Median and Range (Xtilde R)#
The following table lists the properties of the Median and Range (Xtilde R) chart.
| Property | Description |
|---|---|
| Chart | Median and Range ( \(\widetilde{x}\) 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) |
| Median (Indicator 1) | \(\widetilde{x}\) = The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, the median is the average of the two middle values. |
| Range (Indicator 2) | \(r_{n}=Max(x_{1} \cdots x_{n})-Min(x_{1} \cdots x_{n})\) |
| Process Mean | \(\mu=\overline{\widetilde{X}}=\frac{\sum{\widetilde{x_{i}}}}{n}\) |
| 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 |
| Median (Indicator 1) Centerline | \(\overline{\widetilde{X}}=\mu\) |
| Median (Indicator 1) Control Limits | \(UCL = \mu + \frac{3\sigma\cdot e1}{\sqrt{n_i}}\) \(LCL = \mu - \frac{3\sigma\cdot e1}{\sqrt{n_i}}\) |
| 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: Median and Range chart properties
The statistical table used to compute the control limits can be consulted in the Control Chart Constants page.