Simple Averages and Standard Deviation

Consider a sample population, for concreteness the one given in the fourth column of the table given in the Example Table (some of you may surmise what these data represent; any changes since Sunday September 10 will not be inlcuded. See the page at Complete Stats for updated statistics).

Finding the mean, variance and standard deviation of these numbers is easily done with the available software on Athena, and of course on your handheld calculator. One great advantage of Xess over a handheld calculator is that all of the data are displayed simultaneously; another is the graphing capabilities.

For an explanation of the formulas that are used, see Sampling a Population.

To use Xess for this example, you need to follow a few basic steps:

• First, load Xess; this is done by either
  • from the Athena Dash, select Numerical/Math -> Spreadsheets -> Xess, or
  • at the Athena prompt, do
    athena% add xess
    athena% xess&
• When the Xess window appears, resize to your preference. To load the data, you can do one of:
  • Copy the data from Example Table by hand (tedious).
  • Download (with either shift-left-button or via the menu on the right mouse button) on Xess Data Table.
  • Cut and paste the data from Data Table directly into the Xess window (make sure that the first row tabs properly).
• The third column of the Example Table is, as you may have discovered, the ratio of the number of each row in Column B to the number in Column A; to have Xess do this for you, click on cell C1 and activate the Edit Window by either using the Edit menu to select ``Edit Cell'', using the ``F2'' key, or clicking in the Edit window (the widest window at the top of the spreadsheet). The Edit window is activated when the green checkmark shows.
  To find the ratio, the command is not surprising; enter =B1/A1 in the Edit window (the entries are not case-sensitive).
  To find the other ratios, all you need to do is to copy the formula by selecting cell C1, then Control-f, then copying into the remaining cells desired, in this case C2 to C23. The copying may be done in one fell swoop with the mouse.
  For aesthetic reasons, you may not like the way the result is displaced (such ratios are rarely announced to this precision). To get the more conventional format, select the entire column by clicking on the ``C'' at the top of the column, then selecting ``Format -> Cell Fomat'' from the menu. Change the Cell Format from ``Default'' to ``Fixed'' and change the number of Decimal Places from 0 to 3.
• To do the statistics: Select a cell, and in the Edit Window enter =@AVG(C1..C23). (The ``@'' symbol tells the program that you are using one of the known functions, in this case from the Statistics package). Select another cell, and enter =@STD(C1..C23) to find the the ``population standard deviation'', the square root of the variance. The find the standard deviation, you would enter =@STDS(C1..C23) (Xess calls this the ``Sample Standard Devation'', as does 8.01X). You should see that this is slightly higher than the variance.
• Xess can do much more. For instance, if a graphical display, in this case a histogram, is desired, select column C and, from the menu, select Graph -> New Graph -> Histogram. You get a histogram without working hard at all. To change the bin size, select, from the figure window, Options -> X Axis, go to the the window labelled ``Major'' and change the numer of ``ticks''. For instance, for a bin size of 0.02, disable the ``Autoscale'', select a ``Max'' of 0.4 (not any higher this year, we think), and enter 20 for the number of ticks.
  Other features, including colors, titles, legends, and saving the graph may be changed from the different menu items. The thing to do is to try as many as you feel like.
  For presentation purposes, you can save the graph and insert the graph into the sheet by selecting a cell and then Graph -> Insert in Sheet.
• To see a worksheet that contains much of the above, download (with either shift-left-button or via the menu on the right mouse button) the Basic Average Template

Least Squares

Back to the 8.01X - Fall 2001 Home Page

Last updated September 17, 2001.