Length of run in a run chart
Length of run in a run chart
A shift is present if any run of consecutive data points on the same side of the median is longer than its upper 95% prediction limit. Calculate as log(n) + 3 where n is the number of data points and "+3" allows for the 95% upper limit. The result is rounded to the nearest integer.
Mobile phones, if turned sideways into landscape orientation, display their inbuilt calculator as a Scientific Calculator with many more functions. To calculate log2(n), use log(n)/log(2). For 24 points and log2(),
or
log2(24) = ln(24) / ln(2) = (3.17805… / 0.69314…) = 4.5849625…
∴ length limit = log2(24) + 3 = 7.5849625…
then rounded to nearest integer = 8 . A run of more than 8 indicates a shift.
Most software packages have a LOG() function to calculate values using a different base;
Excel or LibreOffice Calc or PHP use LOG(number, [base]) . For example, 24 data points and log2(),
length limit = log2(24) + 3 = 7.5849625…
then rounded to nearest integer = 8
Javascript has Math.log2() function already programmed.
Otherwise, consult the following tables:
n | UL | n | UL | n | UL | n | UL | n | UL |
---|---|---|---|---|---|---|---|---|---|
10 | 6 | 20 | 7 | 30 | 8 | 40 | 8 | 50 | 9 |
11 | 6 | 21 | 7 | 31 | 8 | 41 | 8 | 51 | 9 |
12 | 7 | 22 | 7 | 32 | 8 | 42 | 8 | 52 | 9 |
13 | 7 | 23 | 8 | 33 | 8 | 43 | 8 | 53 | 9 |
14 | 7 | 24 | 8 | 34 | 8 | 44 | 8 | 54 | 9 |
15 | 7 | 25 | 8 | 35 | 8 | 45 | 8 | 55 | 9 |
16 | 7 | 26 | 8 | 36 | 8 | 46 | 9 | 56 | 9 |
17 | 7 | 27 | 8 | 37 | 8 | 47 | 9 | 57 | 9 |
18 | 7 | 28 | 8 | 38 | 8 | 48 | 9 | 58 | 9 |
19 | 7 | 29 | 8 | 39 | 8 | 49 | 9 | 59 | 9 |
n | UL | n | UL | n | UL | n | UL | n | UL |
---|---|---|---|---|---|---|---|---|---|
60 | 9 | 70 | 9 | 80 | 9 | 90 | 9 | 100 | 10 |
61 | 9 | 71 | 9 | 81 | 9 | 91 | 10 | 101 | 10 |
62 | 9 | 72 | 9 | 82 | 9 | 92 | 10 | 102 | 10 |
63 | 9 | 73 | 9 | 83 | 9 | 93 | 10 | 103 | 10 |
64 | 9 | 74 | 9 | 84 | 9 | 94 | 10 | 104 | 10 |
65 | 9 | 75 | 9 | 85 | 9 | 95 | 10 | 105 | 10 |
66 | 9 | 76 | 9 | 86 | 9 | 96 | 10 | 106 | 10 |
67 | 9 | 77 | 9 | 87 | 9 | 97 | 10 | 107 | 10 |
68 | 9 | 78 | 9 | 88 | 9 | 98 | 10 | 108 | 10 |
69 | 9 | 79 | 9 | 89 | 9 | 99 | 10 | 109 | 10 |
References
- Anhøj J (2014) A run chart is not a run chart is not a run chart. Understanding variation using runs analysis https://nhsrcommunity.com/blog/ …
Are there too few crossings in run chart?
Are there too few crossings in run chart?
The total number of crossings of the midline in a run chart has a binomial distribution X ~ B(n-1, p), where n is the number of data points and 0.5 is the success probability (either above or below the midline). Use the lower fifth percentile (0.05) of the cumulative binomial distribution to calculate the critical vale for the lower limit of crossings. A shift is identified when the number of crossings is less than the calculated number expected from n random numbers. [1]
Since 2010, most software packages have a BINOM.INV() function to calculate values for the inverse of the cumulative binomial distribution.
Excel and LibreOffice Calc use BINOM.INV(trials, probability)s, alpha).
For example, 24 data points,
Fewer than 8 crossings would be unusual and suggest that the process is shifting (non-random variation).
Otherwise, consult the following tables:
The values in the table can be approximated by using: [2]
kbar = arithmetic mean of data points = (k + 2)/2
Sk = standard deviation of data points
= SQRT[ k/2 * (k/2-1) / (k-1)]
Lower Limit: LL = kbar - tbar × Sk → ROUND(LL, 0)
Upper Limit: UL = kbar + tbar × Sk → ROUND(UL, 0)
(For k≤60, use tbar = 1.96; For k>60, use tbar = 2.0).
