# Calculating Control Limits in Average and Range Charts

K

#### kaczmar

Hello,

I've got maybe easy to answer doubt. According to the definition of control limits in average and range charts, they should be located 3*sigma from center line. But when we use formula UCL = X + A2*R it's give as something different. So what exactly sigma is?, is it normally calculated standard deviation for all samples?

#### Statistical Steven

##### Statistician
Staff member
Super Moderator
Re: Calculating control limits

Hello,

I've got maybe easy to answer doubt. According to the definition of control limits in average and range charts, they should be located 3*sigma from center line. But when we use formula UCL = X + A2*R it's give as something different. So what exactly sigma is?, is it normally calculated standard deviation for all samples?

The formula say Xbar +/- A2Rbar. So your estimate of variability is Rbar not the standard deviation of the data. If you want to use an Xbar/S chart then use B2.

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
Alternativley, the OP may be confused about the theoretical formula for the control limits on the subgroup averages vs the one to use in practice (average + A2Rbar)

In theory, the process stream is homogenous and therefor the variation between subgroup averages (which is what the average chart is monitoring: the sigma that is referred to in the OP's statement "within 3*sigma of the centerline" is sigma_Xbar) is small compared to the within subgroup variation AND it is equal to the total standard deviation of the individual values (includes within and between subgroup variation)/ square root of the subgroup size .

IF the process is not homogenous OR if the subgrouping scheme is not rational AND/OR if the process under study is statistically out of control, THEN the limits calculated using A2Rbar or A3Sbar will be different than the limits calculated from sigma Xbar (the standard deviation of the subgroup averages).

D

#### Darius

Hello,

I've got maybe easy to answer doubt. According to the definition of control limits in average and range charts, they should be located 3*sigma from center line. But when we use formula UCL = X + A2*R it's give as something different. So what exactly sigma is?, is it normally calculated standard deviation for all samples?

Sample size? Nop, SIGMA is NOT total variation estiamate (aka calculated standard deviation).

TOTAL variation = WITHIN SUB_GOUP variation + BETWEEN SUB_GROUP variation + {IMHO trend}

the traditional STDEV is TOTAL variation, the control limits are fixed with WITHIN variation (the least variation estimate for your process)

And remember something important, they are just "ESTIMATES" not an absolute value

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