UCL (Upper Control Limit) Calculations - Sample Problem

[QE]

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]

Where is this question from?

[QE]

Dear Marc,

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

[Miner]

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(n))

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]

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(n)]

d2= 2.326 => A2 = 0.577

[Miner]

In Reply to Parent Post by Darius

Not sure..,

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

A2 = 3/[d2*sqrt(n)]

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(n)

[Darius]

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

[rajanraj]

In Reply to Parent Post by Miner

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(n))

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(n)".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(n)

where X stands for statistic for respective confidence level

Hope this helps