# XmR Trend Chart Formulas

The XmR Trend Chart can help you evaluate process performance using variable data when there is only one measurement per period and the data may contain trends (e.g. increasing cost due to inflation, decreasing cost due to technology, etc).

**Here are the formulas used to calculate the XmR Trend Chart in QI Macros.**

### Range Chart

### Trend Chart

The center line (CL) is calculated using linear regression to give you the slope of the trend and a y-intercept value: "b", calculated as follows:

time(i) = ti = 1,2,3,4....k for each X value

m = slope

b = intercept

The XmR Trend then calculates the linear correlation coefficient (Ryx) for the degrees of freedom (df = k-2).

If Ryx is greater than the probability for this degree of freedom, you have a "**significant correlation**" between x and y. (Probability that you will conclude there is no correlation when there is one = alpha = 0.05).

If Ryx2 is greater than 0.80, then the correlation indicates a "**useful fit**."

What does this mean?

**Significant Correlation** - A measure of x versus y. Is the relationship between x and y statistically significant? This is a measure of how well the trend line reflects the relationship between x and y.

**Useful Fit** - Even if there is a significant correlation above this asks the question - Is it useful? Can I make an assumption or prediction about y based on past history?

A measure of the variation in x vs the variation in y.

This is a measure of how the points vary within the control limits.

**Probability** - This value comes from the "critical values for the linear correlation coefficient." Said probabilities are derived from the following table for alpha = 0.05. If a calculated value of Ryx is larger or equal to said table value, there is a statistically significant correlation between x and y. And alpha is the probability that we will conclude that there is not a correlation when there actually is one. degrees of freedom = n-2, where n = number of paired samples:

df | Probability | df | Probability |

1 | .997 | 11 | .553 |

2 | .950 | 12 | .532 |

3 | .878 | 13 | .514 |

4 | .811 | 14 | .497 |

5 | .754 | 15 | .482 |

6 | .707 | 16 | .468 |

7 | .666 | 17 | .456 |

8 | .632 | 18 | .444 |

9 | .602 | 19 | .433 |

10 | .576 | 20 | .423 |

df | Probability | df | Probability |

21 | .413 | 35 | .325 |

22 | .404 | 40 | .304 |

23 | .396 | 45 | .288 |

24 | .388 | 50 | .273 |

25 | .381 | 60 | .250 |

26 | .374 | 70 | .232 |

27 | .367 | 80 | .217 |

28 | .361 | 90 | .205 |

29 | .355 | 100 | .195 |

30 | .349 |

(Source: Statistical Methods for the Process Industries, W. McNeese, ASQ Quality Press, Milwaukee)

**Other Resources**

- Other Control Chart Formulas

The **XmR Trend Chart** is just one of the tools included in the QI Macros for Excel SPC Software for Excel.

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