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 | \(\sigma=\overline{S}=\frac{\sum{S_i}\cdot\frac{c_4(n_i)}{1 - c_4(n_i)^{2}}}{\sum{\frac{c_4(n_i)^{2}}{1 - c_4(n_i)^{2}}}}\) Where \(S_i\) is the standard deviation for the data point as given by: \(s=\sqrt{\frac{\sum(x_i-\overline{x})^2}{n-1}}\) |
| Average (Indicator 1) Centerline | \(\mu\) |
| Average (Indicator 1) Control Limits | \(UCL = \mu + \frac{3\sigma}{\sqrt{n_i}}\) \(LCL = \mu - \frac{3\sigma}{\sqrt{n_i}}\) |
| Standard Deviation (Indicator 2) Centerline | \(\overline{S}_i = c_4(n_i)\cdot\sigma\) |
| Standard Deviation (Indicator 2) Control Limits | \(UCL = \overline{S}_i + 3\sigma \cdot c_5(n_i)\) \(LCL = max(\overline{S}_i - 3\sigma \cdot c_5(n_i);0)\) |
Table: Average and Standard Deviation chart properties
The statistical table used to compute the control limits can be consulted in the Control Chart Constants page.