UCL (Upper Control Limit) Calculations - Sample Problem

QE · Sep 29, 2009

X (double bar) = 35.0 and sigma = 5.0. n = 5....what is the upper control limit for the process ?

options a) 37.24 b) 37.89 c)41.71 d)52.50

My approach: UCL = X(2 bar) +/- 0.58 (5.0) / 2.326.
Answer from calculations = ~36

please assist why the answer is not one of the given options.

Marc · Sep 29, 2009

Where is this question from?

QE · Sep 29, 2009

Dear Marc,

It is from the sample exam available on the ASQ website. Please see the attached file.

Miner · Sep 29, 2009

You are using constants for Rbar instead of the StdDev.

Since StdDev = Rbar/d2, you can calculate the answer as follows:

UCL = X2Bar + (A2* d2 * StdDev)

UCL = 35.0 + (0.577 * 2.316 * 5.0) = 41.68;

However, ASQ appears to be using the equation for the population StdDev:

UCL = X2Bar + (3 * StdDev / SQRT

)

UCL = 35.0 + (3 * 5.0 / SQRT(5)) = 41.71

Note that the answers are very close, presumably because the process was in a state of control. If the process had not been stable, the second method would have inflated the control limits.

Darius · Sep 30, 2009

Since StdDev = Rbar/d2, you can calculate the answer as follows:

UCL = X2Bar + (A2* d2 * StdDev)

UCL = 35.0 + (0.577 * 2.316 * 5.0) = 41.68;

Not sure..,

UCL = X_bar + A2 * SIGMA = 35 + 0.577 * 5 = 37.89

A2 = 3/[d2*sqrt

]

d2= 2.326 => A2 = 0.577

Miner · Sep 30, 2009

Darius said:
Not sure..,

UCL = X_bar + A2 * SIGMA = 35 + 0.577 * 5 = 37.89

A2 = 3/[d2*sqrt]

d2= 2.326 => A2 = 0.577

UCL = X_bar + A2 * Rbar (not SIGMA)

Since SIGMA = Rbar / d2; Rbar = d2 * SIGMA

Therefore, UCL = X_bar + A2 * d2 * SIGMA

Side note: You could also simplify this to UCL = X_bar + 3 * SIGMA / sqrt

Darius · Oct 1, 2009

ok,:truce: I missed that it was not Range and I didn't recognized your initial formula, the simplificated formula looks fine.

rajanraj · Nov 30, 2009

Miner said:
You are using constants for Rbar instead of the StdDev.

Since StdDev = Rbar/d2, you can calculate the answer as follows:

UCL = X2Bar + (A2* d2 * StdDev)

UCL = 35.0 + (0.577 * 2.316 * 5.0) = 41.68;

However, ASQ appears to be using the equation for the population StdDev:

UCL = X2Bar + (3 * StdDev / SQRT)

UCL = 35.0 + (3 * 5.0 / SQRT(5)) = 41.71

Note that the answers are very close, presumably because the process was in a state of control. If the process had not been stable, the second method would have inflated the control limits.

Miner,

Your derivation is exactly right.Not just ASQ uses this formula but everyone

Because whatever we use as constants (like A2...) are the simplified form of "3/sqrt

".These values are given in the table assuming 99.7% of variation is captured between LCL and UCL

Once we know this ,it demystifies the notion of "constants".

I will always recommend the formula X(doulble bar)+-(x) sigma/sqrt

where X stands for statistic for respective confidence level

Hope this helps

Bev D · Nov 30, 2009

rajanraj said:
Miner,

Your derivation is exactly right.Not just ASQ uses this formula but everyone

Because whatever we use as constants (like A2...) are the simplified form of "3/sqrt".These values are given in the table assuming 99.7% of variation is captured between LCL and UCL

Once we know this ,it demystifies the notion of "constants".

I will always recommend the formula X(doulble bar)+-(x) sigma/sqrt

where X stands for statistic for respective confidence level

Not 'everyone' uses that formula. In fact Shewhart didn't use it and for a very specific reason: The formula sited holds for homogenous process streams that are rationally subgrouped. It is a safety factor for the necessary assumptions.

If your process has significant subgroup to subgroup variation (such as will be seen with out of control but otherwise homogenous processes or with processes that are not homogenous by their very nature) and you calculate the overall standard deviation based on the individual values (which the above formula implies) then you will have limits that are far too wide and you will potentially miss a poor subgrouping scheme and true out of control conditions resulting in a process with less capability.

The 'factors' were derived to 'force' the user to create limits on the subgroup averages based on the within subgroup variation: this is a fundamental aspect of SPC; it is not an inconvenience to explain to people. (and since science and engineering are rife with constants, anyone in charge of setting control limits should be more than comfortable with concept)

A homogenous process that is properly subgrouped will have very little subgroup to subgroup variation compared to the within subgroup variation. and so sigma_total ~ sigma_within and sigma_Xbar = sigma_within/sqrt

.

When we have a stable, well characterized process we can then safely use the formula sigma_total/sqrt

for the control limits.

It's also important to remember that the mulitplier for control limits (usually 3) is not related to a confidence level or confidence interval. It is an economic choice not a statistical one. Control charts are not a series of confidence intervals or hypothesis tests...

rajanraj · Dec 1, 2009

Bev D said:
Not 'everyone' uses that formula. In fact Shewhart didn't use it and for a very specific reason: The formula sited holds for homogenous process streams that are rationally subgrouped. It is a safety factor for the necessary assumptions.

If your process has significant subgroup to subgroup variation (such as will be seen with out of control but otherwise homogenous processes or with processes that are not homogenous by their very nature) and you calculate the overall standard deviation based on the individual values (which the above formula implies) then you will have limits that are far too wide and you will potentially miss a poor subgrouping scheme and true out of control conditions resulting in a process with less capability.

The 'factors' were derived to 'force' the user to create limits on the subgroup averages based on the within subgroup variation: this is a fundamental aspect of SPC; it is not an inconvenience to explain to people. (and since science and engineering are rife with constants, anyone in charge of setting control limits should be more than comfortable with concept)

A homogenous process that is properly subgrouped will have very little subgroup to subgroup variation compared to the within subgroup variation. and so sigma_total ~ sigma_within and sigma_Xbar = sigma_within/sqrt.

When we have a stable, well characterized process we can then safely use the formula sigma_total/sqrt for the control limits.

It's also important to remember that the mulitplier for control limits (usually 3) is not related to a confidence level or confidence interval. It is an economic choice not a statistical one. Control charts are not a series of confidence intervals or hypothesis tests...

Your are correct about "3" being economical reason

...but not to miss...most of the "statistical constrain" we use have major impact of economy.It is very well applicable for confidence intervals/accuracy..etc

If 3 is used by less-demanding product (considering 99.7% of variations to be "inherent",then a "2" can very well be used by highly critical product.All boils down to "Quality" and "Cost"

I almost forgot to mention that 3/sqrt

works well for more than 25 sample size.Lesser size has to use cohen's correction (sorry if the name is wrong...just trying to recollect).

Again..."Relative range" also jumps in for much lesser sample size (maybe less than 5)

I think we all are here to clarify(get clarified) on why we are using "constants" rather than just refering to tables in "Appendix" of any book

Love to see more of such discussions

Cheers

UCL (Upper Control Limit) Calculations - Sample Problem

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