Given the following data from an xbar and r chart mathematic

Given the following data from an xbar and r chart. mathematical analysis required:

x double bar = 23 pounds r bar ==4 pounds, n=5

calculate the control limits for the average and the range portion of the xbar and r chart. round your answer to 2 decimal places.

explain exactly what these control limits represent

Solution

x double bar=23 pounds   and r bar=4 pounds and sample size=n=5

the upper control limit of Xbar chart is xbar+3sigma/sqrt(n)

and lower control control limit of Xbar chart is xbar-3sigma/sqrt(n)

now sigma is unknown. so estimated by rbar

rbar/d2 is an unbiased estimator of sigma   [d2 is a constant for fixed n]

so UCL=xbar+3rbar/[d2*sqrt(n)]=xbar+A2*rbar    where A2=3/[d2*sqrt(n)]

hence LCL=xbar-A2*rbar

where xbar=23 rbar=4   A2 is calculated from statistical table for n=5 which is 0.577

so UCL=23+0.577*4=25.31 and LCL=23-0.577*4=20.70 [answer]

by similar arguments

for R chart UCL=rbar+3Dsigma and LCL=rbar-3Dsigma      [D is a constant for fixed n]

now sigma is unknown and is estimated by rbar/d2

so UCL=rbar+3Drbar/d2=rbar(1+3D/d2)=D4rbar   and hence LCL=rbar(1-3D/d2)=D3rbar

where rbar=4 and D3,D4 are calculated from statistical tables for n=5

which are D3=0 D4=2.115

so UCl=2.115*4=8.46 and LCL=0   [answer]

these control limits actually represents a boundary for the average values and ranges of the samples taken from the process. if the values lies within these limits then taht indicates that the process is in control.

any deviation of the process from its initial state would result in values fallling outside these control limits.

these limits are called 3sigma limits.

from chebysheb\'s inequality P[mean-3sigma<X<mean+3sigma]>8/9 irrespective of the distribution of X.