Overview
Introduction
Example
Recap
Maximum and Minimum Slopes
Wilfrid Laurier University
Terry Sturtevant
Wilfrid Laurier University
December 12, 2014
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Overview
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Overview
In this document, you’ll learn:
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Overview
In this document, you’ll learn:
how to determine the uncertainties in the slope and
y -intercept when graphs have “big” error bars;
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Overview
In this document, you’ll learn:
how to determine the uncertainties in the slope and
y -intercept when graphs have “big” error bars;
how this depends on whether the slope is positive or negative
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Introduction
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Introduction
When you have points on a linear graph, sometimes they will
line up pretty well.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Introduction
When you have points on a linear graph, sometimes they will
line up pretty well.
In this case, the uncertainties in the slope and y -intercept will
be determined by the error bars on the graph.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Following is an example of a graph with a positive slope.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Here’s a simple graph with error bars.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
The line of best fit would look something like this.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Note that the line of best fit crosses all of the error bars.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
This is the steepest line that crosses all of the error bars.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
(Actually it touches each of the error bar rectangles, which is the
important thing.)
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
It’s called the line of maximum slope.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Since it has a positive slope, it will have a minimum y -intercept.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
This is the least steep line that crosses all of the error bars.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
It’s called the line of minimum slope.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Since it has a positive slope, it will have a maximum y -intercept.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Here’s another set of points, with a similar line of best fit.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
A line drawn from the corners of the first and last points like before
looks like this.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
It’s the line of maximum slope, as before.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Here’s an attempt to draw the line of minimum slope as before.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
It misses the two middle points, so it doesn’t work.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
This line is the line of minimum slope, since it crosses all of the error
bars.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
In general, which points give the lines of maximum and minimum slope will depend on the data.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Also, the same points won’t be used for both lines.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
In cases like this, there are no lines which cross all error bars.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Uncertainties have to be determined using standard errors.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
For a graph with a negative slope, things change slightly.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Here’s a graph with a negative slope.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
Using the same corners as before to get the line of maximum (negative) slope doesn’t work.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Negative slope
You need to use different corners for maximum and minimum negative slopes.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Recap
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Recap
If you can draw lines which cross all of the error bars on a
graph, then you use maximum and minimum slopes to
determine the uncertainties in the slope and y -intercept;
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Recap
If you can draw lines which cross all of the error bars on a
graph, then you use maximum and minimum slopes to
determine the uncertainties in the slope and y -intercept;
If you can’t draw lines which cross all of the error bars on a
graph, then you use standard errors to determine the
uncertainties in the slope and y -intercept.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Recap
If you can draw lines which cross all of the error bars on a
graph, then you use maximum and minimum slopes to
determine the uncertainties in the slope and y -intercept;
If you can’t draw lines which cross all of the error bars on a
graph, then you use standard errors to determine the
uncertainties in the slope and y -intercept.
In general, which points give the lines of maximum and
minimum slope will depend on the data.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Recap
If you can draw lines which cross all of the error bars on a
graph, then you use maximum and minimum slopes to
determine the uncertainties in the slope and y -intercept;
If you can’t draw lines which cross all of the error bars on a
graph, then you use standard errors to determine the
uncertainties in the slope and y -intercept.
In general, which points give the lines of maximum and
minimum slope will depend on the data.
In general, the same points won’t be used for both the
maximum and minimum slope lines.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University
Overview
Introduction
Example
Recap
Recap
If you can draw lines which cross all of the error bars on a
graph, then you use maximum and minimum slopes to
determine the uncertainties in the slope and y -intercept;
If you can’t draw lines which cross all of the error bars on a
graph, then you use standard errors to determine the
uncertainties in the slope and y -intercept.
In general, which points give the lines of maximum and
minimum slope will depend on the data.
In general, the same points won’t be used for both the
maximum and minimum slope lines.
You need to use different corners for maximum and minimum
slope lines, depending on whether the slope is positive or
negative.
Terry Sturtevant
Maximum and Minimum Slopes Wilfrid Laurier University