Locked lesson.

Upgrade

About this lesson

The Cumulative Sum (CUSUM) Chart is more sensitive to subtle shifts in the process than the traditional variable data charts. The look of this control chart is unique among the commonly used control charts. This lesson explains how the data is calculated and interpreted on the chart. The lesson describes how to create this control chart in both Microsoft Excel and using Minitab. The lesson will include practice creating the chart.

Exercise files

Download this lesson’s related exercise files.

CUSUM Chart.xlsx
10.2 KB CUSUM Chart - Solution.docx
64.3 KB

Quick reference

CUSUM Chart

The CUSUM chart (Cumulative Summation) is a control chart that relies on historic process data when determining the values to plot.   It can be easily created in either Microsoft Excel or Minitab.

When to use

Use the CUSUM Chart when monitoring a process for small shifts in the mean.  The CUSUM chart requires variable data.  The CUSUM chart also relies on a target value, so if the process has a target value that it is trying to achieve for business reasons, this chart will monitor deviations from that target.  (Typically the target value used is the process mean.)

Instructions

The CUSUM chart is an accumulation of the deviation of the subgroup mean from an adjusted target value.  The idea of a cumulative summation is a very simple one, but this chart uses an important twist to the summation that often causes it to reset to zero.  This twist is based upon the calculation of the deviation from the adjusted target value. 

The key to this chart is understanding how this adjustment to the target value works.  This adjustment is a sensitivity zone above and below the target value that is essentially considered to be equivalent to the target value for the purpose of calculating the deviation.  This zone is typically plus or minus one half standard deviation above and below the target value. 

The second feature of this control chart is that it is tracking both a high side and low side accumulation.  These are referred to as one-sided accumulations since they are directional in nature.  The control chart plot will include both summations on the same plot.  Therefore, there are two data lines, but only one set of control limits.

There is a variation of this chart that calculates a two-sided summation and applies a V-mask to the chart.  This variation is different in form from the Shewhart control charts and therefore, I do not recommend using it. I use the one-sided approach.  The one-sided chart will be much closer to what your operators are used to seeing and working with.

Because of the way the cumulative summation is working, this chart will be able to detect small shifts to the mean faster than a Shewhart control chart.  In particular, this chart is able to detect long term trends that might be lost in the noise of a system with high common cause variation until the mean had shifted dramatically.

The math for this chart is simple, but it is very important to keep the two cumulative sums separated and not combine or confuse them.

Within Minitab, control charts are created by using the “Stat” pull down menu, then selecting “Control Charts.”  Within the Control Charts window, select “Time Weighted” and then finally select “CUSUM”  In the Minitab CUSUM Chart panel, you will need to select the data columns with your data and set the target value.  So for an existing process, you may need to first determine the mean value and then use that as your target.

If creating the CUSUM Chart in Excel:

  1. Determine the sensitivity zone and target value.  The sensitivity zone is normally .5 times the adjusted standard deviations.  The target value is often the mean of the process once it is stable.  The chart is being used to ensure the process does not drift.  The target can be a value other than the mean, but if it is not close to the mean of the data, the chart will quickly go to an out of control situation.
  2. Measure the attribute for the first item in the subgroup sample and record the data in a column in Excel.  Then measure the next item in the subgroup sample and record that in the next column.  By doing this, each row in Excel represents a subgroup.
  3. In an adjacent column, calculate the Mean for each subgroup and then in the following column calculate the subgroup standard deviation. If the subgroup size was one, this step is unnecessary. 
  4. Calculate the adjusted standard deviation.  The method used depends upon the size of the subgroup.  The subgroup size will also determine the value of the constant.
    1. If the subgroup size is one, determine the absolute value of the Moving Range for each point (See the I-MR Chart).  Take the mean of this value and divide it by the d2 constant. 

    1. If the subgroup size is greater than one,  Take the average of the subgroup standard deviations divided by the c4 constant and then divide that by the d2 constant. 

  1. Calculate the sensitivity zone by multiplying the sensitivity factor times the adjusted standard deviation.  The sensitivity factor is normally 0.

  1. Calculate the deviation value that will be added to the cumulative sum.
    1.  For the high side cumulative sum, reduce the target value by one half the standard deviation when calculating the deviation value.  Add that to the cumulative high side deviation.  If the new cumulative sum value is less than zero, the new high side accumulation is zero.

Where K is the sensitivity zone – normally one half the adjusted standard deviation.

    1. For the low side cumulative sum, increase the target by one half the standard deviation when calculating the deviation value.  Add that to the cumulative low side deviation.  If the new cumulative sum value is greater than zero, the new low side accumulation is zero.

