# 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 ±  A2
[A2 = 3/d2√n]

Therefore,
UCLx̄ = x̄̄ + A2
LCLx̄ = x̄̄ - A2
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 + A1
LCLx̄ = x̄̄ - {3/√(nC2)}.S̄
= x̄̄ - A1
Here, the factor A1 = 3/√(nC2) has given in the table for different values of n from 2 to 25.