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

xmr trend range chart formula

Trend Chart

xmr trend formula

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:

xmr trend center line formula

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

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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