Welcome to this statistical process control
tutorial on Control chart for x-bar. We will be constructing an x-bar chart from
process data, and then determine if the process is in statistical control. If you’ve seen the first video in this series
on R-chart, you know that a control chart comprises of a centerline, a lower control
limit, and an upper control limit. Well, the centerline of the x-bar chart is
x-double bar. That is, the mean of the sample means. Now if standard deviation of the process is
known, this formula is used to calculate the control limits.
However, we will be using the range in this video. In our case, the upper control limit, UCL,
is x-double bar plus A2 R-bar. And the lower control limit, LCL, is x-double
bar minus A2 R-bar. R-bar is the average of the sample ranges,
and A2 is found on the control chart factors table. We will be using this process data, consisting
of samples of size 5, collected every day for 10 days. Our objective is to determine if the process
mean is in statistical control. As required in the control limit formulas,
we will first obtain the sample ranges and means. The range is largest minus smallest, so for
the first day or first sample, the range is 509 minus 496 which gives 13.
We do the same for the rest of the samples. The mean for the first sample can also be
obtained by adding up the values and dividing by 5. And that gives 502.
We do the same for the rest of the samples. Next we calculate R-bar and x double bar.
The sum of these ranges is 231. So R-bar, is 231/10 which gives 23.1.
The sample means add up to 4,978 so their mean, x-double bar, is 497.8 Looking again at the formulas for the control
limit here, we have everything but A2. To find A2, we simply go to the control limit
factors table and check under sample size 5. The corresponding A2 value is 0.577. Now we have A2, x-double bar, and R bar. So UCL equals 497.8 + 0.577(23.1) which gives
511.1. And for LCL, we have 497.8 – 0.577(23.1) which
gives 484.5. For the chart let’s first draw the sample
points for the data. And then draw the control limits. And finally, draw out the run chart. The x-bar-chart is now complete.
We can see that the sample mean for day 5 is clearly below the lower control limit. Therefore, the process mean is not in statistical
control, or is out of control. As a result we could recommend that the activities
of day 5 be investigated to determine the special cause of variation and to take necessary
corrective actions. And that’s x-bar chart. Thanks for watching.