x-bar Chart Example
Climacus Company's most popular wooden ladder is guaranteed to
up to 300 pounds. To give a safety margin, the manufacturing
was designed so that the average ladder would have a breaking strength
Each week, four ladders are randomly selected and subjected to
increasing force until they break.
Natural variations in the wood grain and other random
mean that not all ladders are equally strong. After taking many
of four ladders each, subtracting the strength of the weakest of the
away from that of the strongest of the four, and averaging across many
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
distributed) just by natural variations, but if something important has
in the materials, manufacturing, or measurement process, then the
of a sample mean outside of these boundaries will be much higher.
The observed mean strength on Week 10 might seem alarmingly low it
out of context, but it is within the limits of what could be expected
the natural random variations. On Week 20, we see an extremely
mean strength. We might be tempted to think of this as something
out ladders are even stronger than we designed them to be!
further thought reveals some possibilities that need looking into:
our measuring process is at fault and we have lost control of our data
maybe the observation signals an increase in the standard deviation so
dangerously weak ladders could also be being produced, maybe the
has gone out of adjustment producing ladders that are stronger than
need to be but also heavier and more expensive to produce and ship,
maybe there is a real improvement that we need to take advantage of.
control chart cannot tell us which of these is the case, but it alerts
Now let's imagine that these same observations came from aluminum
instead of wooden ones. As a manufactured material, aluminum is
variable than wood, so the standard deviation would be smaller, the
lines closer together, and the low strength in Week 10 would be truly
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: