Mimimized by the Mean
The mean minimizes the sum of squared error. The figure on the left, represents the data in the frequency distribution on the right, and can be used to illustrate this property of the mean. Like the other interactive figures in this chapter, the red line at the bottom is used to change the value used to represent the data. The red line is drawn at a default value of 2. If you calculate the mode for this data set, the value is 50. The value of the median is 60. The value of the mean is 67. The different colored squares in the figure represent error defined as the distance from the value given by the red line to the data value squared. The green bar at the top of the figure displays the sum of these squares. Now drag the red line to other possible values. Be sure to stop at the mode, median, and mean. Can you set the error value lower than that given with the red line's value is 67 (mean)? Sum of squared error values are used when one wants to set a high penalty for being further away from the data value. Here is an example. Suppose the representative value missed each of two data values by 1 and 4 points respectively. Using the sum of squared errors, the error is increased dramatically for the second value. Instead of having the error only increase by the 4 points missed, it is increased by the square of that miss (16 points).