# Quality Digest

## An Underrated Test for Run Charts

### The total number of runs above and below the median proves revealing.

Suppose you had 16 months of data on an important process (as plotted on the run chart seen in figure 1). For improvement purposes, an intervention was made after the sixth observation to lower this key process indicator. This intervention is equivalent to creating a special cause for a desired effect.

There is no trend as defined statistically (despite the trend downward of length five from observations 5 to 9; with 16 data points, one would need length six, or five successive decreases). Neither is there any run of length eight either all above or below the median. So, would you want to conclude that the intervention had no effect? I doubt it!

Think of the median as a reference, and consider each data point as a 50-50 coin flip (heads = above the median; tails = below the median). The question is: Would you expect to flip a coin 16 times and obtain the specific pattern of seven heads (run of length seven above the median), immediately followed by seven tails (run of length seven below the median), then a head (run of length one above the median) and, finally, a tail (run of length one below the median)? Intuition would seem to say, “No.” Is there a statistical way to prove it?

There is a third test used in conjunction with the trend and runs-of-length-eight tests. It is based on the total number of runs observed above and below the median in a data collection. A run is counted from the first data point until it is broken by crossing the median (points that are literally on the median are ignored for this analysis). From the graph, the 16 data points yielded only four runs total (of lengths 7, 7, 1, and 1)--two above and two below the median.

Looking in the left column of the table in figure 2, under “Number of data points,” at 16 and reading across: 5-12 runs are expected to occur if the variation is only random (common cause). Four is below the expected lower limit of five. Hence, with low risk, the conclusion is that the special cause intentionally imposed after the sixth observation most probably did create the desired effect.

Generally, a successful intervention will tend to create fewer than the expected number of runs. It is relatively rare to obtain more than the expected number of runs. In my experience, this usually indicates that two process inputs are being reflected by them being unintentionally sampled alternately.

I also encountered a situation where someone had fudged data by trying to make it look random--and did too good a job. There is such a thing as being “too random!”

So, unless one gets fewer than the number of runs, one must assume, based on the current data, that there is no evidence of process change. Ott’s, Schilling’s, and Neubauer’s book, Process Quality Control: Troubleshooting and Interpretation of Data, Fourth Edition (ASQ Quality Press, 2005), and Acheson J. Duncan’s classic, Quality Control and Industrial Statistics, Fifth Edition (Richard D. Irwin, 1986), discuss runs analysis extensively and contain more tables. My table uses p < 0.05.

Subsequent plot data may tell a different story, providing a choice to either abandon the current strategy or try a different intervention. But, then again, because of no special cause signals, one could also proceed to making an I-chart of the data, which has a few more powerful tests to apply to detect special causes. I will show an example of this next month.

### Davis Balestracci

Davis Balestracci is a past chair of ASQ’s statistics division. He has synthesized W. Edwards Deming’s philosophy as Deming intended—as an approach to leadership—in the second edition of Data Sanity (Medical Group Management Association, 2015), with a foreword by Donald Berwick, M.D. Shipped free or as an ebook, Data Sanity offers a new way of thinking using a common organizational language based in process and understanding variation (data sanity), applied to everyday data and management. It also integrates Balestracci’s 20 years of studying organizational psychology into an “improvement as built in” approach as opposed to most current “quality as bolt-on” programs. Balestracci would love to wake up your conferences with his dynamic style and entertaining insights into the places where process, statistics, organizational culture, and quality meet.

### I developed a table and regression for Num of runs

UL = Round( (3.376 + 0.5604 Num),0) excel or minitab

LL = Round( (- 2.409 + 0.4398 Num),0) excel or minitab

Num              LL               UL

2
0
4

3
0
5

4
0
6

5
0
6

6
0
7

7
1
7

8
1
8

9
2
8

10
3
8

11
3
9

12
3
10

13
4
10

14
4
11

15
4
12

16
5
12

17
5
13

18
6
13

19
6
14

20
6
15

21
7
15

22
7
16

23
8
16

24
8
17

25
9
17

26
9
18

27
9
19

28
10
19

29
10
20

30
11
20

31
11
21

32
11
22

33
11
22

34
12
23

35
13
23

36
13
24

37
14
24

38
14
25

39
15
25

40
15
26

41
16
26

42
16
27

43
17
27

44
17
28

45
17
29

46
18
29

47
18
30

48
19
30

49
19
31

50
19
32

51
20
32

52
20
33

53
21
33

54
21
34

55
22
34

56
22
35

57
23
35

58
23
36

59
24
36

60
24
37

61
24
38

62
25
38

63
25
39

64
26
39

65
26
40

66
27
40

67
27
41

68
27
41

69
28
42

70
28
43

71
29
43

72
29
44

73
30
44

74
30
45

75
31
45

76
31
46

77
31
47

78
32
47

79
32
48

80
33
48

81
33
49

82
34
49

83
34
50

84
35
50

85
35
51

86
35
52

87
36
52

88
36
53

89
37
53

90
37
54

91
38
54

92
38
55

93
38
55

94
39
56

95
39
57

96
40
57

97
40
58

98
41
58

99
41
59

100
42
59

101
42
60

102
42
61

103
43
61

104
43
62

105
44
62

106
44
63

107
45
63

108
45
64

109
46
64

110
46
65

111
46
66

112
47
66

113
47
67

114
48
67

115
48
68

116
49
68

117
49
69

118
49
70

119
50
70

120
51
70

121
50
71

122
51
72

123
51
72

124
52
73

125
52
73

126
53
74

127
53
75

128
53
75

129
54
76

130
54
76

131
55
77

132
55
77

133
56
78

134
56
78

135
57
79

136
57
80

137
57
80

138
58
81

139
58
81

140
59
82

141
59
82

142
60
83

143
60
84

144
60
84

145
61
85

146
61
85

147
62
86

148
62
86

149
63
87

150
63
87

151
64
88

152
64
89

153
64
89

154
65
90

155
65
90

156
66
91

157
66
91

158
67
92

159
67
92

160
68
93

161
68
94

162
68
94

163
69
95

164
69
95

165
70
96

166
70
96

167
71
97

168
71
98

169
71
98

170
72
99

171
72
99

172
73
100

173
73
100

174
74
101

175
74
101

176
75
102

177
75
103

178
75
103

179
76
104

180
76
104

181
77
105

182
77
105

183
78
106

184
78
106

185
79
107

186
79
108

187
79
108

188
80
109

189
80
109

190
81
110

191
81
110

192
82
111

193
82
112

194
82
112

195
83
113

196
83
113

197
84
114

198
84
114

199
85
115

200
85
115

### Other Methods of Detecting This Non-Randomness?

Wanting to understand this test better, and being the sort of person who prefers automatic solutions to manual ones, I am wondering if there are other ways of detecting this non-randomness (i.e. already implemented in software that I have available). Would any other tools provide the same test? It seems to me that an EWMA or CUSUM chart, with the right parameters, would detect number of runs below the lower limit, because the runs would be long, but on the high end?