Control limits for X̄-Chart

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Control Limits for the X̄-chart:

Case I: When standards are given i.e. both μ and σ are known. 

The 3 - σ control limits for X̄-chart is given by E (X̄̄) ± 3 S.E (X̄̄) = μ ± 3σ/√n = μ ± Aσ 

[A = 3/√n]

If μ' & σ' are known on specified values of μ and σ respectively, then 

UCLx̄ = μ' + Aσ'

LCLx̄ = μ' - Aσ'

Case II: When standards are not given. 

If both μ and σ are unknown, then using their estimates X̄̄ (X-bar-bar) and σ given in (ii) & (iii) respectively, we get the 3- σ control limits on the X̄-chart as-

X̄̄ ± (3R̄/d₂).(1/√n) = X̄̄ ± {3R̄/(d₂√n)} 

= X̄̄ ±  A₂R̄

[A₂ = 3/d₂√n]

Therefore, 

UCLx̄ = x̄̄ + A₂R̄

LCLx̄ = x̄̄ - A₂R̄

Since d₂ is a constant depending on n, A₂ also depends only on n and it's values are found from the table. If the control limits are to be obtained in terms of S̄ (S-bar) rather than R̄, then an estimate of σ can be obtained from the relation -

S̄ (S-bar) = C₂σ

=> σ = S̄/C₂

=> C2 = √(2/n). [{(n-2)/2}!/{(n-3)/2}!]

Therefore, 

UCLx̄ = x̄̄ + {3/√(nC₂)}.S̄

= x̄̄ + A₁S̄

LCLx̄ = x̄̄ - {3/√(nC₂)}.S̄

= x̄̄ - A₁S̄

Here, the factor A₁ = 3/√(nC₂) has given in the table for different values of n from 2 to 25.