Formula for calculation in Xbar-R Chart?

Term	Description
$x_{ij}$	jth observation in the ith subgroup
$n_i$	number of observations in subgroup i
$\sum x$	sum of all individual observations
$\sum n$	total number of observations
$μ$	process mean
$k$	parameter for Test 1 (The default is 3)
$σ$	process standard deviation
$d_2 (.)$	value of unbiasing constant d2 that corresponds to the value specified in parentheses
$d_3 (.)$	value of unbiasing constant d3 that corresponds to the value specified in parentheses
$r_i$	range for subgroup i
$m$	number of subgroups
$\overline x_i$	mean of subgroup i
$µ_ν$	mean of the subgroup variances
$c_4 (.)$	value of the unbiasing constant c4 that corresponds to the value that is specified in parentheses
$c_5 (.)$	value of the unbiasing constant c5 that corresponds to the value that is specified in parentheses
$Γ()$	gamma function

 

• Formula for Xbar Chart

  • Plotted Points:

...........................Each plotted point, $\overline x_i$, represents the mean of the observations for subgroup,$i$.

$\overline x_i = \frac{\sum\limits_{j=1}^{n_i} x_{ij}}{n_i}$

• • Center Line:

    The center line represents the process mean (µ).

$\overline{\overline{X}} = \frac{∑x}{∑n}$

• • Control Limits:

    • Lower Control Limit:

      The value of the lower control limit for each subgroup, $i$, is calculated as follows:

$LCL_i= µ- \frac{kσ}{√(n_i )}$

• • • Upper Control Limit:

      The value of the upper control limit for each subgroup, $i$, is calculated as follows:

$UCL_i= µ+ \frac{kσ}{√(n_i )}$

• Formula for R Chart

  • Plotted Points:

    Each plotted point, $r_i$, represents the range for subgroup $i$.

  • Center Line:

    The value of the center line for each subgroup, $\overline R_i$, is calculated as follows:

$\overline R_i= d_2(n_i) × σ$

• • Control Limits:

    • Lower Control Limit:

      The value of the lower control limit for each subgroup $i$ is equal to the greater of the following:

$LCL_i= [d_2(n_i) × σ] - [kσ×d_3(n_i)]$

.......................................or

$LCL_i= 0$

• • • Upper Control Limit:

      The value of the upper control limit for each subgroup $i$ is calculated as follows:

$UCL_i= [d_2(n_i) × σ] + [kσ×d_3(n_i)]$

• Formula for estimation of sigma (standard deviation)

  • Using Rbar Method:

............................Zometric uses the range of each subgroup, $r_i$, to calculate $S_r$, which is an unbiased estimator of σ:

$S_r = \frac{\sum\limits_{i}^{} (\frac{f_i r_i}{d_2 (n_i)})}{\sum\limits_{i}^{} {f_i}}$

............................where

$f_i = \frac{[d_2 (n_i)]^2}{[d_3 (n_i)]^2}$

............................When the subgroup size is constant, the formula simplifies to the following:

$S_r = \frac{\overline R}{d_2 (n_i)}$

............................where

$\overline R = \frac{\sum r_i}{m}$

• • Using pooled standard deviation method:

............................The pooled standard deviation (S_p) is given by the following formula:

$S_p = \sqrt\frac{\sum\limits_{i}^{} \sum\limits_{j}^{} (x_{ij}-\overline x_i)^2 }{\sum\limits_{i}^{} (n_{i}-1)}$

.............................When the subgroup size is constant, S_p can also be calculated as follows:

$S_p = \sqrt µ_ν$

• Formulae for unbiasing constants

  • $d_2 ()$:

............................For values of N from 51 to 100, use the following approximation for d₂(N):

$d_2 (N) = 3.4873 + 0.0250141 × N - 0.00009823 × N^2$

• • $d_3 ()$ and $d_4 ()$:

............................For values of N from 26 to 100, use the following approximations for d₃(N) and d₄(N):

$d_3 (N) = 0.80818 - 0.0051871 × N + 0.00005098×N^2 - 0.00000019 × N^3$

$d_4 (N) = 2.88606 + 0.051313 × N - 0.00049243×N^2 + 0.00000188 × N^3$

• • $c_4 ()$ and $c_5 ()$:

$c_4 (N) = {\sqrt \frac {2}{N-1}}\frac{Γ\frac{N}{2}}{Γ\frac{N-1}{2}}$

$c_5 (N) = \sqrt {1 - c_4(N)^2}$