...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

Once you have calculated the standard deviation for your data, a legitimate question to ask is "How reliable is the calculated standard deviation?". For this situation the Chi Squared distribution can be used to calculate confidence intervals for the standard deviation.

The full example code & sample output is in chi_square_std_dev_test.cpp.

We'll begin by defining the procedure that will calculate and print out the confidence intervals:

void confidence_limits_on_std_deviation( double Sd, // Sample Standard Deviation unsigned N) // Sample size {

We'll begin by printing out some general information:

cout << "________________________________________________\n" "2-Sided Confidence Limits For Standard Deviation\n" "________________________________________________\n\n"; cout << setprecision(7); cout << setw(40) << left << "Number of Observations" << "= " << N << "\n"; cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n";

and then define a table of significance levels for which we'll calculate intervals:

double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };

The distribution we'll need to calculate the confidence intervals is a Chi Squared distribution, with N-1 degrees of freedom:

chi_squared dist(N - 1);

For each value of alpha, the formula for the confidence interval is given by:

Where

is the upper critical value, and

is the lower critical value of the Chi Squared distribution.

In code we begin by printing out a table header:

cout << "\n\n" "_____________________________________________\n" "Confidence Lower Upper\n" " Value (%) Limit Limit\n" "_____________________________________________\n";

and then loop over the values of alpha and calculate the intervals for each: remember that the lower critical value is the same as the quantile, and the upper critical value is the same as the quantile from the complement of the probability:

for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) { // Confidence value: cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); // Calculate limits: double lower_limit = sqrt((N - 1) * Sd * Sd / quantile(complement(dist, alpha[i] / 2))); double upper_limit = sqrt((N - 1) * Sd * Sd / quantile(dist, alpha[i] / 2)); // Print Limits: cout << fixed << setprecision(5) << setw(15) << right << lower_limit; cout << fixed << setprecision(5) << setw(15) << right << upper_limit << endl; } cout << endl;

To see some example output we'll use the gear data from the NIST/SEMATECH e-Handbook of Statistical Methods.. The data represents measurements of gear diameter from a manufacturing process.

________________________________________________ 2-Sided Confidence Limits For Standard Deviation ________________________________________________ Number of Observations = 100 Standard Deviation = 0.006278908 _____________________________________________ Confidence Lower Upper Value (%) Limit Limit _____________________________________________ 50.000 0.00601 0.00662 75.000 0.00582 0.00685 90.000 0.00563 0.00712 95.000 0.00551 0.00729 99.000 0.00530 0.00766 99.900 0.00507 0.00812 99.990 0.00489 0.00855 99.999 0.00474 0.00895

So at the 95% confidence level we conclude that the standard deviation is between 0.00551 and 0.00729.

Similarly, we can also list the confidence intervals for the standard deviation for the common confidence levels 95%, for increasing numbers of observations.

The standard deviation used to compute these values is unity, so the
limits listed are **multipliers** for any
particular standard deviation. For example, given a standard deviation
of 0.0062789 as in the example above; for 100 observations the multiplier
is 0.8780 giving the lower confidence limit of 0.8780 * 0.006728 = 0.00551.

____________________________________________________ Confidence level (two-sided) = 0.0500000 Standard Deviation = 1.0000000 ________________________________________ Observations Lower Upper Limit Limit ________________________________________ 2 0.4461 31.9102 3 0.5207 6.2847 4 0.5665 3.7285 5 0.5991 2.8736 6 0.6242 2.4526 7 0.6444 2.2021 8 0.6612 2.0353 9 0.6755 1.9158 10 0.6878 1.8256 15 0.7321 1.5771 20 0.7605 1.4606 30 0.7964 1.3443 40 0.8192 1.2840 50 0.8353 1.2461 60 0.8476 1.2197 100 0.8780 1.1617 120 0.8875 1.1454 1000 0.9580 1.0459 10000 0.9863 1.0141 50000 0.9938 1.0062 100000 0.9956 1.0044 1000000 0.9986 1.0014

With just 2 observations the limits are from **0.445**
up to to **31.9**, so the standard deviation
might be about **half** the observed value
up to **30 times** the observed value!

Estimating a standard deviation with just a handful of values leaves a very great uncertainty, especially the upper limit. Note especially how far the upper limit is skewed from the most likely standard deviation.

Even for 10 observations, normally considered a reasonable number, the
range is still from 0.69 to 1.8, about a range of 0.7 to 2, and is still
highly skewed with an upper limit **twice**
the median.

When we have 1000 observations, the estimate of the standard deviation is starting to look convincing, with a range from 0.95 to 1.05 - now near symmetrical, but still about + or - 5%.

Only when we have 10000 or more repeated observations can we start to be reasonably confident (provided we are sure that other factors like drift are not creeping in).

For 10000 observations, the interval is 0.99 to 1.1 - finally a really convincing + or -1% confidence.