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###### Confidence Intervals on the Standard Deviation

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_deviation_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.

##### Confidence intervals as a function of the number of observations

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.