STDEV returns the sample standard deviation for a sample
whose values are contained in an Excel worksheet and specified by the argument
or arguments to STDEV.
Syntax
STDEV(value1, value2, value3, ...)
where value1, value2, ..., up to 30 value arguments.
The
most common usage of STDEV includes only 1 value argument specifying a range of
cells that contain the sample, for example, STDEV(A1:B100).
Example Usage
Create a blank Excel worksheet, copy the table below, select cell
A1 in your blank Excel worksheet, and then paste the entries so that the table below fills cells
A1:D17 in your worksheet.
| Data | | | |
| | | |
| 6 | | sample mean | =AVERAGE(A3:A8) |
| 4 | | sample size | =COUNT(A3:A8) |
| 2 | | STDEV | =STDEV(A3:A8) |
| 1 | | pre-Excel 2003 STDEV
v1 | =SQRT((D4*SUMSQ(A3:A8) - (SUM(A3:A8)^2))/(D4*(D4 - 1))) |
| 3 | | pre-Excel 2003 STDEV
v2 | =SQRT((SUMSQ(A3:A8) - (SUM(A3:A8)^2)/D4)/(D4 - 1)) |
| 5 | | Excel 2003 and in later versions of Excel
STDEV | =SQRT(DEVSQ(A3:A8)/(D4 - 1)) |
| | | |
| 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$10 | | STDEV | =STDEV(A12:A17) |
| =A6 + 10^$D$10 | | pre-Excel 2003 STDEV
v1 | =SQRT((D13*SUMSQ(A12:A17) - (SUM(A12:A17)^2))/(D13*(D13 - 1))) |
| =A7 + 10^$D$10 | | pre-Excel 2003 STDEV
v2 | =SQRT((SUMSQ(A12:A17) - (SUM(A12:A17)^2)/D13)/(D13 - 1)) |
| =A8 + 10^$D$10 | | Excel 2003 and in later versions of Excel
STDEV | =SQRT(DEVSQ(A12:A17)/(D13 - 1)) |
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, do one of the following:
- In Microsoft Office Excel 2007, click the Home tab, click Format in the Cells group, and then click AutoFit Column Widths.
- In Excel 2003, point to
Column on the Format menu, and then click
AutoFit Selection.
Cells A3:A8 contain 6 data points
that are used in this example.
Cell D5 contains the value of STDEV for
your current version of Excel. If you use Excel 2003 and later versions of Excel, this value should agree
with the value in cell D8. The value in cell D8 shows the Excel 2003 and later versions of Excel value of
STDEV (regardless of the version of Excel that you are using). Cells D6 and D7
show two approximations to the value of STDEV that were computed by earlier
versions of Excel. The formula in cell D6 is the formula that is shown in the
Help file for Excel 2002 and earlier.
In this example, all versions
return the value 1.870828693. There are no computational problems here that
cause differences in STDEV between 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 not affect the value of sample standard
deviation.
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 STDEV are well-behaved in these 7
cases.
But, if you continue the experiment to try the values 8, 9, and
10 in cell D10, you will notice that the Excel 2003 and later versions of Excel value remains at
1.870828693 (as it should), while the Excel 2002 and earlier Excel version
values change (even though they should remain constant at 1.870828693). 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. Still, the cases used in
this experiment can be viewed as rather extreme.
Results in Earlier Versions of Excel
In extreme cases where there are many significant digits in the
data but, at the same time, 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 specified in the
Help file in earlier versions of Excel.
Results in Excel 2003 and in later versions of Excel
The procedure 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 STDEV in Excel 2003 and in later versions of Excel. Excel 2003 and in later versions of Excel results will
never be less accurate than results in earlier versions.
In most
practical examples, however, you are not likely to see a difference between
Excel 2003 and later versions of Excel results and results in earlier versions of Excel. This issue occurs
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 want to see if
switching to Excel 2003 and later versions of Excel will make a difference, compare the results of
with the results of
SQRT(DEVSQ(values)/(COUNT(values) � 1))
If the results are consistent to level of accuracy that you want,
then switching to Excel 2003 and later will not affect the value of STDEV.
If
you use Excel 2003 and later versions of Excel, and want to see if the computed value of STDEV(values) has
changed from the value that would have been found when you use an earlier
version of Excel, compare
with
SQRT((SUMSQ(values) - (SUM(values)^2)/COUNT(values))/(COUNT(values) - 1))
This comparison gives at least a good approximation to the value
of STDEV as found by earlier versions of Excel.
The procedure of
finding the sum of squared deviations about a sample mean by
- Finding the sample mean,
- Computing each squared deviation,
- Summing the squared deviations
is more accurate than the alternative procedure (frequently
named the "calculator formula" because it was suitable for the use of a
calculator on a small number of data points):
- Find the sum of squares of all observations, the sample
size, and the sum of all observations.
- Compute the sum of squares of all observations minus ((sum
of all observations)^2)/sample size).
There are many other functions that have been improved for Excel 2003 and later versions of Excel by replacing this latter one-pass procedure by the two-pass procedure that
finds the sample mean on the first pass and computes the sum of squared
deviations about it 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.