n | LL | n | LL | n | LL | n | LL | n | LL |
---|---|---|---|---|---|---|---|---|---|
10 | 3 | 20 | 6 | 30 | 11 | 40 | 15 | 50 | 19 |
11 | 3 | 21 | 7 | 31 | 11 | 41 | 15 | 51 | 20 |
12 | 3 | 22 | 7 | 32 | 11 | 42 | 16 | 52 | 20 |
13 | 4 | 23 | 7 | 33 | 12 | 43 | 16 | 53 | 21 |
14 | 4 | 24 | 8 | 34 | 12 | 44 | 17 | 54 | 21 |
15 | 5 | 25 | 8 | 35 | 12 | 45 | 17 | 55 | 22 |
16 | 5 | 26 | 9 | 36 | 13 | 46 | 17 | 56 | 22 |
17 | 5 | 27 | 10 | 37 | 13 | 47 | 18 | 57 | 23 |
18 | 6 | 28 | 10 | 38 | 14 | 48 | 18 | 58 | 23 |
19 | 6 | 29 | 10 | 39 | 14 | 49 | 18 | 59 | 24 |
n | LL | n | LL | n | LL | n | LL | n | LL |
---|---|---|---|---|---|---|---|---|---|
60 | 23 | 70 | 28 | 80 | 32 | 90 | 37 | 100 | 41 |
61 | 24 | 71 | 28 | 81 | 33 | 91 | 37 | 101 | 42 |
62 | 24 | 72 | 29 | 82 | 33 | 92 | 37 | 102 | 42 |
63 | 25 | 73 | 29 | 83 | 33 | 93 | 38 | 103 | 42 |
64 | 25 | 74 | 29 | 84 | 34 | 94 | 38 | 104 | 42 |
65 | 26 | 75 | 30 | 85 | 34 | 95 | 39 | 105 | 43 |
66 | 26 | 76 | 30 | 86 | 35 | 96 | 39 | 106 | 43 |
67 | 26 | 77 | 31 | 87 | 35 | 97 | 40 | 107 | 44 |
68 | 27 | 78 | 31 | 88 | 36 | 98 | 40 | 108 | 45 |
69 | 27 | 79 | 32 | 89 | 36 | 99 | 41 | 109 | 45 |
References
- Anhøj J (2014) A run chart is not a run chart is not a run chart. Understanding variation using runs analysis https://nhsrcommunity.com/blog/ …
- Provost LP, Murray SK. The health care data guide. Learning from data for improvement. www.amazon.com 2011. San Francisco: John Wiley & Sons. p.82
Are there too many crossings in run chart?
Are there too many crossings in run chart?
An unusually high number of crossings, oscillation, is a sign of non-random variation. It which will appear if data are negatively auto-correlated. However, oscillation is not an effect of the process shifting location, but most likely a result of a poorly designed measure or sampling issues.
The values in the table can be approximated by using: [2]
kbar = arithmetic mean of data points
Sk = standard deviation of data points
= SQRT[ k/2 * (k/2-1) / (k-1)]
Lower Limit: LL = kbar - 1.96 × Sk → ROUND(LL, 0)
Upper Limit: UL = kbar + 1.96 × Sk → ROUND(UL, 0)
(For k≤60, use tbar = 1.96; For k>60, use tbar = 2.0).
n | UL | n | UL | n | UL | n | UL | n | UL |
---|---|---|---|---|---|---|---|---|---|
10 | 9 | 20 | 16 | 30 | 21 | 40 | 27 | 50 | 33 |
11 | 10 | 21 | 16 | 31 | 22 | 41 | 27 | 51 | 33 |
12 | 11 | 22 | 17 | 32 | 23 | 42 | 28 | 52 | 34 |
13 | 11 | 23 | 17 | 33 | 23 | 43 | 28 | 53 | 34 |
14 | 12 | 24 | 18 | 34 | 24 | 44 | 29 | 54 | 35 |
15 | 12 | 25 | 18 | 35 | 24 | 45 | 30 | 55 | 35 |
16 | 13 | 26 | 19 | 36 | 25 | 46 | 31 | 56 | 36 |
17 | 13 | 27 | 19 | 37 | 25 | 47 | 31 | 57 | 36 |
18 | 14 | 28 | 20 | 38 | 26 | 48 | 32 | 58 | 37 |
19 | 15 | 29 | 20 | 39 | 26 | 49 | 32 | 59 | 38 |
n | UL | n | UL | n | UL | n | UL | n | UL |
---|---|---|---|---|---|---|---|---|---|
60 | 39 | 70 | 44 | 80 | 50 | 90 | 55 | 100 | 61 |
61 | 39 | 71 | 45 | 81 | 50 | 91 | 56 | 101 | 61 |
62 | 40 | 72 | 45 | 82 | 51 | 92 | 57 | 102 | 62 |
63 | 40 | 73 | 46 | 83 | 52 | 93 | 57 | 103 | 63 |
64 | 41 | 74 | 47 | 84 | 52 | 94 | 58 | 104 | 63 |
65 | 41 | 75 | 47 | 85 | 53 | 95 | 58 | 105 | 64 |
66 | 42 | 76 | 48 | 86 | 53 | 96 | 59 | 106 | 64 |
67 | 43 | 77 | 48 | 87 | 54 | 97 | 59 | 107 | 65 |
68 | 43 | 78 | 49 | 88 | 54 | 98 | 60 | 108 | 65 |
69 | 44 | 79 | 49 | 89 | 55 | 99 | 60 | 109 | 66 |
References
- Anhøj J (2014) A run chart is not a run chart is not a run chart. Understanding variation using runs analysis https://nhsrcommunity.com/blog/ …
- Provost LP, Murray SK. The health care data guide. Learning from data for improvement. www.amazon.com 2011. San Francisco: John Wiley & Sons. p.82
Equations for control chart limits
ˉp±3√ˉp(1-ˉp)ni | ˉu±3√ˉuni |