Where K is the sensitivity zone – normally one half the adjusted standard deviation

  1. Calculate the upper and lower control limits.  These are just four times the standard deviation.
    1. Upper control limit = 4 x adjuseted standard deviation
    2. Lower control limit = -4 x adjusted standard deviation
  2. Plot high side accumulation, the low side accumulation and the upper and lower control limits.  The chart will always have the Y axis centered at zero.
  3. Take appropriate actions to remove special causes or to center your data within the  control limits. 

Hints & tips

  • This uses variable data, not attribute data.
  • I find this chart is helpful for maintaining a process that is already under control.
  • The data for this chart is normally collected in a subgroup size of one.
Login to download
  • 00:04 Hi, I'm Ray Sheen.
  • 00:05 Let's start our discussion of time way to control charts by
  • 00:09 looking at the CUSUM Chart.
  • 00:12 >> The CUSUM Chart gets its name from the expression cumulative summation.
  • 00:17 The chart relies on a cumulative sum
  • 00:19 of the deviation of the subgroup mean from an adjusted target value.
  • 00:24 Now that's a mouthful.
  • 00:25 So let's break that down a little bit to understand it.
  • 00:28 I'll start at the end and work backwards.
  • 00:31 The CUSUM chart will based upon how well your process achieves a target value.
  • 00:36 The target value will be the process mean unless it is specified to be some
  • 00:40 other value.
  • 00:42 But I said it was an adjusted target value,
  • 00:45 the value has a sensitivity slack zone around it.
  • 00:48 This sensitivity adjustment is normally set
  • 00:51 at one-half the adjusted standard deviation.
  • 00:54 So for the purposes of the calculations with the chart, any value that is plus or
  • 00:58 minus one-half of the adjusted standard deviation from the target
  • 01:02 is considered to be on target.
  • 01:04 Why do we care about on target?
  • 01:06 Well, back to the first bullet point.
  • 01:07 I said that we will accumulate the deviations of the subgroup mean
  • 01:11 from the adjusted target.
  • 01:13 So once the accumulation gets to be within the sensitivity slack zone,
  • 01:17 it is set to zero.
  • 01:18 Only when the deviation is greater than one-half the adjusted standard deviation
  • 01:22 when we start to accumulate again.
  • 01:25 Now, I've been saying the subgroup mean but
  • 01:27 almost all the time I've seen the CUSUM chart in use the subgroup size was one.
  • 01:31 So the subgroup mean was actually the measured value.
  • 01:36 The normal way I see CUSUM charts used is with two plots.
  • 01:40 One is with the high side accumulation, the other is the low side accumulation.
  • 01:44 I'll show you what I mean on the next slide.
  • 01:47 But there's an alternative form, which some of the statisticians love,
  • 01:49 that creates a V-mask over the data and has constantly narrowing control limits.
  • 01:54 I found that view is actually confusing for my operators, so
  • 01:57 I stick with the one sided approach.
  • 01:59 And again we use this charts to quickly detect small shifts in the process mean.
  • 02:05 So let's look at this thing.
  • 02:06 We have plots of data with upper control limits and
  • 02:09 lower control limits, the X axis is the same, it's subgroups, but
  • 02:13 there's no mean line, instead it's a center line.
  • 02:16 This is because what we are plotting on the vertical axis
  • 02:19 is the cumulative deviation from the mean or the center.
  • 02:23 So the chart is centered around the case of no cumulative deviation.
  • 02:27 That is the 0 line on the chart.
  • 02:30 And another strange thing about the chart is that there are two data
  • 02:32 plots on the same chart.
  • 02:34 When we did the I-MR or Xbar-R charts, we had two things that we were plotting,
  • 02:39 but we had two charts to do that.
  • 02:41 Here we have two lines on the same chart.
  • 02:44 One of these is the accumulation of the high side deviation and
  • 02:47 the other is accumulation of the low side deviation.
  • 02:51 But we went through the steps of creating a control chart in our previous module,
  • 02:54 but let's look up some specifics about the CUSUM chart.
  • 02:58 We start with the definition of the subgroup size and the sampling plan.
  • 03:01 And as I mentioned, that's usually a subgroup of one.
  • 03:04 Next is to set the target value in the sensitivity slack band.
  • 03:08 A good default for the target is the process mean.
  • 03:11 But if you do have a specific process target for
  • 03:13 optimal performance go ahead and use it.
  • 03:15 I've always seen the sensitivity set at one-half the standard deviation, so
  • 03:19 I recommend we keep it at that point.
