Processing math: 100%


Frequently Asked Questions (FAQ)

Question and Answer area for problems in quality improvement

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(),

log2(24) = log10 (24) / log10 (2) = (1.38021… / 0.30102…) = 4.5849625…
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(),

log2(24) = LOG(24, 2) = 4.584963
length limit = log2(24) + 3 = 7.5849625…
then rounded to nearest integer = 8

Javascript has Math.log2() function already programmed.

Math.log2(24) = 4.584962500721156

Otherwise, consult the following tables:

Table-1. Critical values for run length in run chart.
n = 10 ~ 59
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
Legend: n = Number of useful (not on midline) observations; UL=upper limit for longest run calculated as log2(n)+3

 

Table-2. Critical values for run length in run chart.
n = 60 ~ 109
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
Legend: n = Number of useful (not on midline) observations; UL=upper limit for longest run calculated as log2(n)+3

References

  1. 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,

Lower limit → BINOM.INV(24, 0.5, 0.05) = 8

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]

k = number of data points
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).
Table-1. Critical values for lower limit for number of crossings in run chart.[2].p80
n = 10 ~ 59
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
Legend: n=Number of useful (not on midline) observations; LL=lower limit for random number of crossings calculated as binom.inv(n-1, 0.5, 0.05)

 

Table-2. Critical values for lower limit for number of crossings in run chart.[2].p82
n = 60 ~ 109
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
Legend: n=Number of useful (not on midline) observations; LL=lower limit for random number of crossings calculated as binom.inv(n-1, 0.5, 0.05)

References

  1. 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/ …
  2. 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]

k = number of data points = (k + 2)/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).
Table-1. Critical values for upper limit for number of crossings in run chart.
n = 10 ~ 59
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
Legend: n=Number of useful (not on midline) observations; UL=upper limit for number of crossings (>UL is "too many")
Table-2. Critical values for upper limit for number of crossings in run chart.
n = 60 ~ 109
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
Legend: n=Number of useful (not on midline) observations; UL=upper limit for number of crossings (>UL is "too many")

References

  1. 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/ …
  2. 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±3ˉp(1-ˉp)ni ˉu±3ˉuni





Accept Cookies?
Provided by Web design, Gloucester