, then
the data has a
NEGATIVE
CORRELATION
Correlation Coefficient = r
A number between -1 and 1
This measures how well a line fits a set of data pairs.
Near 1: points are close to a positive
sloping line
Near -1: points are close to a negative
sloping line
Near 0: points do not lie close to any line
• r = close to 1
• r = close to -1
• r = close to 0
Example 1: Correlation
Coefficients
Describe the data as having a positive correlation, a
negative correlation, or approximately no correlation.
Tell whether the correlation coefficient for the data is
closest to -1, -.5, 0, .5, or 1
Example 2: Correlation
Coefficients
Describe the data as having a positive correlation, a
negative correlation, or approximately no correlation.
Tell whether the correlation coefficient for the data is
closest to -1, -.5, 0, .5, or 1
Example 3: Correlation
Coefficients
Describe the data as having a positive correlation, a
negative correlation, or approximately no correlation.
Tell whether the correlation coefficient for the data is
closest to -1, -.5, 0, .5, or 1
Best Fitting Line
The line that lies as close as possible to all
the data points.
Linear Regression is a method for finding
the equation of the best fitting line.
Example 4: Best-Fitting Line
The ordered pairs (x,y) give the height y in feet of a
young tree x years after 2000. Approximate the bestfitting line for the data.
(0, 5.1), (1,6.4), (2,7.7), (3,9),
(4,10.3), (5,11.6), (6,12.9)
Example 4(cont.): Best-Fitting Line
(0, 5.1), (1,6.4), (2,7.7), (3,9), (4,10.3), (5,11.6), (6,12.9)
Step 1: Draw a scatter plot of the data
Step 2: Sketch a line that appears to best fit
the data
Example 4(cont.): Best-Fitting Line
Step 3: Choose two points ON THE LINE
for the line you drew and find the slope of
the line between the points.
Example 4(cont.): Best-Fitting Line
Step 4: Use the slope you just found and
one point on the line to write the equation
of the line.
Remember point-slope form!! y – y1 = m(x – x1)
m = slope; (x1 ,y1) = point on the line. Solve for y
Median-Median Line
Another linear model used to fit a line to a
data set. The line is fit only to summary
points.
Example 5: Median-Median Line
Find the equation of the median-median line for
the data:
(1, 48), (2, 42), (2, 50), (4, 45), (5, 69), (6, 44), (7, 82), (7, 93), (8, 96)
Step 1: Organize the data so the x-values are in
order from least to greatest and separate into 3
equal size groups. If groups are not equal size,
make sure the first and last groups have the
same amount of data points.
Example 5(cont.): Median-Median Line
Step 2: Find the median x value and the median y
value for each group:
Group
#
xymedian median
values values x-value y-value
Summary
Point
1
2
3
Step 3: Create a summary point using the median x
and y values for each group
Example 5(cont.): Median-Median Line
Step 4: Find the slope between the outer
summary points (groups 1 and 3) and use
the first summary point to write the
equation in point-slope form
Hint: y – y1 = m(x – x1)
Example 5(cont.): Median-Median Line
Step 5: Move the equation a third of the way
toward the middle summary point.
To do this, find the predicted value for y at the
middle summary x-value by plugging it into your
equation for x. Subtract this predicted y-value
from the middle summary y value and multiply
by 1/3.
Now add this number to the equation you found in
step 4.
This is the equation of the median-median line!!