# 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 (Sp) 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, Sp 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}