---
alias: user-guide-average-and-range
description: "This documentation details the properties and calculations for an average and range chart"
---

# 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**   | $\sigma=S_r=\frac{\sum{r_i}\cdot\frac{\left [ d_2(n_i) \right ]}{\left [ d_3(n_i) \right ]^{2}}}{\sum{\frac{\left [ d_2(n_i) \right ]^{2}}{\left [ d_3(n_i) \right ]^{2}}}}$<br/>Where $r_i$ is the range for the data point |
| **Average (Indicator 1) Centerline**     | $\mu$ |
| **Average (Indicator 1) Control Limits** | $UCL = \mu + \frac{3\sigma}{\sqrt{n_i}}$<br/>$LCL = \mu - \frac{3\sigma}{\sqrt{n_i}}$<br/>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**   | $UCL = \overline{R}_i + 3\sigma \cdot d_3(n_i)$<br/>$LCL = max(\overline{R}_i - 3\sigma \cdot d_3(n_i);0)$ |

Table: Average and Range chart properties

The statistical table used to compute the control limits can be consulted in the [Control Chart Constants](control_chart_constants.md) page.