Average and Range (Xbar R)#
The following table list the properties of the Average and Range (x̄ R) chart
| Property | Description |
|---|---|
| Chart | Average and Range (x̄ R) |
| Process observation types | Variables |
| Process observations relationships | Independent |
| Sample Size | \< 10, but usually 3 to 5 |
| Distribution type | Normal |
| Size of shift to detect | Large (≥ 1.5σ) |
| Average (Indicator 1) | \(\mu=\frac{x_{1}+x_{2}+x_{3}+\cdots+x_{n}}{n}\) |
| Range (Indicator 2) | \(r_{n}=Max(x_{1} \cdots x_{n})-Min(x_{1} \cdots x_{n})\) |
| Mean for All values | \(\mu=\overline{\overline{X}}=\frac{\sum{x_{ij}}}{\sum{n_{i}}}\) |
| Process Mean | \(\mu\) |
| Process Standard Deviation | \(\sigma=S_r=\frac{\sum{r_i}\cdot\frac{\left [ d_2(n_i) \right ]}{\left [ d_3(n_i) \right ]^{2}}}{\sum{\frac{\left [ d_2(n_i) \right ]^{2}}{\left [ d_3(n_i) \right ]^{2}}}}\) Where \(r_i\) is the range for the data point |
| Average (Indicator 1) Centerline | \(\mu\) |
| Average (Indicator 1) Control Limits | \(UCL = \mu + \frac{3\sigma}{\sqrt{n_i}}\) \(LCL = \mu - \frac{3\sigma}{\sqrt{n_i}}\) Where \(n_i\) = number of observations in sample (subgroup) |
| Range (Indicator 2) Centerline | \(\overline{R_i} = d_2(n_i)\cdot\sigma\) |
| Range (Indicator 2) Control Limits | \(UCL = \overline{R}_i + 3\sigma \cdot d_3(n_i)\) \(LCL = max(\overline{R}_i - 3\sigma \cdot d_3(n_i);0)\) |
Table: Average and Range chart properties
The statistical table used to compute the control limits can be consulted in the Control Chart Constants page.