# X-Bar Chart - CQE Problem to Solve

M

#### Mehtap

Hello,

Could anyone can tell me where I can find the information about this question below? I keep checking the books but I can not find the answer.

An x bar- R chart with n=5 has been plotted for some time and has demonstrated random variation. Upon review of the last 30 plot points, the expected number of runs around the center line on the x bar chart is expected to be approxiametely which number?

Thanks

K

#### kg209

When I was studying for my CQE I had the CQE primer from the Quality Council of Indiana. The answer to this question was in Chapter 10 if I remember correctly. If you are studying for your CQE I highly recommend QCI primers. I studied hard but with the primer's help I was able to pass on my first attempt.

KG

#### Tim Folkerts

Super Moderator
If I had to make an off-the-cuff answer, I would say 15 runs.

If the data is random, then whichever side of center the current point is, there is a 50-50 chance the next the next point will be on the opposite side of the center. So there is a 50-50 chance that the next point will start a new run on the other side of the center. That would suggest 15 runs.

Or try with smaller numbers and look for a pattern.

With a single point, there can only be one run With two points, you get either 1 or 2 runs. They could be symbolized as
·· .. (1 run)
·. .· (2 runs)
The average is 1.5 runs

With three points, you can have
... ··· (1 run)
..· .·· ·.. ··. (2 runs)
·.· .·. (3 runs)
the average is (1+1+2+2+2+2+3+3)/8 = 2

By extension, it appears the average is (n+1)/2.
So 30 points would have an average of (30+1)/2 = 15.5 runs.