VARP(value1, value2, value3, ...)
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 usageCreate 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.
|1||pre-Excel 2003 VARP v1:||=(D4*SUMSQ(A3:A8) - (SUM(A3:A8)^2))/(D4^2)|
|3||pre-Excel 2003 VARP v2:||=(SUMSQ(A3:A8) - (SUM(A3:A8)^2)/D4)/D4|
|5||VARP in Excel 2003 and in later versions of Excel:||=DEVSQ(A3:A8)/D4|
|Modified Data||Power of 10 to add to data:||1|
|=A3 + 10^$D$10||sample mean:||=AVERAGE(A12:A17)|
|=A4 + 10^$D$10||sample size:||=COUNT(A12:A17)|
|=A5 + 10^$D$10||VARP:||=VARP(A12:A17)|
|=A6 + 10^$D$10||pre-Excel 2003 VARP v1:||=(D13*SUMSQ(A12:A17) - (SUM(A12:A17)^2))/(D13^2)|
|=A7 + 10^$D$10||pre-Excel 2003 VARP v2:||=(SUMSQ(A12:A17) - (SUM(A12:A17)^2)/D13)/D13|
|=A8 + 10^$D$10||VARP in Excel 2003 and in later versions of Excel:||=DEVSQ(A12:A17)/D13|
- 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
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 ExcelIn 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 ExcelThe 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.
ConclusionsReplacing 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
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
(SUMSQ(values) - (SUM(values)^2)/COUNT(values))/COUNT(values)
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).
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.
Article ID: 826393 - Last Review: Apr 29, 2008 - Revision: 1