Squared Differences Demonstration

Learning Objectives
1. Understand what it means to minimize the sum of squared deviations
2. Learn which measure of central tendency minimizes the sum of squared deviations.

Instructions
This demonstration allows you to examine the sum of squared deviations from a given value. The slide to the right shows the numbers 1, 2, 3, 4, and 5 and their deviations from an arbitrary starting value of 0.254 (the figure displays this rounded to 0.25).

The first number, 1, is represented by a red dot. The deviation from 0.254 is represented by a red line from the red dot to the black line. The value of the black line is 0.254.

Similarly, the number 2 is represented by a blue dot and its deviation from 0.25 is represented by a blue line.

The height of the colored rectangles represents the sum of the absolute deviations from the black line. The sum of the deviations of the numbers 1, 2, 3, 4, and 5 from 0.25 is 0.746 + 1.75 + 2.746 + 3.746 + 4.746 = 13.73.

The area of each rectangle represents the magnitude of the squared deviation of a point from the black line. For example, the red rectangle has an area of 0.746 x 0.746 = 0.557. The sum of all the areas of the rectangles is 47.70. The sum of all the areas represents the sum of the squared deviations.

In this demonstration, you see the effects of moving the black bar as well as individual points.

Your goal for this demonstration is to discover a rule for determining what value will give you the smallest sum of squared deviations.

Slide Show