  • 03:22 Now collect the data points, and if the target is to be the process mean,
  • 03:26 calculate the mean.
  • 03:27 Decide if you're going to do one-sided analysis or two sided analysis.
  • 03:31 Don't let the terms confuse you, either way you will look for
  • 03:35 shifts in the high side or low side.
  • 03:37 One-sided has two lines for high and
  • 03:40 low while the two-sided has one line funneling in along the V-mask.
  • 03:45 Now we calculate our adjusted deviation and if necessary accumulate
  • 03:48 each of the two sums, one on the next slide with the formulas.
  • 03:53 You are ready now to plot the two cumulative sum values,
  • 03:57 calculate the control limits and plot them.
  • 04:00 Of course if the chart shows an out of control condition, stop, investigate and
  • 04:04 take appropriate action.
  • 04:06 Let's look at how we do the calculations manually or in Excel.
  • 04:10 As with the other modules, the formulas are shown on the right, and
  • 04:13 I'll talk you through them on the left.
  • 04:15 First, if there's a subgroup with more than one data point,
  • 04:18 calculate the mean and standard deviation for the subgroup.
  • 04:21 Now get your target value.
  • 04:23 If it's a process mean you'll want to calculate a global mean for the process.
  • 04:28 Then we set the sensitivity and threshold constants.
  • 04:31 Unless you have a good reason to change them, the k value is 0.5 and
  • 04:35 the h value is 4.
  • 04:36 For the high side accumulation, we take a measured value and
  • 04:40 subtract the target adjusted down by one-half the adjusted standard deviation.
  • 04:45 We then add that to the previous value of high side accumulation.
  • 04:49 If that is a positive number, that is our new high side accumulation value.
  • 04:54 If that is a negative number, we reset it to zero.
  • 04:57 And for the low side accumulation, we'll do something similar.
  • 05:00 We take the measured value minus the target value, but
  • 05:04 now we add the adjustment of one-half the adjusted standard deviation.
  • 05:08 We then add the previous value of the low side accumulation, and
  • 05:12 finally if that is a negative number, that’s our new side accumulation.
  • 05:16 If it’s a positive number, the low side accumulation value is reset to zero.
  • 05:21 The control limits are easy calculations.
  • 05:24 The upper control limit is the h value times the adjusted standard deviation, and
  • 05:28 the lower control limit is minus the h value times
  • 05:31 the adjusted standard deviation.
  • 05:33 If you're using Excel, you can plot your accumulated data values and
  • 05:36 your upper-lower control limits using the line chart graphics option.
  • 05:41 Now I've talked about this adjusted standard deviation.
  • 05:44 Let's see how the adjustment works.
  • 05:46 When the subgroup size is 1, we'll create the adjusted standard deviation from
  • 05:50 the average of the moving range values.
  • 05:53 If you have an IMR chart, you can just read the value from the chart.
  • 05:56 Otherwise you will need to calculate it by determining the moving range between each
  • 06:01 data point, taking the average of those and then dividing by the constant d2.
  • 06:06 When the subgroup size is greater 1, we'll modify the subgroups standard deviations.
  • 06:11 First calculate a mean of the subgroup standard deviations
  • 06:15 then divide that mean by the c4 constant for that size subgroup.
  • 06:19 Now, divide that by the d2 constant for the subgroup size, and
  • 06:23 that is your adjusted standard deviation.
  • 06:26 Now let's look at creating this chart in Minitab.
  • 06:29 Go to the Stat menu > Control Charts > Time Weighted Charts > CUSUM Chart.
  • 06:34 When you do that, you should get a panel that looks like this.
  • 06:38 Tell Minitab how to read your data,
  • 06:40 then place your cursor in the variables window to activate the column display.
  • 06:44 Highlight the columns where your data is located then click the Select button.
  • 06:48 Next, you have to set the target value.
  • 06:50 If you intend to use your mean value, you'll need to check the descriptive
  • 06:54 statistics first to know which value to set here.
  • 06:57 The k prime and h values can be changed from 0.5 and 4 if you select the button
  • 07:02 CUSUM options and then select the tab Plan/Type.
  • 07:06 Now click on the OK button in the bottom of the panel and
  • 07:09 Minitab will Generate your chart.
  • 07:12 >> The CUSUM Chart will quickly show when shifts occur to the mean.
  • 07:17 It's more complex than the standard Shewhart Charts, but
  • 07:20 sometimes it's just what you need.

Lesson notes are only available for subscribers.

PMI, PMP, CAPM and PMBOK are registered marks of the Project Management Institute, Inc.

Gift this course