#### 2. Minimizing the Spread

The mean has a special characteristic that makes it more useful for certain advanced statistical analyses than the median and the mode. That characteristic has to do with the distance of each individual observation from the mean of all observations included in calculating the mean.

Suppose you have a list of ten numbers—say, the ages of all your close relatives. Pluck another number out of the air. Subtract that number from each of the ten ages and square the result of each subtraction. Now, find the total of all ten squared differences.

If the number that you chose, the one that you subtracted from each of the ten ages, happens to be the mean of the ten ages, then the total of the squared differences is minimized, thus the term least squares. That total is smaller than it would be if you chose any number other than the mean. This outcome probably seems a strange thing to care about, but it turns out to be an important characteristic of many statistical analyses, as you’ll see in later chapters of this book.

Here’s a concrete example. Figure 4 shows the height of each of ten people in cells A2:A11.

##### Figure 4. Columns B, C and D are reserved for values that you supply.

You should fill in columns B, C, and D as described later in this section. The cells B2:B11 in Figure 4 will then contain a value—any numeric value—that’s different from the actual mean of the ten observations in column A. You will see that if the mean is in column B, the sum of the squared differences in cell D13 is smaller than if any other number is in column B.

To see that, you will need to have made Solver available to Excel.

##### About Solver

Solver is an add-in that comes with Microsoft Excel. You can install it from the factory disc or from the software that you downloaded to put Excel on your computer. Solver helps you backtrack to underlying values when you want them to result in a particular outcome.

For example, suppose you have ten numbers on a worksheet, and their mean value is 25. You want to know what the tenth number must be in order for the mean to equal 30 instead of 25. Solver can do that for you. Normally, you know your inputs and you’re seeking a result. When you know the result and want to find the necessary values of the inputs, Solver provides one way to do so.

The example in the prior paragraph is trivially simple, but it illustrates the main purpose of Solver: You specify the outcome and Solver determines the input values needed to reach the outcome.

You could use another Excel tool, Goal Seek, to solve the latter problem. But Solver offers you many more options than does Goal Seek. For example, using Solver, you can specify that you want an outcome maximized or minimized, instead of solving for a particular outcome. That’s relevant here because we want to find a value that minimizes the sum of the squared differences.

##### Finding and Installing Solver

It’s possible that Solver is already installed and available to Excel on your computer. To use Solver in Excel 2007 or 2010, click the Ribbon’s Data tab and find the Analysis group. If you see Solver there you’re all set. (In Excel 2003 or earlier, check for Solver in the Tools menu.)

If you don’t find Solver on the Ribbon or the Tools menu, take these steps in Excel 2007 or 2010:

1. | Click the Ribbon’s File tab and choose Options. |

2. | Choose Add-Ins from the Options navigation bar. |

3. | At
the bottom of the View and Manage Microsoft Office Add-Ins window, make
sure that the Manage drop-down is set to Excel Add-Ins and then click
Go. |

4. | The Add-Ins dialog box appears. If you see Solver Add-in listed, fill its check box and click OK. |

You should now find Solver in the Analysis group on the Ribbon’s Data tab.

If you’re using Excel 2003 or earlier, start by choosing Add-Ins from the Tools menu. Then complete step 4 in the preceding list.

If you didn’t find Solver in the Analysis group on the Data tab (or on the Tools menu in earlier Excel versions), and if you did not see it in the Add-Ins dialog box in step 4, then Solver was not installed with Excel. You will have to re-run the installation routine, and you can usually do so via the Programs item in the Windows Control Panel.

The sequence varies according to the operating system you’re running, but you should choose to change features for Microsoft Office. Expand the Excel option by clicking the plus sign by its icon and then do the same for Add-ins. Click the drop-down by Solver and choose Run from My Computer. Complete the installation sequence. When it’s through, you should be able to make the Solver add-in available to Excel using the sequence of four steps provided earlier in this section.