If we use k = 20 subgroups of size n = 5, we will have 72 d.f. for the average range. Twenty such average ranges are shown in the bottom histogram of Figure 1. Notice that, as the number of degrees of freedom increase, the histogram of the average ranges becomes more concentrated. The variation of the statistics decreases as the degrees of freedom increase. A traditional measure of just how much variation is present in any measure is the coefficient of variation, which is defined as: CV = standard deviation of measure/mean of measure Examining Figure 1, we can see that as the degrees of freedom go up, the coefficient of variation for the average range goes down. This relationship holds for all those statistics that we use to estimate the standard deviation of the data. In fact, there is a simple equation that shows the relationship. For any estimate of the standard deviation of X: CV = 1/sq.rt.(2d.f.) This relationship is shown in Figure 2. So just what can you learn from Figure 2? The curve shows that when you have very few degrees of freedom  say less than 10  each additional degree of freedom that you have in your computations results in a dramatic reduction in the coefficient of variation for your limits. Since degrees of freedom are directly related to the number of data used, Figure 2 suggests that when we have fewer than 10 d.f., we will want to revise and update our limits as additional data become available. The curve in Figure 2 also shows that there is a diminishing return associated with using more data in computing limits. Limits based upon 8 d.f. will have half of the variation of limits based upon 2 d.f., and limits based upon 32 d.f. will have half of the uncertainty of limits based upon 8 d.f. Each 50percent reduction in variation for the limits requires a fourfold increase in degrees of freedom. As may be seen from the curve, this diminishing return begins around 10 degrees of freedom, and by the time you have 30 to 40 d.f., your limits will have solidified. So, if you have fewer than 10 degrees of freedom, consider the limits to be soft, and recompute the limits as additional data become available. With Shewhart's charts, 10 degrees of freedom require about 15 to 24 data. You may compute limits using fewer data, but you should understand that such limits are soft. (While I have occasionally computed limits using as few as two data, the softest limits I have ever published were based on four data!) When you have fewer than 10 d.f. for your limits, you can still say that points which are comfortably outside the limits are potential signals. Likewise, points comfortably inside the limits are probable noise. With fewer than 10 d.f., only those points close to the limits are uncertain. Thus, with an appreciation of the curve in Figure 2, you no longer must be a slave to someone's arbitrary guideline about how much data you need. Now you can use whatever amount of data may be available. You know that with fewer than 10 d.f., your limits are soft, and with more than 30 d.f., your limits are fairly solid. After all, the important thing is not the limits but the insight into the process behavior that they facilitate. The objective is not to get the "right" limits but rather to take the appropriate actions on the process. So use the amount of data the world gives you, and get on with the job of separating potential signals from probable noise. About the author Donald J. Wheeler is an internationally known consulting statistician and author of Understanding Variation: The Key to Managing Chaos and Understanding Statistical Process Control, 2nd Edition. 

