# X, R variable control chart calculations example

Example 3.1

In this example, a critical dimension for a part is measured as it is being inspected in a machining operation. To set up the control chart, four measurements were taken every day for 25 successive days, to approximate the daily production variability. These measurements were then used to calculate the limits of the control charts. The measurements are shown in Table 3.2.

It should be noted that the value n used in Equation 3.5 is equal to 4, which is the number of observations in each sample. This is not to be confused with the 25 sets of subgroups or samples for the historical record of the process. If the 25 samples are taken daily, they represent approximately a one-month history of production.

During the first day, four samples were taken, measuring 9,12,11, and 14 thousands of an inch. These were recorded in the top of the four columns of sample #1. The average, or X was calculated and entered in column 5, and the R is entered in column 6.

X Sample 1 = (9 + 12 + 11 + 14)/4 = 11.50

The range, or R, is calculated by taking the highest reading (H in this case), minus the lowest reading (9 in this case).

R Sample 1 = 14 - 9 = 5

The averages of X and R are calculated by dividing the column totals of X and R by the number of subgroups.

X = (SUM OFXs)/number of subgroups

X=315.50/25 = 12.62

R = (SUM OF R's) number of subgroups

R=111/25 = 4.44

Using the control chart (Table 3.1), the control limits can be calculated using n = 4 as follows:

X Control limits

UCKX = X + A2 R = 12.62 + 0.73 • 4.44 = 15.86

UCLX =1-A2^= 12.62 - 0.73 • 4.44 = 9.38

R Control limits

Upper control limit (UCL^) = D4R = 2.28 * 4.44 = 10.12

Lower control limit (LCL/f) = = 0

Since the measurements were recorded in thousands of an inch, the centerline of the-X" control chart is 0.01262 and the control limits for X are 0.01586 and 0.00938. For the R chart, the centerline is set at 0.00444 and the limits are 0.01012 and 0.

These numbers form the control limits of the control chart. After the limits have been calculated, the control chart is ready for use in production. Each production day, four readings of the part dimension are to be taken by the responsible operators, with the average of the four readings plotted on the X chart，and the range or difference be- tween the highest and_ lowest reading to be plotted on the R chart. The daily numbers of X and R should plot within the control limits. If they plot outside the limits, the production process is not in control， and immediate corrective action should be initiated.