x-bar Chart Example

Climacus Company's most popular  wooden ladder is guaranteed to support up to 300 pounds.  To give a safety margin, the manufacturing process was designed so that the average ladder would have a breaking strength of 400 pounds. 
Each week, four ladders are randomly selected and subjected to gradually increasing force until they break. 
Natural variations in the wood grain and other random variation mean that not all ladders are equally strong.  After taking many samples of four ladders each, subtracting the strength of the weakest of the four away from that of the strongest of the four, and averaging across many samples, we find that  R-bar is 32.9 pounds.
Since the sample size is 4, we read from the table that a2 = 0.73.  If the manufacturing process is "in control,"  sample means will randomly be either below 400-0.73*32.9 = 376 pounds or above
400+0.73*32.9 = 424 pounds about a quarter of a percent of the time (assuming strength is normally distributed) just by natural variations, but if something important has changed in the materials, manufacturing, or measurement process, then the probability of a sample mean outside of these boundaries will be much higher.

The observed mean strength on Week 10 might seem alarmingly low it taken out of context, but it is within the limits of what could be expected from the natural random variations.  On Week 20, we see an extremely high mean strength.  We might be tempted to think of this as something good; out ladders are even stronger than we designed them to be!  However, further thought reveals some possibilities that need looking into: maybe our measuring process is at fault and we have lost control of our data collection, maybe the observation signals an increase in the standard deviation so that dangerously weak ladders could also be being produced, maybe the process has gone out of adjustment producing ladders that are stronger than they need to be but also heavier and more expensive to produce and ship,  Finally, maybe there is a real improvement that we need to take advantage of.  The control chart cannot tell us which of these is the case, but it alerts us to look.



Now let's imagine that these same observations came from aluminum ladders instead of wooden ones.  As a manufactured material, aluminum is less variable than wood, so the standard deviation would be smaller, the control lines closer together, and the low strength in Week 10 would be truly alarming:  If the mean range in the strength of aluminum ladders among samples of 4 is 22 pounds, the control limits would be 384 pounds and 416 pounds: