Least Square Fitting & Regression

When you try to fit a line to a number of points, you add the distances from the points to the line and minimize the sum of the squares of the distances. In mathematics the distance between a point and a line is defined as the shortest distant, i.e. the perpendicular distance. When you do numerical calculations however, there are various ways of defining the distance.

Perpendicular fitting

The segments are perpendicular to the red line. Move the line and try to get an optimal fitting. The line is optimal when the sum of the squares is minimized. Do not show the line of best fit until you have found a best fit by yourself.

Find a minimum perpendicular fitting!

Vertical fitting

The segments are vertical. Move the line and try to find an optimal fitting. Do not show the line of best fit until you have found a best fit by yourself.

Find a minimum vertical fitting!

The green line is made with the tool Icon Best Fit Line. Which method is used by GeoGebra, a perpendicular or vertical fitting?

Regression in GeoGebra

Regression is used to fit a function to data.

There are several commands for regression in GeoGebra that work on list of points. To create a list of points, do this:

  • Create some points.
  • Click on the tool Icon Create List. Drag the mouse over the graphics view to select points.
  • Write Fit in the input bar and choose the command you want to use.
Move the red points! Change the degree of the polynomial!
Note that high degree polynomials tend to have extreme values.

Women mathematicians and how math is taught in school

The data about the relative number of women mathematicians from some European countries, is taken from Statistics on Women in Mathematics / Catherine Hobbs & Esmyr Koomen.

The data about the relative percentage of pure+applied mathematics that is pure mathematics, in math education of different countries, is taken from OECD: Equations and Inequalities ‐ Making Mathematics Accessible to All.

Country Pure math (%) Women mathematicians (%)
Denmark 64.94 9.7
Finland 74.39 14.5
Norway 76.46 11.9
Sweden 67.06 12.4
Austria 77.05 9.2
Belgium 83.24 24.9
France 80.15 23
Germany 81.73 14.6
Ireland 77.3 10.4
Netherlands 90.48 9.8
Switzerland 77.99 6.7
United Kingdom 76.73 17.9
Czech Republic 83.13 26.4
Estonia 77.54 35.2
Italy 85.97 35
Spain 80.72 26.3
Portugal 86.8 47.6

Copy the numbers in the table above into GeoGebra. Selecting the entire table and then Copy/Paste to a GeoGebra spreadsheet should work.

Select the data in columns B and C and click on Icon Two Variable Regression Analysis.

regression analysis

In the window that pops up, click on Analyze. Here you can find mean, standard deviation, correlation coefficients, and other data.

In order to show the data in the graphics view, having the same data selected, right-click and choose Create -> List of points. A list \(L_1\) is created. You can use the command FitLine(L_1) to fit a line to the data. You can use the command CorrelationCoefficient(L_1) to find the correlation coefficient.

If you want to see the name of the country at each point, enter the word Label i cell C1. Write Text(A2, (B2, C2)) in cell C2. Make relative copies along column C. Select the text objects and make sure that they are shown.

women and pure math
A correlation coefficient 0.47 is "moderate".
However, correlation does not imply causation.

Polynomial fitting

A (n-1)-degree polynomial will fit n data points exactly, i.e. each of the points will lie on the curve. High order polynomials can, however, be impractical for various reasons; they can be highly oscillatory and difficult to handle.

When making a polynomial fitting in GeoGebra you use two parameters; the first one is the list and the second is the degree of the polynomial. Try the command FitPoly[list1,n].

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License