Statistical FAQs
How are control limits calculated for the different moving average control charts?
EWMA Charts:
The following notation is used for the formulas:
| Ei | exponentially weighted moving average for the i th subgroup |
| r | EWMA weight parameter (0 < r ≤ 1) |
| μ | process mean (expected value of the population of measurements) |
| σ | process standard deviation (standard deviation of the population of measurements) |
| xij | jth measurement in i th subgroup, with j=1,2,3,. . .,ni |
| ni | sample size of i th subgroup |
 |
mean of measurements in i th subgroup. If ni=1, then the subgroup mean reduces to the single observation in the subgroup |
 | weighted average of subgroup means |
 (·) | inverse standard normal function |
You can compute the limits in the following ways:
- as a specified multiple (k) of the standard error of Ei above and below the central line. The default limits are computed with k=3 (these are referred to as 3σ limits).
- as probability limits defined in terms of α, a specified probability that Ei exceeds the limits.
The following table presents the formulas for the limits:
These formulas assume that the data are normally distributed. If standard values μ0 and σ0
are available for μ and σ, respectively, replace with μ0 and
with σ0 in the above formula. Note that the limits vary with both ni
and i.
UWMA Charts:
The following notation is used for the formulas:
| Ai | uniformly weighted moving average for the i th subgroup |
| w | span parameter (number of terms in moving average) |
| μ | process mean (expected value of the population of measurements) |
| σ | process standard deviation (standard deviation of the population of measurements) |
| xij | jth measurement in i th subgroup, with j=1,2,3,. . .,ni |
| ni | sample size of i th subgroup |
| mean of measurements in i th subgroup. If ni=1, then the subgroup mean reduces to the single observation in the subgroup |
| weighted average of subgroup means |
| (·) | inverse standard normal function |
You can compute the limits in the following ways:
- as a specified multiple (k) of the standard error of Ai above and below the central line. The default limits are computed with k=3 (these are referred to as 3σ limits).
- as probability limits defined in terms of α, a specified probability that Ai exceeds the limits.
The following table presents the formulas for the limits:
Control Limits
LCL=-k(/min(i, w))*sqrt((1/ni) + (1/ni-1) +...+
(1/n1+max(i-w,0)))
UCL=+k(/min(i, w))*sqrt((1/ni) + (1/ni-1) +...+
(1/n1+max(i-w,0)))
Probability Limits
LCL=-(1-α/2)(/min(i, w))*sqrt((1/ni) + (1/ni-1) +...+
(1/n1+max(i-w,0)))
UCL=+(1-α/2)(/min(i, w))*sqrt((1/ni) + (1/ni-1) +...+
(1/n1+max(i-w,0)))
These formulas assume that the data are normally distributed. If standard values μ0 and σ0
are available for μ and σ, respectively, replace with μ0 and replace
with σ0 in the above formula. Note that the limits vary with both ni
and i.
FAQ # 2094
Last Updated: 2005 Apr 13
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