D Critical Values for Longest Runs and Number of Crossings

In SPC charts, a run is one or more consecutive data points on the same side of the centre line. Data points that lie directly on the centre line are not counted, nor do they break a run if the next data point is on the same side as the previous one.

A crossing occurs when two consecutive data points lie on opposite sides of the centre line, ignoring any data points directly on the centre line.

To perform the runs analysis, first count the number of useful observations in the chart. Then count the number of data points in the longest run and the number of crossings, and compare these values with the critical values in the table.

For example, in an SPC chart with 24 useful observations, a run of more than eight data points on the same side of the centre line or fewer than eight crossings of the centre line would suggest one or more sustained shifts in the data.

In random data sequences, these two rules used together have a false signal rate of around 5%, regardless of the number of useful observations (Anhøj and Olesen 2014; Anhøj 2015; Anhøj and Wentzel-Larsen 2018).

Number of useful observations	Upper limit for longest run	Lower limit for number of crossings
10	6	2
11	6	2
12	7	3
13	7	3
14	7	4
15	7	4
16	7	4
17	7	5
18	7	5
19	7	6
20	7	6
21	7	6
22	7	7
23	8	7
24	8	8
25	8	8
26	8	8
27	8	9
28	8	9
29	8	10
30	8	10
31	8	11
32	8	11
33	8	11
34	8	12
35	8	12
36	8	13
37	8	13
38	8	14
39	8	14
40	8	14
41	8	15
42	8	15
43	8	16
44	8	16
45	8	17
46	9	17
47	9	17
48	9	18
49	9	18
50	9	19
51	9	19
52	9	20
53	9	20
54	9	21
55	9	21
56	9	21
57	9	22
58	9	22
59	9	23
60	9	23
61	9	24
62	9	24
63	9	25
64	9	25
65	9	25
66	9	26
67	9	26
68	9	27
69	9	27
70	9	28
71	9	28
72	9	29
73	9	29
74	9	29
75	9	30
76	9	30
77	9	31
78	9	31
79	9	32
80	9	32
81	9	33
82	9	33
83	9	34
84	9	34
85	9	34
86	9	35
87	9	35
88	9	36
89	9	36
90	9	37
91	10	37
92	10	38
93	10	38
94	10	39
95	10	39
96	10	39
97	10	40
98	10	40
99	10	41
100	10	41

References

Anhøj, Jacob. 2015. “Diagnostic Value of Run Chart Analysis: Using Likelihood Ratios to Compare Run Chart Rules on Simulated Data Series.” PLoS ONE, ahead of print. https://doi.org/10.1371/journal.pone.0121349.

Anhøj, Jacob, and Anne Vingaard Olesen. 2014. “Run Charts Revisited: A Simulation Study of Run Chart Rules for Detection of Non-Random Variation in Health Care Processes.” PLoS ONE, ahead of print. https://doi.org/10.1371/journal.pone.0113825.

Anhøj, Jacob, and Tore Wentzel-Larsen. 2018. “Sense and Sensibility: On the Diagnostic Value of Control Chart Rules for Detection of Shifts in Time Series Data.” BMC Medical Research Methodology, ahead of print. https://doi.org/10.1186/s12874-018-0564-0.