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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/d

_{2}).(1/√n) = X̄̄ ± {3R̄/(d_{2}√n)}
= X̄̄ ± A

_{2}R̄
[A

_{2}= 3/d_{2}√n]**Therefore,**

UCLx̄ = x̄̄ + A

_{2}R̄
LCLx̄ = x̄̄ - A

_{2}R̄**Since**d

_{2}is a constant depending on n, A

_{2}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) = C

_{2}σ
=> σ = S̄/C

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

**Therefore**,

UCLx̄ = x̄̄ + {3/√(nC

_{2})}.S̄
= x̄̄ + A

_{1}S̄
LCLx̄ = x̄̄ - {3/√(nC

_{2})}.S̄
= x̄̄ - A

_{1}S̄
Here, the factor A

_{1}= 3/√(nC_{2}) has given in the table for different values of n from 2 to 25.
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