Run Charts ﹝趨勢圖、推移圖﹞ 彰化基督教醫院 陶阿倫 部長

Variation in Process A normal pattern for a process in control is one of randomness If only common causes of variation exist in your process, the data will exhibit random behavior Two tests for randomness in run charts Number of runs about the median Number of runs up or down

Run Charts Simple to construct Easy to use Do not require assumptions about the distribution from which data are obtained Capture the rhythm of your process for analysis of variation Because time plots are concerned with long-term data patterns, at least 24 data points are needed

Run Charts Dynamic display of performance Performance plotted on the ‘y’ axis Time plotted on the ‘x’ axis Use the MEDIAN as the centerline Helps distinguish between ‘signals’ and ‘noise’ Identifies most special causes

A typical run chart that examines performance for measure “X” from January 1999 to June 2000 (Fig.6 p.28)

Step 1: Determine the number of data runs in the chart (Fig.7 p.29) 18 data points 非中位數 非中位數 7 3 5 1 6 2 4 At least one or more data points will fall on the median whenever there is an odd number of data points. If even, the median will always fall between two points → Ignore points that fall on the median A data run consists of one or more consecutive data points on the same side of the baseline (median)

Step 2: Determine the number of useful observations (Fig.7 p.29) 18 data points 非中位數 非中位數 7 3 5 1 6 2 4 Number of data points that fall on the median: = 0 Total number of data points on the chart: = 18 Total number of useful observations = 18 – 0 = 18

Num Min Max 15 4 12 28 10 19 16 5 29 20 17 13 30 11 18 6 31 21 14 32 22 33 7 34 23 35 8 36 24 37 25 9 38 26 39 27 40 7 vs 6~13

判讀 (p.31) There are a total of 18 data points Since no data points fall on the median, they are all useful The number of useful observations is 18 With 18 useful observations, there should be a minimum of 6 and a maximum of 13 data runs. As there are 7 data runs in the run chart, therefore this step does not identify any special cause variation

Step 3: Determine if there are too many data points in any data run → a shift in the process (Fig.9 p.31) Useful observations Consecutive Data Points in Single Run < 20 7 or more 20 or more 8 or more 2 4 2 1 4 3 1 Since there are no more than 4 consecutive data points in any single data run, therefore this step does not identify special-cause variation

Step 4: Determine if there is a series of consecutive data points that steadily increases or decreases → a trend in the process (Fig.10 p.32) The chart identifies 4 data points that steadily increase, but because there are 18 data points, a trend would be present only if there were 6 or more consecutive points. Therefore, there is no special cause variation attributable to a trend in this example. 4 4 Data points on chart Consecutive Data Points that Increase/Decrease 5 ~ 8 5 or more 9 ~ 20 6 or more 21 ~ 100 7 or more Unlike for ‘useful observations’, DO count data points that fall on the median

Step 5: Determine if there is a sequence of consecutive data pints that alternate up and down → a zigzag or saw tooth pattern (Fig.11 p.33) A series of 14 or more consecutive data points whose values alternate up and down (zigzag, saw-tooth). This pattern occurs when there is meddling in the process. 6 There are no more than 6 consecutive data points, therefore this step does not identify any special-cause variation.

Interpreting [1] Determine the number of data runs in the chart Determine the number of useful observations Look for shifts / trends / data oscillations

Interpreting [1] Data runs Ignore points on the median A ‘data run’ = [1] one or more [2] consecutive data points on [3] the same side of the baseline Median How many data runs: 11 ?

Interpreting [1] Useful Observations Count the total number of data points on the chart: N = 26 Count the total number of data points on the median: M = 4 Useful = N – M = 26 – 4 = 22 Median

Interpreting [1] Data runs & Useful Observations Common cause ? or Special cause? Median Table of Min-Max

Num Min Max 15 4 12 28 10 19 16 5 29 20 17 13 30 11 18 6 31 21 14 32 22 33 7 34 23 35 8 36 24 37 25 9 38 26 39 27 40 22 Vs 11?

Interpretation [1] With 22 useful observations, there should be a minimum of 7 and a maximum of 16 data runs. As there are 11 data runs in the run chart, this step does not identify any special-cause variation.

Interpreting [2] Shift A data run with too many consecutive points demonstrates a shift in the process Useful Data obs. Points --------- --------- <20 7 or more >=20 8 or more Cause? Too many data points in a single run indicates a ‘special-cause’

Interpreting [2] Trends A series of consecutive data points that steadily increases or decreases Data Required Points Points --------- --------- 5~8 5 or more 9~20 6 or more 21~100 7 or more Cause? A trend indicates ‘special-cause’ variation

Interpreting [3] Oscillation A series of 14 or more consecutive data points that alternate on opposite sides of the baseline Chart exhibits a ‘zigzag’ or ‘sawtooth’ pattern Occurs where there is meddling with the process Oscillation indicates ‘special-cause’ variation

Interpreting: MINITAB Test for randomness Condition Indicates Number of runs about the median More runs observed than expected Mixed data from two populations; overcompensation or poor data-collection Fewer runs observed than expected Clustering of data: a cycle or long-term trend Number of runs up or down Oscillation: data varies up and down rapidly Trending of data

Theory: MINITAB With both tests, the null hypothesis is that the data is a random sequence The software converts the observed number of runs into a test statistic that is approximately standard normal, then uses the normal distribution to obtain p-values. The two P-values correspond to the one-sided probabilities associated with the test statistic. When either p-value is smaller than you’re a-value, also known as the significance level, you should reject the hypothesis of randomness. The a-value is the probability that you will incorrectly reject the hypothesis of randomness when the hypothesis is true. Assume the test is significant at an a -value of 0.05

趨勢圖練習

Practice-1

Practice-3

MINITAB Software

http://www.winona.msus.edu/teamscouncil/Education/timeplot.htm