## Statistical Process Control

Read this chapter on the basics of statistical process control (SPC). SPC is a standard tool for monitoring whether a process is performing as expected and, if not, where problems occur. While reading, consider how this kind of tool factors in process capacity management.

### Example 1

Assume that in the manufacture of 1 kg Mischmetal ingots, the product weight varies with the batch. Below are a number of subsets taken at normal operating conditions (subsets 1-7), with the weight values given in kg. Construct the -Bar, -charts, and -charts for the experimental data (subsets 8-11). Measurements are taken sequentially in increasing subset number.

Subset # | Values (kg) |

1 (control) | 1.02, 1.03, 0.98, 0.99 |

2 (control) | 0.96, 1.01, 1.02, 1.01 |

3 (control) | 0.99, 1.02, 1.03, 0.98 |

4 (control) | 0.96, 0.97, 1.02, 0.98 |

5 (control) | 1.03, 1.04, 0.95, 1.00 |

6 (control) | 0.99, 0.99, 1.00, 0.97 |

7 (control) | 1.02, 0.98, 1.01, 1.02 |

8 (experimental) | 1.02, 0.99, 1.01, 0.99 |

9 (experimental) | 1.01, 0.99, 0.97, 1.03 |

10 (experimental) | 1.02, 0.98, 0.99, 1.00 |

11 (experimental) | 0.98, 0.97, 1.02, 1.03 |

Solution:

First, the average, range, and standard deviation are calculated for each subset.

Subset # | Values (kg) | Average (cc) |
Range (R) |
Standard Deviation(s) |

1 (control) | 1.02, 1.03, 0.98, 0.99 | 1.0050 | 0.05 | 0.023805 |

2 (control) | 0.96, 1.01, 1.02, 1.01 | 1.0000 | 0.06 | 0.027080 |

3 (control) | 0.99, 1.02, 1.03, 0.98 | 1.0050 | 0.05 | 0.023806 |

4 (control) | 0.96, 0.97, 1.02, 0.98 | 0.9825 | 0.06 | 0.026300 |

5 (control) | 1.03, 1.04, 0.95, 1.00 | 1.0150 | 0.09 | 0.040509 |

6 (control) | 0.99, 0.99, 1.00, 0.97 | 0.9875 | 0.03 | 0.022583 |

7 (control) | 1.02, 0.98, 1.01, 1.02 | 1.0075 |
0.04 | 0.028930 |

8 (experimental) | 1.02, 0.99, 1.01, 0.99 | 1.0025 | 0.03 | 0.025000 |

9 (experimental) | 1.01, 0.99, 0.97, 1.03 | 1.0000 | 0.06 | 0.025820 |

10 (experimental) | 1.02, 0.98, 0.99, 1.00 | 0.9975 | 0.04 | 0.027078 |

11 (experimental) | 0.98, 0.97, 1.02, 1.03 | 1.0000 | 0.06 | 0.029409 |

Next, the grand average , average range , and average standard deviation are computed for the subsets taken under normal operating conditions, and thus the centerlines are known. Here .

-Bar limits are computed (using ).

-Bar limits are computed (using ).

*Note: Since (a relatively small subset size), both and can be used to accurately calculate the UCL and LCL.*

The individual points in subsets 8-11 are plotted below to demonstrate how they vary with in comparison with the control limits.

**Figure E-1**: Chart of individual points in subsets 8-11.

The subgroup averages are shown in the following -Bar chart:

**Figure E-2**: -Bar chart for subsets 8-11.

**Figure E-3:** -chart for subsets 8-11.

**Figure E-4**: -chart for subsets 8-11.

The experimental data is shown to be in control, as it obeys all of the rules given above.