Excel statistical functions: VARP

The purpose of this article is to describe the VARP function in Microsoft Office Excel 2003 and in later versions of Excel, to illustrate how the function is used, and to compare results of the function for Excel 2003 and for later versions of Excel with the results of VARP when used in earlier versions of Excel.

VARP returns the population variance for a population whose values are contained in an Excel worksheet and are specified by the argument or arguments to VARP.

Syntax

where value1, value2, ..., up to 30 value arguments.

The most common usage of VARP includes only 1 value argument that specifies a range of cells that contains the population, for example, VARP(A1:B100).

Example of usage

Create a blank Excel worksheet, and copy the following table. Select cell A1 in your blank Excel worksheet, and then paste the entries so that the table fills cells A1:D17 in your worksheet.After you paste this table into your new Excel worksheet, click the Paste Options button, and then click Match Destination Formatting. With the pasted range still selected, use one of the following procedures, as appropriate for the version of Excel that you are running:

• In Microsoft Office Excel 2007, click the Home tab, click Format in the Cells group, and then click AutoFit Column Width.
• In Excel 2003, point to Column on the Format menu, and then click AutoFit Selection.

Cell D5 contains the value of VARP for your current version of Excel. If you use Excel 2003 or a later version of Excel, this value should agree with the value in cell D8. Cell D8 shows the value of VARP in Excel 2003 and in later versions of Excel (regardless of the version of Excel that you are using). Cells D6 and D7 show two approximations to the value of VARP that was computed by earlier versions of Excel. The formula in cell D6 is the formula that is shown in the Help file for Microsoft Excel 2002 and earlier versions of Excel.

In this example, all versions return the value 2.9166667. There are no computational problems here that cause differences in VARP among the versions of Excel.

You can use rows 10 to 17 to experiment with modified data by adding a constant (in this case a power of 10 is added) to each data point. It is well-known that adding a constant to each data point will have no affect on the value of population variance.

If you change the value in cell D10 (for example: to 1, 2, 3, 4, 5, 6, or 7), you can see the revised data values in cells A12:A17, and you can also see that all versions of VARP are well-behaved in these 7 cases.

However, if you continue the experiment to try the values 8, 9, and 10 in cell D10, you will notice that the value for Excel 2003 and for later versions of Excel remains at 2.9166667 (as it should), while the values for Excel 2002 and for earlier versions of Excel change (even though they should remain constant at 2.9166667). This would not occur if computations could be done with infinite precision.

Earlier versions of Excel exhibit wrong answers in these cases because the effects of round-off errors are more profound with the computational formula that is used by these versions of Excel. Still, the cases that are used in this experiment could be viewed as rather extreme.

Results in earlier versions of Excel

In extreme cases where there are many significant digits in the data but a small variance, the old computational formula leads to inaccurate results. Earlier versions of Excel used a single pass through the data to compute the sum of squares of the data values, the sum of the data values, and the count of the data values (sample size). These quantities were then combined into the computational formula that is specified in the Help file in earlier versions of Excel.

Results in Excel 2003 and in later versions of Excel

The procedure that is used in Excel 2003 and in later versions of Excel uses a two-pass process through the data. First, the sum and count of the data values are computed and from these the sample mean (average) can be computed. Then, on the second pass, the squared difference between each data point and the sample mean is found and these squared differences are summed.

In the numeric examples, even with a high power of 10 in cell D10, these squared differences are not affected and the results of the second pass are independent of the entry in cell D10. Therefore, the results in Excel 2003 and in later versions of Excel are more stable numerically.

Conclusions

Replacing a one-pass approach by a two-pass approach guarantees better numeric performance of VARP in Excel 2003 and in later versions of Excel. Results in Excel 2003 and in later versions of Excel will never be less accurate than results in earlier versions of Excel.

In most practical examples, however, you are not likely to see a difference between the results in later versions of Excel and the results in earlier versions of Excel. This is because typical data is unlikely to exhibit the kind of unusual behavior that this experiment illustrates.

Numeric instability is most likely to appear in earlier versions of Excel when data contains a high number of significant digits combined with relatively little variation between data values.

If you use an earlier version of Excel, and if you want to determine whether switching to Excel 2003 or to a later version of Excel will make a difference, compare the results of with the results of If the results are consistent with the accuracy that you want, then switching to Excel 2003 or a later version of Excel will not affect the value of VAR.

If you use Excel 2003 or a later version of Excel, and if you want to determine whether the computed value of VARP(values) has changed from the value that would have been found when you used an earlier version of Excel, compare withThis comparison gives at least a good approximation of the value of VARP as found by earlier versions of Excel.

The procedure of finding the sum of squared deviations about a mean (average) by finding the mean, by computing each squared deviation, and by summing the squared deviations is more accurate than the alternative procedure. The alternative procedure is frequently named the "calculator formula" because it was suitable for the use of a calculator on a small number of data points. The calculator formula uses the following procedures:

• Find the sum of squares of all observations, the number of observations, and the sum of all observations.
• Compute the sum of squares of all observations minus ((sum of all observations)^2)/number of observations).

There are many other functions that have been improved for Excel 2003 and for later versions of Excel. These functions were improved by replacing the one-pass procedure with the two-pass procedure that finds the mean on the first pass and then computes the sum of squared deviations about the mean on the second pass.

A short list of such functions includes VAR, VARP, STDEV, STDEVP, DVAR, DVARP, DSTDEV, DSTDEVP, FORECAST, SLOPE, INTERCEPT, PEARSON, RSQ, and STEYX. Similar improvements were made in each of the three Analysis of Variance tools in the Analysis ToolPak.