Operations and Supply Chain Management The Core

326 OPERATIONS AND SUPPLY CHAIN MANAGEMENT
How to Construct ​¯ X ​- ​ and R-Charts
If the standard deviation of the process distribution is known, the X ​ ¯ ​-chart may be defined:
where
UCL X ¯ ​ = ​ X ​ + z ​S X ¯ ​
S X ¯ ​ = s / ​√ __
n ​ = Standard deviation of sample means​
s = Standard deviation of the process distribution
and ​LCL X ¯ ​ = ​ X ​ − z​S X ¯ ​ [10.12]
n = Sample size
¯ X ​ ​ = Average​ of sample means or a target value set for the process
z = Number of standard deviations for a specific confidence level (typically, z = 3)
An ​ X ¯ ​-chart is simply a plot of the means of the samples that were taken from a process.
X ¯ ​ ​ is the average of the means.
In practice, the standard deviation of the process is not known. For this reason, an
approach that uses actual sample data is commonly used. This practical approach is
described in the next section.
An R-chart is a plot of the average of the range within each sample. The range is the
difference between the highest and the lowest numbers in that sample. R values provide an
easily calculated measure of variation used like a standard deviation. R ​¯ ​is the average of
the range of each sample. More specifically defined, these are
where
¯ X ​ = Mean​ of the sample
where
i = Item number
n = Total number of items in the sample
n
​∑​
​ ​X​ i ​
X ¯ i=1
​ = ______ ​ ​ [Same as 10.1]
n
m
​
X ​ = The​ average of the means of the samples
j = Sample number
m = Total number of samples
where
​∑​ ​ X ¯ ​ j
j=1
X ​ = ​ _____ ​ [10.13]
m
m
​
​∑​ ​ ​R​ j ​
R ¯
j=1
​ = ______ ​ ​ [10.14]
m
R j = Difference between the highest and lowest measurement in the sample
¯ R ​ = Average​ of the measurement differences R for all samples