---
alias: user-guide-average-and-standard-deviation
description: "This documentation details the properties and calculations for an Xbar-s control chart"
---

# Average and Standard Deviation (Xbar s)

The following table lists the properties of the Average and Standard Deviation (x̄ s) chart.

| Property                                            | Description                                                       |
|-----------------------------------------------------|--------------------------------------------------------------------|
| **Chart**                                           | Average and Standard Deviation *(x̄ s)*                |
| **Process observation types**                       | Variables                                                         |
| **Process observations relationships**              | Independent                                                       |
| **Sample Size**                                     | Usually \>=10                                                     |
| **Distribution type**                               | Normal                                                            |
| **Size of shift to detect**                         | Large (≥ 1.5σ)                                                    |
| **Average (Indicator 1)**                         | $\overline{x}=\frac{\sum_i\overline{x}_i}{i}$                     |
| **Standard Deviation (Indicator 2)**              | $s=\sqrt{\frac{\sum(x_i-\overline{x})^2}{n-1}}$ |
| **Mean for All values**   | $\mu=\overline{\overline{X}}=\frac{\sum{x_{ij}}}{\sum{n_{i}}}$ |
| **Process Mean**   | $\mu$ |
| **Process Standard Deviation**   | $\sigma=\overline{S}=\frac{\sum{S_i}\cdot\frac{c_4(n_i)}{1 - c_4(n_i)^{2}}}{\sum{\frac{c_4(n_i)^{2}}{1 - c_4(n_i)^{2}}}}$<br/>Where $S_i$ is the standard deviation for the data point as given by:<br/>$s=\sqrt{\frac{\sum(x_i-\overline{x})^2}{n-1}}$ |
| **Average (Indicator 1) Centerline**                | $\mu$ |
| **Average (Indicator 1) Control Limits**            | $UCL = \mu + \frac{3\sigma}{\sqrt{n_i}}$<br/>$LCL = \mu - \frac{3\sigma}{\sqrt{n_i}}$ |
| **Standard Deviation (Indicator 2) Centerline**     | $\overline{S}_i = c_4(n_i)\cdot\sigma$                     |
| **Standard Deviation (Indicator 2) Control Limits** | $UCL = \overline{S}_i + 3\sigma \cdot c_5(n_i)$<br/>$LCL = max(\overline{S}_i - 3\sigma \cdot c_5(n_i);0)$ |

Table: Average and Standard Deviation chart properties

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