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Z-Stats / Basic Statistics

Break All the Rules - Answers to Part Two

The answers to Part 2 of our QC basic interpretation series. Now we see what happens when we have 2 controls, and need to interpret across materials, across levels, and across runs. 

Break All the Rules! (Answers to Part Two)

Sten Westgard, MS
November 2015

We're continuing in our effort to break all the "Westgard Rules." As we'll soon see, Part One was easy - just one control to worry about. Now that we've got two controls, we're going to have to start looking in many directions to detect the errors.

The data - converted to z-scores

Recall that low control has a mean of 47 and an SD of 3 and a high control has a mean of 71 and an SD of 8. Rather than deal in the "raw" numbers, we're going to convert those values into z-scores, which tell you how many standard deviations each point is from the mean. It's easier to pick up the rule violations this way.

Control Measurement

Value 1

Z-value 1

Value 2

Z-value 2

Rule Broken

 1  51.5  1.5  82.2  1.4  
 2  38.9  -2.7  51  -2.5 2:2s on both
3 47.6  0.2 74.2  0.4  
4 46.0  -0.33 91.8  2.6  
5 50.9  1.3 74.2  0.4  
6 53.9  2.3  69.0  -0.25  
7 52.7  1.9  58.2  -1.6  
8 49.1  0.7  95.5  3.06  1:3s on 2
9 48.5  0.5  65.4  -0.7  
10 37.0  -3.33  90.0  2.375 1:3s on 1 and
R:4s on both
11 47.0  0  63.0  -1  
12 54.0  2.33  82.2  1.4  
13 53.3  2.1  65.4  -0.7  2:2s on 1
14 40.0  -2.33  86.2  1.9  R:4s by calc
15 55.4  2.8  70.2  -0.1  
16 49.4  0.8  88.6  2.2  
17 46.7 -0.1  87.8  2.1 2:2s on 2
18 45.8  -0.4  85.0  1.75  
19 52.7  1.9 76.6  0.7
20 51.1  1.4  65.4  -0.7
21 54.0  2.33  86.2  1.9
22 53.0  2  71.0  0  4:1s on 1 and
2:2s on 1
23 49.7  0.9  81.4  1.3  
24 48.2  0.4 90.2  2.4  
25 46.4  -0.2  80.6  1.2  
26 54.2  2.4  54.0  -2.12  R:4s
27 51.2  1.4  53  -2.25  
28 53.9  2.3  76.6  0.7  
29 44.3  -0.9  72.6  0.2  
30 48.5  0.5  84.6  1.7  

To make it a little easier to see the errors, we've marked them on a Levey-Jennings chart with z-scores for these first 30 values:

BreakAllRules N2 Answers 1 30

Control Measurement

Value 1

Z-value 1

Value 2

Z-value 2

Rule Broken

31 48.5 0.5 85.4 1.8
32 48.8 0.6 86.2 1.9
33 46.7 -0.1 80.0  1.12
34 46.1  -0.3 79.9  1.11 4:1s on 2
35 46.0 -0.33 83.8 1.6  
36 53.9 2.3 68.0 -0.37  
37 43.0  -1.33 64.6  -0.8
38 42.0  -1.67 71.0  0
39 40.0  -2.33 83.0  1.5
40 42.5  -1.5 67.0  -0.5  4:1s on 1
41 50.3  1.1 74.2  0.4  
42 44.3  -0.9 46.0 -3.12 1:3s on 2 
43 51.8  1.6 71.8  0.1  
44 41.9  -1.7 59.8  -1.4  
45 40.0  -2.33 57.0 -1.75 4:1s on both 
46 48.8  0.6 86.2  -2  
47 53.0  2 55.0 -0.12  
48 52.4 1.8 70.0 1  
49 38.0  -3 79.0 -0.12 1:3s on 1
50 49.1  0.7 70.0  -0.25  
51 44.0  -1 69.0  -0.25  
52 48.8  0.6 75.0  0.5
53 50.6  1.2 76.6  0.7
54 43.7  -1.1 72.6  0.2
55 47.6 0.2 80.0  1.12
56 45.5  -0.5 79.0  1
57 44.3  -0.9 74.0  0.37
58 52.4  1.8 82.0  1.37
59 53.9  2.3 72.0  0.12 8:x on 2

Here is the second set of Levey-Jennings charts for values 31 through 59:

BreakAllN2 Answers 31 59

Control Measurement

Value 1

Z-value 1

Value 2

Z-value 2

Rule Broken

60 45.8 -0.4 74.2 0.4  
61 50.6  1.2 64.6  -0.8
62 48.0  0.33 71.0  0
63 49.0  0.67 83.0  1.5
64 51.0  1.33 67.0  -0.5
65 53.0  2 74.2  0.4
66 48.0  0.33 80.6  1.2
67 49.0  0.67 71.8  0.1
68 50.0  1 69.4  -0.2  8:x on 1
69 47.8  0.26 81.4  1.3  
70 45.5 -0.5 65.4  -0.7  
71 46.0  -0.33 70.0  -0.12  
72 45.0  -0.67 69.0  -0.25  
73 46.0  -0.33 66.0  -0.62 8:x on both
74 47.9 0.3 68.6  -0.3  
75 48.5  0.5 86.2  1.9  
76 50.9  1.3 59.8  -1.4  
77 46.7  -0.1 67.8  -0.4  
78 47.8  0.26 86.2  1.9
79 50.3  1.1 72.6  0.2
80 52.0  1.67 81.4  1.3
81 52.0  1.67 74.2  0.4  8:x on both
82 48.5 0.5 80.6  1.2  
83 44.3  -0.9 79.8  1.1  
84 49.4  0.8 75.0  0.5  
85 46.7  -0.1 76.6  0.7  
86 52.7  1.9 72.6  0.2  
87 45.2  -0.6 55.8  -1.9  
88 48.8 0.6 79.0  1  
89 46.4  -0.2 70.2  -0.1  
90 49.7  0.9 82.2  1.4  

Finally, here is the third set of Levey-Jennings charts for the values 60 through 90:

BreakAllRules N2 Answers 60 90

Remember the caveats:

  • We are running TWO controls per run; assume each pair of values represents one run
  • We're NOT going to look for 2s violations. Let's break that habit right now. In any case, I didn't generate these data points randomly, so we should not expect to see a certain number "natural false rejections" in the form of 2s outliers
  • We ARE going to try to find outliers of the following "Westgard Rules": 1:3s/2:2s/R:4s/4:1s/8:x 
  • We're NOT going to sink into the swamp that is the repeating of controls. So don't assume anything you see is a repeated control - just stick with "in" or "out"

So, at the end of this, can you answer the following questions?

  • How many rule violations were there? 17 violations
  • Which rules were violated and how many times?
    • 1:3s (2 times)
    • 2:2s (4 times)
    • R:4s (3 times)
    • 4:1s (4 times)
    • 8:x violations (4 times)
  • What would be different if I told you that only the "Westgard Rules" 1:3s/2:2s/R:4s were necessary?
    • Ignore the 4:1s and 8:x violations because they aren't significant errors
  • What would be different if I told you that only the 1:3s control rule was necessary?
    • Only 3-4 violations
  • What's the extra little error that exists in these charts and data points?

[Did you miss something? Do you still feel like you're missing something? Please don't feel afraid to ask for more explanations.]